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and Statistical Analysis
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Jerome L. Myers
Arnold D. Well
University of Massachusetts
LAWRENCE ERLBAUM ASSOCIATES, PUBLISHERS
2003 Mahwah, New Jersey London
Senior Editor: Debra Riegert
Textbook Marketing Manager: Marisol Kozlovski
Editorial Assistant: Jason Planer
Cover Design: Kathryn Houghtaling Lacey
Textbook Production Manager: Paul Smolenski
Full-Service Compositor: TechBooks
Text and Cover Printer: Hamilton Printing Company
This book was typeset in 10/12 pt. Times, Italic, Bold, and Bold Italic.
The heads were typeset in Futura, Italic, Bold, and Bold Italic.
Copyright © 2003 by Lawrence Erlbaum Associates, Inc.
All rights reserved. No part of this book may be reproduced in
any form, by photostat, microfilm, retrieval system, or any
other means, without prior written permission of the publisher.
Lawrence Erlbaum Associates, Inc., Publishers
10 Industrial Avenue
Mahwah, New Jersey 07430
Library of Congress Cataloging-in-Publication Data
Myers, Jerome L.
Research design and statistical analysis / Jerome L. Myers, Arnold D. Well.- 2nd ed.
ISBN 0-8058-4037-0 (case only : alk. paper)
1. Experimental design. 2. Mathematical statistics. I. Well, A. (Arnold) II. Title.
QA279 .M933 2002
519.5-dc21
Books published by Lawrence Erlbaum Associates are printed on
acid-free paper, and their bindings are chosen for strength and durability.
Printed in the United States of America
To Nancy and Susan
Preface xi
CHAPTER 1 INTRODUCTION 1
1.1 Variability and the Need for Statistics 1
1.2 Systematic Versus Random Variability 3
1.3 Error Variance Again 5
1.4 Reducing Error Variance 5
1.5 Overview of the Book 7
1.6 Concluding Remarks 7
CHAPTER 2 LOOKING AT DATA: UNIVARIATE
DISTRIBUTIONS 10
2.1 Introduction 10
2.2 Exploring a Single Sample 11
2.3 Comparing Two Data Sets 18
2.4 Other Measures of Location and Spread: The Mean and Standard
Deviation 20
2.5 Standardized (z) Scores 27
2.6 Measures of the Shape of a Distribution 28
2.7 Concluding Remarks 33
CHAPTER 3 LOOKING AT DATA: RELATIONS BETWEEN
QUANTITATIVE VARIABLES 37
3.2 Some Examples 37
3.3 Linear Relations 43
3.4 The Pearson Product-Moment Correlation Coefficient 44
viii CONTENTS
3.5 Linear Regression 51
3.6 The Coefficient of Determination, r2 54
3.7 Influential Data Points and Resistant Measures of Regression 55
3.8 Describing Nonlinear Relations 56
CHAPTER 4 PROBABILITY AND THE BINOMIAL
DISTRIBUTION 61
4.2 Discrete Random Variables 62
4.3 Probability Distributions 63
4.4 Some Elementary Probability 67
4.5 The Binomial Distribution 75
4.6 Means and Variances of Discrete Distributions 79
4.7 Hypothesis Testing 80
4.8 Independence and the Sign Test 86
4.9 More About Assumptions and Statistical Tests 89
4.10 Concluding Remarks 89
CHAPTER 5 ESTIMATION AND HYPOTHESIS TESTS: THE
NORMAL DISTRIBUTION 100
5.1 Introduction 100
5.2 Continuous Random Variables 100
5.3 The Normal Distribution 102
5.4 Point Estimates of Population Parameters 104
5.5 Inferences About Population Means: The One-Sample Case 112
5.6 Inferences About Population Means: The Correlated-Samples Case 117
5.7 The Power of the z Test 119
5.8 Hypothesis Tests and CIs 122
5.9 Validity of Assumptions 123
5.10 Comparing Means of Two Independent Populations 125
5.11 The Normal Approximation to the Binomial Distribution 128
5.12 Concluding Remarks 129
CHAPTER 6 ESTIMATION, HYPOTHESIS TESTS, AND EFFECT
SIZE: THE t DISTRIBUTION 140
6.2 Inferences About a Population Mean 141
6.3 The Standardized Effect Size 145
6.4 Power of the One-Sample t Test 147
6.5 The t Distribution: Two Independent Groups 152
6.6 Standardized Effect Size for Two Independent Means 156
6.7 Power of the Test of Two Independent Means 157
6.8 Assumptions Underlying the Two-Group t Test 158
6.9 Contrasts Involving More than Two Means 161
6.10 Correlated Scores or Independent Groups? 165
CONTENTS ix
CHAPTER 7 THE CHI-SQUARE AND F DISTRIBUTIONS 173
7.2 The x2 Distribution 174
7.3 Inferences About the Population Variance 175
7.4 The F Distribution 179
7.5 Inferences About Population Variance Ratios 182
7.6 Relations Among Distributions 185
7.7 Concluding Remarks 186
CHAPTER 8 BETWEEN-SUBJECTS DESIGNS: ONE FACTOR 191
8.2 Exploring the Data 193
8.3 The Analysis of Variance 195
8.4 The Model for the One-Factor Design 201
8.5 Assessing the Importance of the Independent Variable 207
8.6 Power of the F Test 212
8.7 Assumptions Underlying the F Test 216
CHAPTER 9 CONTRASTS AMONG MEANS 233
9.2 Definitions and Examples of Contrasts 234
9.3 Calculations of the t Statistic for Testing Hypotheses
About Contrasts 235
9.4 The Proper Unit for the Control of Type 1 Error 241
9.5 Planned Versus Post Hoc Contrasts 243
9.6 Controlling the FWE for Families of K Planned Contrasts 244
9.7 Testing All Pairwise Contrasts 247
9.8 Comparing a — 1 Treatment Means with a Control: Dunnett's Test 255
9.9 Controlling the Familywise Error Rate for Post Hoc Contrasts 256
9.10 The Sum of Squares Associated with a Contrast 258
CHAPTER 10 TREND ANALYSIS 267
10.1 Introduction 267
10.2 Linear Trend 268
10.3 Testing Nonlinear Trends 274
10.4 Concluding Remarks 280
CHAPTER 11 MULTIFACTOR BETWEEN-SUBJECTS DESIGNS:
SIGNIFICANCE TESTS IN THE TWO-WAY CASE 284
11.2 A First Look at the Data 285
11.3 Two-Factor Designs: The ANOVA 288
11.4 The Structural Model and Expected Mean Squares 295
11.5 Main Effect Contrasts 297
X CONTENTS
11.6 More About Interaction 298
11.7 Simple Effects 302
11.8 Two-Factor Designs: Trend Analysis 305
FURTHER DEVELOPMENTS 315
12.2 Measures of Effect Size 315
12.3 Power of the F Test 318
12.4 Unequal Cell Frequencies 319
12.5 Three-Factor Designs 324
12.6 More than Three Independent Variables 332
12.7 Pooling in Factorial Designs 332
12.8 Blocking to Reduce Error Variance 335
CHAPTER 13 REPEATED-MEASURES DESIGNS 342
13.2 The Additive Model and Expected Mean Squares for the
S x A Design 345
13.3 The Nonadditive Model for the S x A Design 352
13.4 Hypothesis Tests Assuming Nonadditivity 355
13.6 Multifactor Repeated-Measures Designs 363
13.7 Fixed or Random Effects? 371
13.8 Nonparametric Procedures for Repeated-Measures Designs 372
CHAPTER 14 MIXED DESIGNS: BETWEEN-SUBJECTS AND
WITHIN-SUBJECTS FACTORS 386
14.2 One Between-Subjects and One Within-Subjects Factor 386
14.3 Rules for Generating Expected Mean Squares 392
14.5 Power Calculations 396
14.6 Contrasting Means in Mixed Designs 397
14.7 Testing Simple Effects 401
14.8 Pretest-Posttest Designs 402
14.9 Additional Mixed Designs 403
14.10 Concluding Remarks 407
CHAPTER 15 USING CONCOMITANT VARIABLES
TO INCREASE POWER: BLOCKING AND
ANALYSIS OF COVARIANCE 412
15.2 Example of an ANCOVA 415
CONTENTS xi
15.3 Assumptions and Interpretation in an ANCOVA 422
15.4 Testing Homogeneity of Slopes 427
15.5 More About ANCOVA Versus Treatments x Blocks 428
15.6 Estimating Power in an ANCOVA 430
15.7 ANCOVA in Higher-Order Designs 431
15.8 Some Extensions of the ANCOVA 431
CHAPTER 16 HIERARCHICAL DESIGNS 436
16.2 Groups Within Treatments 437
16.3 Groups Versus Individuals 443
16.4 Extensions of the Groups-Within-Treatments Design 445
16.5 Items Within Treatments 449
CHAPTER 17 LATIN SQUARES AND RELATED DESIGNS 457
17.2 Selecting a Latin Square 459
17.3 The Single Latin Square 461
17.4 The Replicated Latin Square Design 469
17.5 Balancing Carry-Over Effects 474
17.6 Greco-Latin Squares 476
CHAPTER 18 MORE ABOUT CORRELATION 480
18.2 Further Issues in Understanding the Correlation
Coefficient 481
18.3 Inference About Correlation 489
18.4 Partial Correlations 501
1 8.5 Other Measures of Correlation 504
CHAPTER 19 MORE ABOUT BIVARIATE REGRESSION 519
19.2 Regression Toward the Mean 520
19.3 Inference in Linear Regression 522
19.4 An Example: Regressing Cholesterol Level on Age 532
19.5 Checking for Violations of Assumptions 534
19.6 Locating Outliers and Influential Data Points 542
19.7 Testing Independent Slopes for Equality 548
19.8 Repeated-Measures Designs 549
19.9 Multilevel Modeling 551
xii CONTENTS
CHAPTER 20 MULTIPLE REGRESSION 562
20.2 A Regression Example with Several Predictor Variables 563
20.3 The Nature of the Regression Coefficients 572
20.4 The Multiple Correlation Coefficient and the Partitioning of Variability
in Multiple Regression 573
20.5 Inference in Multiple Regression 580
20.6 Selecting the Best Regression Equation for Prediction 591
20.7 Explanation Versus Prediction in Regression 593
20.8 Testing for Curvilinearity in Regression 598
20.9 Including Interaction Terms in Multiple Regression 601
20.10 Multiple Regression in Repeated-Measures Designs 607
CHAPTER 21 REGRESSION WITH CATEGORICAL AND
QUANTITATIVE VARIARLES: THE GENERAL
LINEAR MODEL 614
21.2 One-Factor Designs 615
21.3 Regression Analyses and Factorial Designs 621
21.4 Using Categorical and Continuous Variables in the Same Analysis 630
21.5 Coding Designs with Within-Subjects Factors 634
Appendix A Notation and Summation Operations 641
Appendix B Expected Values and Their Applications 649
Appendix C Statistical Tables 653
Answers to Selected Exercises 685
Endnotes 721
Author Index 743
Subject Index 749
In writing this book, we had two overriding goals. The first was to provide a textbook from
which graduate and advanced undergraduate students could really learn about data analysis.
Over the years we have experimented with various organizations of the content and have
concluded that bottom-up is better than top-down learning. In view of this, most chapters
begin with an informal intuitive discussion of key concepts to be covered, followed by the
introduction of a real data set along with some informal discussion about how we propose to
analyze the data. At that point, having given the student a foundation on which to build, we
provide a more formal justification of the computations that are involved both in exploring
and in drawing conclusions about the data, as well as an extensive discussion of the relevant
assumptions. The strategy of bottom-up presentation extends to the organization of the
chapters. Although it is tempting to begin with an elegant development of the general linear
model and then treat topics such as the analysis of variance as special cases, we have found
that students learn better when we start with the simpler, less abstract, special cases, and then
work up to more general formulations. Therefore, after we develop the basics of statistical
inference, we treat the special case of analysis of variance in some detail before developing
the general regression approach. Then, the now-familiar analyses of variance, covariance,
and trend are reconsidered as special cases. We feel that learning statistics involves many
passes; that idea is embodied in our text, with each successive pass at a topic becoming
more general.
Our second goal was to provide a source book that would be useful to researchers.
One implication of this is an emphasis on concepts and assumptions that are necessary
to describe and make inferences about real data. Formulas and statistical packages are not
enough. Almost anybody can run statistical analyses with a user-friendly statistical package.
However, it is critically important to understand what the analyses really tell us, as well as
their limitations and their underlying assumptions. No text can present every design and
analysis that researchers will encounter in their own research or in their readings of the
research literature. In view of this, we build a conceptual foundation that should permit the
reader to generalize to new situations, to comprehend the advice of statistical consultants,
xiv PREFACE
and to understand the content of articles on statistical methods. We do this by emphasizing
such basic concepts as sampling distributions, expected mean squares, design efficiency,
and statistical models. We pay close attention to assumptions that are made about the
data, the consequences of their violation, the detection of those violations, and alternative
methods that might be used in the face of severe violations. Our concern for alternatives
to standard analyses has led us to integrate nonparametric procedures into relevant design
chapters rather than to collect them together in a single last chapter, as is often the case.
Our approach permits us to explicitly compare the pros and cons of alternative data analysis
procedures within the research context to which they apply.
Our concern that this book serve the researcher has also influenced its coverage. In our
roles as consultants to colleagues and students, we are frequently reminded that research
is not just experimental. Many standard textbooks on research design have not adequately
served the needs of researchers who observe the values of independent variables rather
than manipulate them. Such needs are clearly present in social and clinical psychology,
where sampled social and personality measures are taken as predictors of behavior. Even in
traditionally experimental areas, such as cognitive psychology, variables are often sampled.
For example, the effects of word frequency and length on measures of reading are often of
interest. The analysis of data from observational studies requires knowledge of correlation
and regression analysis. Too often, ignorant of anything other than analysis of variance,
researchers take quantitative variables and arbitrarily turn them into categorical variables,
thereby losing both information and power. Our book provides extensive coverage of these
research situations and the proper analyses.
MAJOR CHANGES IN THE SECOND EDITION
This second edition of Research Design and Statistical Analysis is a major revision of the
earlier work. Although it covers many of the same research designs and data analyses as
the earlier book, there have been changes in content and organization. Some new chapters
have been added; some concepts not mentioned in the first edition have been introduced,
and the coverage of some concepts that were previously discussed has been expanded. We
have been motivated in part by our sense that data analysis too often consists of merely
tabling means or correlation coefficients, and doing time-honored analyses on them without
really looking at the data. Our sense that we can learn more from our data than we often do
has been reinforced by the recent publication of the American Psychological Association's
guidelines for statistical methods (Wilkinson, 1999). Among other things, these guidelines
urge researchers to plot and examine their data, to find confidence intervals, to use power
analyses to determine sample size, and to calculate effect sizes. We illustrate these, and
other, procedures throughout this book. It may be helpful to consider the changes from the
first to the second edition in greater detail.
Statistics and Graphics. One change from the first edition is the expansion of the
section, Sample Distributions: Displaying the Data, into two chapters in the present edition.
Because it would take an entire volume to do justice to the array of statistics and graphic
devices available in many statistical computer packages, Chapters 2 and 3 provide only
some of the more basic ways of displaying univariate and bivariate data. However, these
should provide more insight into data than is usually the case. Furthermore, we believe that
PREFACE xv
an important contribution of the present text is that we then present such displays in many
subsequent chapters, using them to inform subsequent decisions about, and interpretation
of, the data analyses.
Confidence Intervals. Although we presented confidence intervals and discussed
their interpretation in the first edition, we now emphasize them in two ways. First, in our
chapters on inferences based on normal and t distributions, we present confidence intervals
before we present hypothesis tests. This is in accord with our belief that they deserve priority
because—as we point out—they provide the information available from hypothesis tests,
and more. Furthermore, they focus on the right question: What is the size of the effect?
rather than Is there an effect? Second, we make the calculation of confidence intervals a part
of the data analysis process in many of the subsequent chapters, illustrating their application
in various designs and with various statistics.
Standardized Effect Size. The calculation of standardized effect sizes has been
urged by several statisticians, most notably Cohen (1977). The standardized effect, in con-
trast to the raw effect, permits comparisons across experiments and dependent variables,
and it is a necessary input to power analyses. This new edition introduces the standardized
effect size early in the book (Chapter 6), and then it routinely illustrates its calculation in
subsequent chapters featuring different research designs and analyses.
Power Analyses. Power analyses, both to determine the required sample size and
to assess the power of an experiment already run, were discussed in the earlier edition.
There, we relied on charts that provided approximate power values. Currently, however,
several statistical software packages either provide direct calculations of power or provide
probabilities under noncentral distributions, which in turn allow the calculation of power.
Individuals lacking access to such programs can instead access software available on the
Internet that is easy to use and is free. We use two such programs in illustrations of power
analyses. In view of the ready accessibility of exact power analyses in both commercial
packages such as SAS, SPSS, and SYSTAT and in free programs such as GPOWER and
UCLA's statistical calculators, we have dropped the power charts, which are cumbersome to
use and at best provide approximate results. As with graphic displays, confidence intervals,
and effect size calculations, we present several examples of power calculations in the present
Tests of Contrasts. We believe that much research is, or should be, directed at focused
questions. Although we present all the usual omnibus tests of main effects and interactions,
we deal extensively with contrasts. We discuss measures of effect size and power analyses
for contrasts, and how to control Type 1 errors when many contrasts are considered. We
illustrate the calculation of tests of contrasts earlier (Chapter 6), presenting such tests as
merely a special case of t tests. We believe this simplifies things, paving the way for
presenting calculations for more complex designs in later chapters.
Elementary Probability. We have added a chapter on probability to review basic
probability concepts and to use the binomial distribution to introduce hypothesis testing. For
some students, reviewing the material in Chapter 4 may be unnecessary, but we have found
that many students enter the course lacking a good understanding of basic concepts such
xvi PREFACE
as independence, or of the distinction between p(A\B) and p(B\A). The latter distinction
is particularly important because a, b, statistical power, and the p values associated with
hypothesis tests are all examples of conditional probabilities. The chapter also serves the
purpose of introducing hypothesis testing in a relatively transparent context in which the
student can calculate probabilities, rather than take them as given from some table.
Correlation and Regression. The section on correlation and regression has been
reorganized and expanded. The basic concepts are introduced earlier, in Chapter 3, and are
followed up in Chapters 18-21. A major emphasis is placed on the kinds of misinterpre-
tations that are frequently made when these analyses are used. The treatment of power for
correlation and regression, and of interaction effects in multiple regression, is consider-
ably expanded. Significance tests for dependent correlations have been addressed both by
calculations and by software available on the Internet. Trend analysis and analysis of co-
variance are presented in Chapters 10 and 15 in ways that require only a limited knowledge
of regression, and then they are revisited as instances of multiple regression analyses in
Chapters 20 and 21. Nonorthogonal analysis of variance is first addressed in Chapter 12,
and then it is considered within the multiple regression framework in Chapter 21. We be-
lieve that the coverage of multiple regression can be more accessible, without sacrificing
the understanding of basic concepts, if we develop the topic without using matrix notation.
However, there is a development that uses matrix notation on the accompanying CD.
Data Sets. The CD-ROM accompanying the book contains several real data sets in
the Data Sets folder. These are provided in SPSS (.sav), SYSTAT (.syd), and ASCII (.txt)
formats, along with readme files (in Word and ASCII formats) containing information about
the variables in the data sets. The Seasons folder contains a file with many variables, as
well as some smaller files derived from the original one. The file includes both categorical
variables (e.g., sex, occupation, and employment status) and continuous variables (e.g.,
age, scores in each season on various personality scales, and physical measures such as
cholesterol level). The Royer folder contains files with accuracy and response time scores
on several arithmetic skills for boys and girls in first to eighth grades. The Wiley_Voss folder
contains a number of measures from an experiment that compares learning from text with
learning from Web sites. The Probability Learning folder contains a file from an experiment
that compares various methods of teaching elementary probability. In addition, there is an
Exercises folder containing artificial data sets designed for use with many of the exercises
in the book.
The "real-data" files have provided examples and exercises in several chapters. They
should make clear that real data often are very different from idealized textbook exam-
ples. Scores are often missing, particularly in observational studies, variables are often not
normally distributed, variances are often heterogeneous, and outliers exist. The use of real
data forces us to consider both the consequences of violations of assumptions and the re-
sponses to such violations in a way that abstract discussions of assumptions do not. Because
there are several dependent variables in these files, instructors may also find them useful in
constructing additional exercises for students.
Supplementary Material. We have also included three files in the Supplementary
Materials folder of the accompanying CD to supplement the presentation in the text. As we
note in Chapter 6, confidence intervals can be obtained for standardized effect sizes. We
PREFACE xvii
provided references to recently published articles that describe how to find these confidence
intervals in the text, and we illustrate the process in the "Confidence Intervals for Effect
Sizes" file in the Supplementary Materials folder. In addition, as we note in Chapter 20,
although not necessary for understanding the basic concepts, matrix algebra can greatly
simplify the presentation of equations and calculations for multiple regression. To keep the
length of the book within bounds, we have not included this material in the text; however; we
have added a file, "Chapter 20A, Developing Multiple Regression Using Matrix Notation,"
to the folder. Finally, when we discussed testing for the interaction between two quantitative
variables in multiple regression in the text, we mentioned that if we do not properly specify
the model, we might wind up thinking that we have an interaction when, in fact, we have
curvilinearity. We discuss this issue in the "Do We Have an Interaction or Do We Have
Curvilinearity or Do We Have Both?" file.
Chapter Appendices. Although we believe that it is useful to present some deriva-
tions of formulas to make them less "magical" and to show where assumptions are required,
we realize that many students find even the most basic kinds of mathematical derivations
intimidating and distracting. In this edition, we still include derivations. However, most
have been placed in chapter appendices, where they are available for those who desire a
more formal development, leaving the main text more readable for those who do not.
Instructors' Solutions Manual. In the "Answers to Selected Exercises" contained
in the text, we usually have provided only short answers, and we have done that only for
the odd-numbered exercises. The Solutions Manual contains the intermediate steps, and in
many cases further discussion of the answers, and does so for all exercises.
Many individuals have influenced our thinking of, and teaching of, statistics. Discussions
with our colleague Alexander Pollatsek have been invaluable, as has been the feedback of our
teaching assistants over the years. Most recently these have included Kristin Asplin, Mary
Bryden-Miller, Joseph DiCecco, Patricia Collins, Katie Franklin, Jill Greenwald, Randall
Hansen, Pam Hardiman, Celia Klin, Susan Lima, Jill Lohmeier, Laurel Long, Robert Lorch,
Edward O'Brien, David Palmer, Jill Shimabukuro, Scott Van Manen, and Sarah Zemore. We
would also like to thank the students in our statistics courses who encouraged us in this effort
and made useful suggestions about earlier drafts of the book. Special thanks go to those
individuals who reviewed early chapters of the book and made many useful suggestions that
improved the final product: Celia M. Klin, SUNY Binghamton; Robert F. Lorch, University
of Kentucky; Jay Maddock, University of Hawaii at Manoa; Steven J. Osterlind, University
of Missouri at Columbia; and Thomas V. Petros, University of North Dakota.
We wish to thank Mike Royer for making the Royer data available, Jenny Wiley and Jim
Voss for the Wiley_Voss data, and Ira Ockene for permission to use the Seasons data. The
Seasons research was supported by National Institutes of Health, National Heart, Lung, and
Blood Institute Grant HL52745 awarded to University of Massachusetts Medical School,
Worcester, Massachusetts.
We would like to express our gratitude to Debra Riegert, a senior editor at Erlbaum,
who encouraged us in this work and provided important editorial assistance. We also wish
xviii PREFACE
to thank the American Statistical Association, the Biometric Society, and the Biometrika
Trustees for their permission to reproduce statistical tables.
As in all our endeavors, our wives, Nancy and Susan, have been very supportive and
patient throughout the writing of this book. We gratefully acknowledge that contribution to
our work.
1.1 VARIABILITY AND THE NEED FOR STATISTICS
Empirical research is undertaken to answer questions that often take the form of whether,
and to what extent, several variables of interest are related. For example, an educator may
be interested in whether the whole language method of teaching reading is more effective
than another method based mostly on phonics; that is, whether reading performance is
related to teaching method. A political scientist may investigate whether preference for a
political party is related to gender. A social psychologist may want to determine the relation
between income and attitude toward minorities. In each case, the researcher tries to answer
the question by first collecting relevant measures and then analyzing the data. For example,
the educator may decide to measure the effectiveness of reading training by first obtaining
scores on a standard test of reading comprehension and then determining whether the scores
are better for one of the teaching methods than for the other.
A major problem in answering the research question is that there is variability in the
scores. Even for a single teaching method, the reading comprehension scores will differ
from one another for all sorts of reasons, including individual differences and measurement
errors. Some children learn to read faster than others, perhaps because they are brighter,
are more motivated, or receive more parental support. Some simply perform better than
others on standardized tests. All this within-treatment variability presents a number of
major challenges. Because the scores differ from one another, even within a single treatment
group, the researcher has to consider how to describe and characterize sets of scores before
they can be compared. Considerable attention will be given in this book to discussing how
best to display, summarize, and compare distributions of scores. Usually, there are certain
summary measures that are of primary interest. For example, the educational researcher
may be primarily interested in the average reading test score for each method of teaching
reading. The political scientist may want to know the proportion of males and females who
vote for each political party. The social psychologist may want a numerical index, perhaps
2 1 / INTRODUCTION
a correlation or regression coefficient, that reflects the relation between income and some
attitude score. Although each of these summary statistics may provide useful information,
it is important to bear in mind that each tells only part of the story. In Chapters 2 and 3,
we return to this point, considering statistics and data plots that provide a fuller picture of
treatment effects.
A major consequence of all the within-treatment variability is that it causes us to
refine the research question in a way that distinguishes between samples and populations. If
there was no within-treatment variability, research would be simple. If we wanted to compare
two teaching methods, we would only have to find the single reading comprehension score
associated with each teaching method and then compare the two scores. However, in a world
awash with variability, there is no single score that completely characterizes the teaching
method. If we took two samples of students who had been taught by one of the methods, and
then found the average reading comprehension score for each sample, these averages would
differ from one another. The average of a sample of comprehension scores is an imperfect
indicator of teaching effectiveness because it depends not only on the teaching method but
also on all the sources of variability that cause the scores to differ from one another. If we
were to find that a sample of scores from students taught by one teaching method had a
higher average than a sample from students taught by the other, how could we tell whether
the difference was due to teaching method or just to uncontrolled variability? What score
could be used to characterize reading performance for each teaching method to answer the
We generally try to answer the research question by considering the populations of
scores associated with each of the teaching methods; that is, all the scores that are relevant to
the question. To answer the question about teaching methods, we would ideally like to know
the comprehension scores for all the students who might be taught by these methods, now
and in the future. If we knew the population parameters, that is, the summary measures
of the populations of scores, such as the average, we could use these to answer questions
about the effectiveness of the teaching methods.
Obviously, we usually do not have access to the entire population of scores. In the
current example, the populations of comprehension scores are indefinitely large, so there is
no way that we can measure the population means directly. However, we can draw inferences
about the population parameters on the basis of samples of scores selected from the relevant
populations. If the samples are appropriately chosen, summary measures of the sample—the
sample statistics—can be used to estimate the corresponding population parameters. Even
though the sample statistics are imperfect estimators of the population parameters, they do
provide evidence about them. The quality of this evidence depends on a host of factors,
such as the sizes of the samples and the amount and type of variability. The whole field of
inferential statistics is concerned with what can be said about population parameters on
the basis of samples selected from the population. Most of this book is about inferential
It should be emphasized that, for population parameters to be estimated, the sam-
ples must be chosen appropriately. The statistical procedures we discuss in this book as-
sume the use of what are called simple random samples; these samples are obtained
by methods that give all possible samples of a given size an equal opportunity to be se-
lected. If we can assume that all samples of a given size are equally likely, we can use
the one sample we actually select to calculate the likelihood of errors in the inferences we
make.
SYSTEMATIC VERSUS RANDOM VARIABILITY 3
Even when randomly selected, the sample is not a miniature replica of the population.
As another example, consider a study of the change in arithmetic skills of third graders who
are taught arithmetic by use of computer-assisted instruction (CAI). In such a study, we are
likely to want to estimate the size of the change. We might address this by administering two
tests to several third-grade classes. One test would be given at the beginning of third grade,
and one would follow a term of instruction with CAI. The sample statistic of interest, the
average change in the sample, is unlikely to be exactly the same as the population parameter,
the average change that would have been observed if measurements were available for the
entire population of third graders. This is because there will be many sources of variability
that will cause the change scores to vary from student to student. Some students are brighter
than others and would learn arithmetic skills faster no matter how they were taught. Some
may have had experience with computers at home, or may have a more positive attitude
toward using a computer. If the variability of scores is large, even if we choose a random
sample, then the sample may look very different from the population because we just may
happen, by chance, to select a disproportionate number of high (or low) scores. We can
partly compensate for variability by increasing sample size, because larger samples of data
are more likely to look like the population. If there were no, or very little, variability in the
population, samples could be small, and we would not need inferential statistical procedures
to enable us to draw inferences about the population.
Because of variability, the researcher has a task similar to that of someone trying to
understand a spoken message embedded in noise. Statistical procedures may be thought of
as filters, as methods for extracting the message in a noisy background. No one procedure is
best for every, or even for most, research questions. How well we understand the message
in our data will depend on choosing the research design and method of data analysis most
appropriate in each study. Much of this book is about that choice.
1.2 SYSTEMATIC VERSUS RANDOM VARIABILITY
In the example of the study of CAI, the researcher might want to contrast CAI with a more
traditional instructional method. We can contrast two different types of approaches to the
research: experimental and observational. In an experiment, the researcher assigns subjects
to the treatment groups in such a way that there are no systematic differences between the
groups except for the treatment. One way to do this is to randomly assign students to each
of the two instructional methods. In contrast, in an observational or correlational study,
the researcher does not assign subjects to treatment conditions, but instead obtains scores
from subjects who just happen to be exposed to the different treatments. For example, in
an observational approach to the study of CAI, we might examine how arithmetic is taught
in some sample of schools, finding some in which CAI is used, others where it is not,
and comparing performances across the two sets of schools. In either the experimental or
the observational study, the instructional method is the independent variable. However,
in an experiment, we say that the independent variable is manipulated, whereas in an
observational study, we say the independent variable is observed. The dependent variable
in both approaches would be the score on a test of arithmetic skills. A problem with the
observational approach is that the treatment groups may differ systematically from one
another because of factors other than the treatment. These systematic differences often
make it very difficult or impossible to assess the effect of the treatment.
As we previously indicated, variables other than the independent variable could influ-
ence the arithmetic test scores. In both the experimental and the observational approaches,
the groups might differ by chance in ability level, exposure to computers outside of the
classroom, or parental encouragement. We will refer to these as nuisance variables. Al-
though they influence performance, and may be of interest in other studies, they are not
the variables of current interest and will produce unwanted, nuisance, variability. In an
experiment, we might account for the influence of nuisance variables by assigning students
to the teaching methods by using randomization; that is, by employing a procedure that
gave each student an equal chance of being assigned to each teaching method. Random
assignment does not perfectly match the experimental groups on nuisance variables; the
two groups may still differ on such dimensions as previous experience with computers,
or ability level. However, random assignment does guard against systematic differences
between the groups. When assignment to experimental conditions is random, differences
between groups on nuisance variables are limited to "chance" factors. If the experiment is
repeated many times, in the long run neither of the instructional methods will have an ad-
vantage caused by these factors. The statistical analyses that we apply to the data have been
developed to take chance variability into account; they allow us to ask whether differences
in performance between the experimental groups are more than would be expected if they
were due only to the chance operation of nuisance variables. Thus, if we find very large
differences on the arithmetic skills test, we can reasonably conclude that the variation in
instructional methods between experimental groups was the cause.
In an observational study we observe the independent variable rather than manipulate
it. This would involve seeking students already being taught by the two teaching methods
and measuring their arithmetic performance. If we did this, not only would the instructional
groups differ because of chance differences in the nuisance variables, it is possible that
some of them might vary systematically across instructional conditions, yielding systematic
differences between groups that are not readily accounted for by our statistical procedures.
For example, school districts that have the funds to implement CAI may also have smaller
class sizes, attract better teachers with higher salaries, and have students from more affluent
families, with parents who have more time and funds to help children with their studies. If
so, it would be difficult to decide whether superior performance in the schools using CAI
was due to the instructional method, smaller class size, more competent teachers, or greater
parental support. We describe this situation by saying that CAI is confounded with income
level. Because there is often greater difficulty in disentangling the effects of nuisance and
independent variables in observational studies, the causal effects of the independent variable
are more readily assessed in experiments.
Although we can infer causality more directly in experiments, observational studies
have an important place in the research process. There are many situations in which it is
difficult or impossible to manipulate the independent variable of interest. This is often the
case when the independent variable is a physical, mental, or emotional characteristic of
individuals. An example of this is provided in a study conducted by Rakkonen, Matthews,
Flory, Owens, and Gump (1999). Noting that ambulatory blood pressure (BP) had been
found to be correlated with severity of heart disease, they investigated whether it in turn
might be influenced by certain personality characteristics, specifically, the individual's level
of optimism or pessimism and general level of anxiety. These two predictor variables were
assessed by tests developed in earlier studies of personality. The dependent variable, BP, was
monitored at 30-minute intervals over 3 days while the 50 male and 50 female participants
REDUCING ERROR VARIANCE 5
went about their usual activities. An important aspect of the study was that participants kept
diaries that enabled the investigators to separate out the effects of several nuisance variables,
including mood, physical activity, posture (sitting and standing versus reclining), and intake
of caffeinated beverages such as coffee. By doing so, and by applying sophisticated statistical
procedures to analyze their data, the investigators were able to demonstrate that stable
personality characteristics (optimism, pessimism, and general anxiety level) influenced
BP beyond the transient effects of such variables as mood. Thus, it is possible to collect
data on all the important variables and to test causal models. However, such analyses are
more complicated and inferences are less direct than those that follow from performing an
1.3 ERROR VARIANCE AGAIN
Let's review some of the concepts introduced in Section 1.1, using some of the terms we
introduced in Section 1.2. Even if subjects have been randomly assigned to experimental
conditions, the presence of nuisance variables will result in error variance, variability
among scores that cannot be attributed to the effects of the independent variable. Scores
can be thought of as consisting of two components: a treatment component determined by
the independent variable and an error component determined by nuisance variables. Error
components will always exhibit some variability, even when scores have been obtained
under the same experimental treatment. This error variance may be the result of individual
differences in such variables as age, intelligence, and motivation. Error variance may also
be the result of within-individual variability when measures are obtained from the same
individuals at different times, and it is influenced by variables such as attentiveness, practice,
and fatigue.
Error variance tends to obscure the effects of the independent variable. For example,
in the CAI experiment, if two groups of third graders differ in their arithmetic scores,
the difference could be due, at least in part, to error variance. Similarly, if BP readings
are higher in more pessimistic individuals, as Rakkonen et al. (1999) found, we must ask
whether factors other than pessimism could be responsible. The goal of data analysis is to
divide the observed variation in performance into variability attributable to variation in the
independent variable, and variability attributable to nuisance variables. As we stated at the
beginning of this chapter, we have to extract the message (the effects of the independent
variable) from the noise in which it is embedded (error variance). Much of the remainder
of this book deals with principles and techniques of inferential statistics that have been
developed to help us decide whether variation in a dependent variable has been caused by
the independent variable or is merely a consequence of error variability.
1.4 REDUCING ERROR VARIANCE
If we can reduce error variance through the design of the research, it becomes easier for
us to assess the influence of the independent variable. One basic step is to attempt to hold
nuisance variables constant. For example, Rakkonen et al. (1999) took BP measurements
from all subjects on the same 3 days of the week; 2 were workdays and 1 was not. In this
way, they minimized any possible effects of the time at which measurements were taken.
In a study such as the CAI experiment, it is important that teachers have similar levels of
competence and experience, and, if possible, classes should be similar in the distribution of
ability levels. If only one level of a nuisance variable is present, it cannot give any advantage
to any one level of the independent variable, nor can it contribute to the variability among
the scores. Each research study will have its own potential sources of error variance, but,
by careful analysis of each situation, we can eliminate or minimize many of them.
We can also minimize the effects of error variance by choosing an efficient research
design; that is, we can choose a design that permits us to assess the contribution of one or
more nuisance variables and therefore to remove that contribution from the error variance.
One procedure that is often used in experiments is blocking, sometimes also referred to as
stratification. Typically, we divide the pool of subjects into blocks on the basis of some
variable whose effects are not of primary interest to us, such as gender or ability level. Then
we randomly assign subjects within each block to the different levels of the independent
variable. In the CAI experiment, we could divide the pool of third graders into three levels of
arithmetic skill (low, medium, and high) based on a test administered at the start of the school
year. We might then randomly assign students at each skill level to the two instructional
methods, yielding six combinations of instruction and initial skill level. The advantage of
this design is that it permits us to remove some of the contribution of initial skill level from
the total variability of scores, thus reducing error variance. The blocking design is said to
be more efficient than the design that randomly assigns subjects to instructional methods
without regard to ability level. Chapter 12 presents the analysis of data when a blocking
design has been used. For some independent variables (instructional method is not one of
them), even greater efficiency can be achieved if we test the same subject at each level of the
independent variable. This repeated-measures design is discussed in Chapter 13. Other
designs that enable us to remove some sources of error variance from the total variability
are the Latin Squares of Chapter 17.
Often, blocking is not practical. Morrow and Young (1997) studied the effects of
exposure to literature on reading scores of third graders. Although reading scores were
obtained before the start of the school year (pretest scores), the composition of the third-
grade classes was established by school administrators prior to the study. Therefore, the
blocking design we just described was not a possibility. However, the pretest score could
still be used to reduce error variance. Morrow and Young adjusted the posttest scores,
the dependent variable, essentially removing that portion of the score that was predictable
from the pretest score. In this way, much, though not all, of the variability caused by the
initial level of ability was removed from the final data set. This statistical adjustment, called
analysis of covariance, is presented in Chapter 15. Both blocking designs and analysis of
covariance use measures that are not affected by the independent variable but are related
to the dependent variable to reduce the error variance, thus making it easier to assess the
variability caused by the independent variable.
Usually the greater efficiency that comes with more complicated designs and analyses
has a cost. For example, additional information is required for both blocking and the analysis
of covariance. Furthermore, the appropriate statistical analysis associated with more efficient
approaches is usually based on more stringent assumptions about the nature of the data.
In view of this, a major theme of this book is that there are many possible designs and
analyses, and many considerations in choosing among them. We would like to select our
design and method of data analysis with the goal of reducing error variance as much as
possible. However, our decisions in these matters may be constrained by the resources and
CONCLUDING REMARKS 7
subjects that are available and by the assumptions that must be made about the data. Ideally,
the researcher should be aware of the pros and cons of the different designs and analyses,
and the trade-offs that must be considered in making the best choice.
1.5 OVERVIEW OF THE BOOK
Although most researchers tend to compute a few summary statistics and then carry out
statistical tests, data analyses should begin by exploring the data more thoroughly than is
usually done. This means not only calculating alternative statistics that tell us something
about the location, variability, and shape of the distribution of data, but also graphing the data
in various ways. Chapter 2 presents useful statistics and methods of graphing for univariate
data, that is, for cases involving a single variable. Chapter 3 does the same for bivariate
data, cases in which the relation between two variables is of interest.
Theoretical distributions play a central role in procedures for drawing inferences about
population parameters. These can be divided into two types: discrete and continuous. A
variable is discrete if it assumes a finite, countable, number of values; the number of indi-
viduals who solve a problem is an example. In contrast, a continuous variable can take on
any value in an interval. Chapter 4 presents an important discrete distribution, the binomial
distribution, and uses it to review some basic concepts involved in testing hypotheses about
population parameters. Chapter 5 provides a similar treatment of an important continuous
distribution, the normal distribution, extending the treatment of inference to concepts in-
volved in estimating population parameters, and intervals in which they may lie. Chapter 6
continues the treatment of continuous distributions and their applications to inferences about
population parameters in the context of the t distribution, and it also introduces the concept
of standardized effect size, a measure that permits comparisons of treatment effects ob-
tained in different experiments or with different measures. Chapter 7 concludes our review
of continuous distributions with a discussion of the chi-square (x 2 ) and F distributions.
As we noted in the preceding section, there are many different experimental designs.
We may assign subjects to blocks on the basis of a pretest score, or age, or gender, or some
other variable. We may test the same subject under several levels of an independent variable.
We may sequence the presentation of such levels randomly or in an arbitrary order designed
to balance practice or fatigue effects across treatments. These various experimental designs,
and the analyses appropriate for each, are discussed in Chapters 8-17.
Most of the analyses presented in the experimental design chapters are usually referred
to as analyses of variance. An analysis of variance, or ANOVA, is a special case of multiple
regression analysis, or MRA, a general method of analyzing changes in the dependent
variable that are associated with changes in the independent variable. Chapters 18-21
develop this regression framework, including estimation and statistical tests, and its relation
to ANOVA.
1.6 CONCLUDING REMARKS
In the initial draft of a report of a special task force of the American Psychological
Association (Task Force on Statistical Inference, 1996, posted at the APA Web site; see
also Wilkinson, 1999), the committee noted that "the wide array of quantitative techniques
and the vast number of designs available to address research questions leave the researcher
with the non-trivial task of matching analysis and design to the research question." The goal
of this book is to aid in that task by providing the reader with the background necessary to
make these decisions. No text can present every design and analysis that researchers will
encounter in their own work or in the research literature. We do, however, consider many
common designs, and we attempt to build a conceptual framework that permits the reader to
generalize to new situations and to comprehend both the advice of statistical consultants and
articles on statistical methods. We do this by emphasizing basic concepts; by paying close
attention to the assumptions on which the statistical methods rest and to the consequences
of violations of these assumptions; and by considering alternative methods that may have
to be used in the face of severe violations.
The special task force gave their greatest attention to "approaches to enhance the quality
of data usage and to protect against potential misrepresentation of quantitative results."
One result of their concern about this topic was a recommendation "that more extensive
descriptions of the data be provided. . . ." We believe this is important not only as a way
to avoid misrepresentation to reviewers and readers of research reports, but also as the
researcher's first step in understanding the data, a step that should precede the application
of any inferential procedure. In the next two chapters, we illustrate some of the descriptive
methods that are referred to in the report.
Boldfaced terms in the text are important to understand. In this chapter, many concepts were
only briefly introduced. Nevertheless, it will be useful to have some sense of them even at
a basic level. They are listed here for review.
within-treatment variability population
sample population parameter
sample statistic inferential statistics
random sample experiment
observational study independent variable
dependent variable nuisance variables
random assignment treatment component
error component error variance
blocking design efficiency
repeated-measures design analysis of covariance
discrete variable continuous variable
1.1 A researcher requested volunteers for a study comparing several methods to reduce
weight. Participants were told that if they were willing to be in the study, they would be
assigned randomly to one of three methods. Thirty individuals agreed to this condition
and participated in the study,
(a) Is this an experiment or an observational study?
EXERCISES 9
(b) Is the sample random? If so, characterize the likely population.
(c) Describe and discuss an alternative research design.
1.2 A study of computer-assisted learning of arithmetic in third-grade students was carried
out in a private school in a wealthy suburb of a major city.
(a) Characterize the population that this sample represents. In particular, consider
whether the results permit generalizations about CAI for the broad population of
third-grade students. Present your reasoning.
(b) This study was done by assigning one class to CAI and one to a traditional method.
Discuss some potential sources of error variance in this design.
1.3 Investigators who conducted an observational study reported that children who spent
considerable time in day care were more likely than other children to exhibit aggressive
behavior in kindergarten (Stolberg, 2001). Although this suggests that placement in
day care may cause aggressive behavior—either because of the day-care environment
or because of the time away from parents—other factors may be involved.
(a) What factors other than time spent in day care might affect aggressive behavior in
the study cited by Stolberg?
(b) If you were carrying out such an observational study, what could you do to try to
understand the effects on aggression of factors other than day care?
(c) An alternative approach to establishing the effects of day care on aggressive be-
havior would be to conduct an experiment. How would you conduct such an
experiment and what are the pros and cons of this approach?
1.4 It is well known that the incidence of lung cancer in individuals who smoke cigarettes
is higher than in the general population.
(a) Is this evidence that smoking causes lung cancer?
(b) If you were a researcher investigating this question, what further lines of evidence
would you seek?
1.5 In the Seasons study (the data are in the Seasons file in the Seasons folder on the
CD accompanying this book), we found that the average depression score was higher
for men with only a high school education than for those with at least some college
education. Discuss the implications of this finding. In particular, consider whether the
data demonstrate that providing a college education will reduce depression.
1.6 In a 20-year study of cigarette smoking and lung cancer, researchers recorded the
incidence of lung cancer in a random sample of smokers and nonsmokers, none of
whom had cancer at the start of the study.
(a) What are the independent and dependent variables?
(b) For each, state whether the variable is discrete or continuous.
(c) What variables other than these might be recorded in such a study? Which of these
are discrete or continuous?
Looking at Data:
Univariate Distributions
This chapter and the next are primarily concerned with how to look at and describe data.
Here, we consider how to characterize the distribution of a single variable; that is, what
values the variable takes on and how often these values occur. We consider graphic displays
and descriptive statistics that tell us about the location, or central tendency, of the distribution,
about the variability of the scores that make up the distribution, and about the shape of
the distribution. Although we present examples of real-life data sets that contain many
different variables, and sometimes compare several of them, in this chapter our focus is on
the description of single variables. In Chapter 3 we consider relations among variables and
present plots and statistics that characterize relations among two or more variables.
Data analyses should begin with graphs and the calculation of descriptive statistics. In
some instances, description is an end in itself. A school district superintendent may wish
to evaluate the scores on a standardized reading test to address various questions, such as
What was the average score? How do the scores in this district compare with those in the
state as a whole? Are most students performing near the average? Are there stragglers who
require added help in learning? If so, which schools do they come from? Do boys and girls
differ in their average scores or in the variability of their scores? We must decide which
statistics to compute to answer these and other questions, and how to graph the data to find
the most salient characteristics of the distribution and to reveal any unusual aspects.
In other instances, we may want to draw inferences about a population of scores on the
basis of a sample selected from it. Summarizing and graphing the data at hand is important
for that purpose as well. The exploration of the data may suggest hypotheses that we might
not otherwise have considered. For example, we may have begun the study with an interest in
comparing treatment averages but find that one treatment causes greater variability than the
others. A close look at our data may also suggest potential problems for the statistical tests
we planned. For example, the validity of many standard statistical tests depends on certain
EXPLORING A SINGLE SAMPLE 11
assumptions about the distributions of scores. Many tests assume that the distributions of
scores associated with each treatment are bell shaped; that is, they have a so-called normal
distribution.1 Some tests make the assumption that the variability of scores is the same for
each treatment. If these assumptions are not supported by the data, we may wish to consider
alternative procedures.
2.2 EXPLORING A SINGLE SAMPE
Suppose we have carried out a study of arithmetic performance by elementary school
students. Given the data set, there are many questions we could ask, but the first might be,
How well are the students doing? One way to answer this is to calculate some average value
that typifies the distribution of scores. "Typifies" is a vague concept, but usually we take as
a typical value the mean or median. These measures provide a sense of the location, or the
central tendency, of the distribution. We calculate the arithmetic mean for our sample by
adding together the students' scores and dividing by the number of students. To obtain the
median value, we rank order the scores and find the middle one if the number of scores is
odd, or we average the two middle scores if the number of scores is even. No matter which
average we decide to calculate, it provides us with limited information. For example, in a
study of the arithmetic skills of elementary school children conducted by Royer, Tronsky,
and Chan (1999; see the Royer data file in the Royer folder on the CD), the mean percentage
correct addition score for 28 second-grade students was 84.607 and the median was 89.2 This
tells us that, on the average, the students did quite well. What it does not tell us is whether
everyone scored close to the mean or whether there was considerable variability. Nor does
the average tell us anything about the shape of the distribution. If most students have scored
near the median but a few students have much lower scores, than we should know this
because it alerts us to the fact that there are children having problems with simple addition.
Table 2.1 presents the scores for the 28 students in the Royer study under the label
"Royer" together with a second set of 28 scores (Y) that we created that has the same mean
and median. A quick glance at the numbers suggests that, despite the fact that the two data
sets have the same means and medians, there are differences between the distributions.
Specifying the differences on the basis of an examination of the numbers is difficult, and
would be even more so if we had not placed them in order, or if the data sets were larger. We
need a way of getting a quick impression of the distributions of the scores—their location.
TABLE 2.1 THE ROYER GRADE 2 ADDITION SCORES AND AN ARTIFICIAL SET (Y ) WITH
THE SAME MEAN AND MEDIAN
Royer 84 85 88 89 89 90 93 94 94 94
Y 87 89 89 89 89 89 89 90 90 91
12 2 / UNIVARIATE DISTRIBUTIONS
Fig. 2.1 Histograms of the data in Table 2.1.
variability, and shape. Histograms, graphs of the frequency of groups of scores, provide one
way to view the data quickly. They can be generated by any one of several statistical (e.g.,
SPSS, SAS, or SYSTAT) or graphic (e.g., Sigma Plot, StatGraphics, or PsiPlot) programs
or spread sheets (e.g., Excel or Quattro Pro). Figure 2.1 presents such histograms for the
two data sets of Table 2.1.
2.2.1 Histograms of the Data
In these histograms, the X axis (the abscissa) has been divided into intervals of 5 points
each. The label on the left-hand Y axis (the ordinate) is the frequency, the number of
scores represented by each bar; the label on the right side is the proportion of the 28 scores
represented by each bar. Important characteristics of each distribution, as well as similarities
and differences among the distributions, should now be more evident than they would be
from a listing of the scores. For example, whereas the modal (most frequent) category in the
Royer data is the interval 96-100, the modal category in the Y data is the interval 86-90,
and the bar corresponding to the Y mode is noticeably higher than that of the Royer mode.
Another difference is that, despite being equal in both means and medians, theY distribution
contains two scores much lower than any in the Royer data.
The gap we observe in both the Y and Royer distributions is typical of many real
data sets, as is the obvious asymmetry in both distributions. Micceri (1989) examined 440
distributions of achievement scores and other psychometric data and noted the prevalence
of such departures from the classic bell shape as asymmetry (or skew), and "lumpiness,"
or more than one mode (the most frequently observed value). Similarly, after analyzing
many data distributions based on standard chemical analyses of blood samples, Hill and
Dixon (1982) concluded that their real-life data distributions were "asymmetric, lumpy,
and have relatively few unique values" (p. 393). We raise this point because the inferential
procedures most commonly encountered in journal reports rest on strong assumptions about
the shape of the distribution of data in the population. It is worth keeping in mind that these
assumptions are often not met, and it is therefore important to understand the consequences
of the mismatch between assumptions and the distribution of data. We consider those
consequences when we take up each inferential procedure.
Most statistical packages enable the user to determine the number of histogram intervals
and their width. There is no one perfect choice for all data sets. We chose to display
Fig. 2.2 Stem-and-leaf plot of the Royer data in
Table 2.1.
14 intervals between 30 and 100, each 5 points wide. We had previously constructed the
histograms with 7 intervals, each 10 points wide. However, this construction lost certain
interesting differences among the distributions. For example, because scores from 91 to 100
were represented by a single bar, the distributions looked more similar than they actually
were at the upper end. It is often helpful to try several different options. This is easily done
because details such as the interval width and number of intervals, or the upper and lower
limits on the axes, can be quickly changed by most computer graphics programs.
Histograms provide only one way to look at our data. For a more detailed look at the
numerical values in the Royer Grade 2 data, while still preserving information about the
distribution's shape, we next consider a different kind of display.
2.2.2 Stem-and-Leaf Displays
Figure 2.2 presents a stem-and-leaf display of the Royer data. The display consists of two
parts. The first part contains five values, beginning with the minimum and ending with the
maximum. This first part is sometimes referred to as the 5-point summary. The minimum
and maximum are the smallest and largest values in the data set. Before we consider the
second part of the display, the actual stem-and-leaf plot, let's look at the remaining 3 points
in the 5-point summary.
The Median. If the number of scores, N, is an odd number, the median is the middle
value in a set of scores ordered by their values. If N is an even number, the median is the
value halfway between the middle two scores. Another way of thinking about this is to
define the position of the median in an ordered set of scores; this is its depth, dM, where
For example, if
Y = 1,3,4,9,12, 13, 18
14 2/ UNIVARIATE DISTRIBUTIONS
then N = 7, dM = 4, and the median is the fourth score, 9. If the preceding set contained
an additional score, say 19, we would have N= 8 and dM = 4.5. This indicates that the
median would be the mean of the fourth and fifth scores, 9 and 12, or 10.5. In the Royer
data, there are 28 scores; therefore dM = 14.5, and, because the 14th and 15th scores are
both 89, the median is 89.
The Hinges. There are many possible measures of the spread of scores that are based
on calculating the difference between scores at two positions in an ordered set of scores. The
range, the difference between the largest and the smallest scores, has intuitive appeal, but its
usefulness is limited because it depends only on two scores and is therefore highly variable
from sample to sample. Other measures of spread are based on differences between other
positions in the ordered data set. The interquartile range, or IQR, is one such measure.
The first quartile is that value which equals or exceeds one fourth of the scores (it is also
referred to as the 25th percentile). The second quartile is the median (the 50th percentile),
and the third quartile is the value that equals or exceeds three fourths of the scores (the 75th
percentile). The IQR is the difference between the first and third quartile. Calculating the
first or third quartile value is often awkward, requiring linear interpolation. For example,
if there are seven scores, the first quartile is the value at the 1.75 position, or three fourths
of the distance between the first and second score. Somewhat simpler to calculate, but close
to the first and third quartile, are the hinges. As an example of their calculation, and of
the interhinge distance, or H spread, consider the Royer data of Table 2.1. Then take the
following steps:
1. Find the location, or depth, of the median dM = (N + l)/2. With 28 scores,
dM = 14.5.
2. When dM has a fractional value—that is, when N is an even number—drop the
fraction. We use brackets to represent the integer; that is, [ d M ] = 14. The lower
and upper hinges are simply the medians of the lower and of the upper 14 scores.
3. Find the depth of the lower hinge, dm. This is given by
In our example, dLH = 7.5; this means that the lower hinge will be the score midway
between the seventh score (76) and the eighth score (82), or 79. The upper hinge
will lie midway between the seventh and eighth scores from the top; this is 94.5 in
the Royer data. The H spread is therefore 94.5 — 79, or 15.5.
The 5-point summary provides a rough sense of the data. The median tells us that at
least half of the Grade 2 students have a good grasp of addition. When we consider the
minimum and maximum together with the median, it is clear that there are some stragglers;
the distance from the minimum to the median is almost four times greater than that of
the maximum to the median. However, that distance could be due to just one student
with a low score. More telling is the comparison of the top and bottom fourths; 25%
of the students have scores between 95 and 100, whereas another 25% fall between 47
and 79.
Most of our readers are probably more familiar with the arithmetic mean than with
the median, and with the variance (or its square root, the standard deviation) than with the
H spread. It is worth noting that the mean and variance are more sensitive to individual
scores than the median and H spread. If we replaced the 47 in the data set with a score of
67, the median and H spread would be unchanged but the mean would be increased and the
variance would be decreased. Because they change less when the values of individual scores
are changed, we say that the median and H spread are resistant statistics. This does not
necessarily mean that they are better measures of location and variability than the mean and
variance. The choice of measures should depend on our purpose. The median and H spread
are useful when describing location and variability. However, the mean and variance play
a central role in statistical inference.
The Stem-and-Leaf Plot. The plot in Fig. 2.2 is essentially a histogram laid on its
side. The length of each row gives a sense of the frequency of a particular range of scores,
just as in the histogram. However, this plot provides somewhat more information because
it allows us to reconstruct the actual numerical values of Table 2.1. The left-hand column
of values, beginning with 4 and ending with 10, is called the stem. For the Royer data,
to obtain a score, we multiply the stem by 10 and add the leaf, the value to the right of
the stem. Thus the first row of the plot represents the score of 47. The next row informs
us that there are two scores of 50. The next two rows contain the scores 69, 72, and 74.
The row following this indicates the score of 76 and has an H between the stem (7) and
the sole leaf (6). The H indicates that the (lower) hinge lies in the range of scores from
75 to 79. Note that it does not mean that the lower hinge is 76; the hinges and the median do
not necessarily correspond to observed scores; in this case, the actual hinge is 79, midway
between the observed scores of 76 and 82.
The stem-and-leaf plot provides a sense of the shape of the distribution, although the
gap between 50 and 71 is not as immediately evident as it was in the histogram. The trade-off
between the histogram and the stem-and-leaf plot is that the former usually provides a more
immediate sense of the overall shape whereas the latter provides more detail about the nu-
merical values. In addition, it provides summary statistics in the hinges and median, and, as
we discuss shortly, it also clearly marks outliers, scores that are very far from most of the data.
The values by which the stem and leaf should be multiplied depend on the numerical
scale. Consider a set of 30 Standardized Achievement Test (SAT) scores, the first 10 of which
are 394, 416, 416, 454, 482, 507, 516, 524, 530, and 542. Figure 2.3 presents SYSTAT's
stem-and-leaf display for the entire data set. To obtain an approximation to the actual scores,
multiply the stem by 100 and the leaf by 10. Thus the first row tells us that there is a score
between 390 and 399, actually 394. The next row tells us that there are two scores between
410 and 419; both are actually 416. Although we cannot tell the exact score from this plot,
we clearly have more information than the histogram would have provided, and we still
have a sense of the shape of the distribution.
Outliers. In both Figs. 2.2 and 2.3, H marks the intervals within which the hinges fall
and M marks the interval that contains the median. The values above the "Outside Values"
line in the Royer plot, and outside the two such lines in Fig 2.3, are called outliers. In the
Royer data, the outliers call our attention to students whose performances are far below
those of the rest of the class; these students may require remedial help. Of course, there
are other possible reasons for outliers in data sets. The students who produced the scores
of 47 and 50 may have been ill on the day the test was administered, or have performed
below their capabilities for other reasons. In some cases, outliers may reflect clerical errors.
In situations in which interest resides primarily in the individuals tested, it is important
to identify outliers and try to ascertain whether the score truly represents the ability or
Fig. 2.3 Stem-and-leaf plot of 30 SAT scores.
characteristic being assessed. In studies in which we are primarily interested in drawing
inferences about a population—for example, in deciding which of two treatments of a
medical condition is superior—outliers should also be considered. If they are due to clerical
errors in transcribing data, they can distort our understanding of the results, and therefore
should be corrected. In some cases, there is no obvious reason for an outlier. It may reflect a
subject's lack of understanding of instructions, a momentary failure to attend, or any number
of other factors that distort the behavioral process under investigation. Unfortunately, there
is no clear consensus about how to deal with such data. Some researchers do nothing about
such nonclerical outliers. Most either replace the outlier by the nearest nonoutlying score,
or just drop the outlier from the data set.
Our present concern is to understand how outliers are defined. The criterion for the
outside values in Fig. 2.3 was calculated in the following steps:
1. Calculate the H spread. In Fig. 2.3, this is 602 - 524, or 78.
2. Multiply the H spread by 1.5. The result is 117.
3. Subtract 117 from the lower hinge and add 117 to the upper hinge. The resulting
values, 407 and 719, are called inner fences. Scores below 407 and above 719 are
outliers.
Equation 2.3 represents these steps: a score, Y, is an outlier if
where HL and Hu are the lower and upper hinges, respectively. Outer fences may be
calculated by multiplying the H spread by 3, rather than 1.5. The lower outer fence would
be 524 - 234, or 290, and the upper outer fence would be 602 + 234, or 836. Values beyond
these two points would be labeled extreme outliers.
Fig. 2.4 Box plots of the data in Table 2.1.
Histograms and stem-and-leaf displays provide considerable information about the
shape of the distribution. A less detailed view, but one that provides a quick snapshot of the
main characteristics of the data, may be obtained by still another type of plot, which we
consider next.
2.2.3 Box Plots
Figure 2.4 presents box plots of the Royer Grade 2 addition accuracy data and of the Y data
of Table 2.1. The top and bottom sides of the "boxes" are at the hinges. The lines somewhat
above the middle of the boxes are at the medians. The lines extending vertically from the
boxes are sometimes referred to as "whiskers." Their endpoints are the most extreme scores
that are not outliers.
For the Royer data, the 5-point summary of Fig. 2.2 informs us that the hinges are at
79 and 94.5. Therefore, the H spread is 15.5, and the lower fence is 79 — 1.5 x 15.5, or
55.75. The asterisks represent outliers, scores below 55; in this example, these are the 47
and the two 50s in Table 2.1. There are no extreme outliers in the Royer data but there are
two in the Y data (scores of 31 and 32; see Table 2.1); these are represented by small circles
rather than asterisks. The bottom whisker in the Royer data extends to 69, the lowest value
in Table 2.1 that was not an outlier. Note that the whisker does not extend to the fence; the
fence is not represented in the plot.
The box plot quickly provides information about the main characteristics of the dis-
tribution. The crossbar within the box locates the median, and the length of the box gives
us an approximate value for the H spread. The box plot for the Royer data tells us that the
distribution is skewed to the left because the bottom whisker is longer than the top, and
there are three low outliers. Thus, at a glance, we have information about location, spread,
skewness, tail length, and outliers. Furthermore, we can see at a glance that the H spread
is much smaller for the Y data, that the two medians are similar, and, with the exception of
the two extreme outliers, the Y data are less variable than the Royer data. To sum up, the
stem-and-leaf and box plots provide similar information. However, the stem-and-leaf plot
gives us numerical values of hinges, medians, and outliers, and it provides a more precise
view of the distribution, In contrast, the box plot makes the important characteristics of
the distribution immediately clear and provides an easy way of comparing two or more
distributions with respect to those characteristics.
2.3 COMPARING TWO DATA SETS
Suppose we have measures of anxiety for male and female samples. We might wish to know
whether men and women differ with respect to this measure. Typically, researchers translate
this into a question of whether the mean scores differ, but there may be more to be learned
by also comparing other characteristics of the distributions. For example, researchers at the
University of Massachusetts Medical School collected anxiety measures, as well as several
other personality measures, during each season of the year, from male and female patients
of various ages.3 We calculated the average of the four seasonal anxiety scores for each
participant in the study for whom all four scores were available. The means for the two
groups are quite similar: 4.609 for female participants and 4.650 for male participants. Nor
is there a marked difference in the medians: 4.750 for female participants and 4.875 for
male participants. However, plotting the distributions suggested that there is a difference.
Figure 2.5 contains box plots and histograms created with the data from 183 female and
171 male participants. If we look first at the box plots, it appears that the H spread (the
length of the box) is slightly greater for women, suggesting greater variability in their
anxiety scores. We further note that there are more outlying high scores for the women.
Turning to the histograms, we confirm this impression. Why this difference in variability
Fig. 2.5 Box plots and histograms of anxiety data.
COMPARING TWO DATA SETS 19
occurs and what the implications are for the treatment of anxiety, if any, is something best
left to the medical investigators. We only note that plotting the data reveals a difference
in the distributions that may be of interest, a difference not apparent in statistics reflecting
This is not to suggest that we disregard measures of location, but that we supplement
them with other statistics and with plots of the data. With respect to measures of location, it
is a good idea to bear in mind that there are many situations in which the mean and median
will differ. In any salary dispute between labor and management, management may argue
that salaries are already high, pointing to the mean as evidence. Workers, recognizing that
a few highly paid individuals have pushed the mean upward, may prefer to negotiate on the
basis of a lower value, the median, arguing that 50% of employees have salaries below that
figure. Similar discrepancies between mean and median also arise with many behavioral
measures. The data in the Seasons file (Seasons folder in the CD) present one illustration.
In examining Beck depression scores for the winter season for men of various ages, we
found a difference between the mean (over seasons) of the youngest group (<40 years,
mean = 6.599) and that of a group between 50 and 59 years old (mean = 5.502). However,
the medians were identical at 4.500.
Plotting the Beck depression data for the two groups is a first step in clarifying why
the means are further apart than the medians. The upper panel of Fig. 2.6 presents box
plots for the two groups. As in all plots of Beck depression scores, most of the scores are
at the low (normal) end of the scale in the histograms for both age groups. We say that
the distributions are skewed to the right because of the straggling right-hand (upper) tails.
The explanation for the difference in means is readily apparent. The younger group has
Fig. 2.6 Box plots and histograms of winter depression scores for two age
groups of men.
several individuals with scores above 18; these are depicted as outliers in the box plot for
that group. Some of the outliers, depicted by open circles, are extreme outliers according
to the definition of Subsection 2.2.2. No score is an outlier in the older group. Although the
medians are identical, the greater number of extreme high scores in the younger group has
moved that mean higher than the mean of the older group. The histograms in the lower panel
of Fig. 2.6 provide a complementary view of the data. Again, the more pronounced right-
hand tail in the younger group is evident. Just why there are more extremely depressed
men in the under-40 group is not clear. It may be due to chance, in which case other
samples from the same population may not exhibit the pattern of medians and means we
have noted. It may be that, for some reason, more depressed older men are not in the pool
of patients included in the study. For now, we withhold judgment about the cause of the
pattern we have observed, while noting that calculating both the mean and the median, and
plotting the data, may be more informative than calculating only one of the two measures of
2.4 OTHER MEASURES OF LOCATION AND SPREAD:
THE MEAN AND STANDARD DEVIATION
Thus far, using stem-and-leaf and box plots has focused our attention on the median and
the H spread. However, there are many other statistics that can aid us in understanding
our data, and these are readily obtained from most statistical packages. For example, a
sequence of clicks ("Analyze," "Descriptive Statistics," and "Explore") in SPSS provides
the statistics in Table 2.2, summarizing the data of Table 2.1. We have already considered
several of these statistics—the median, minimum, maximum, range, and IQR. Others, such
as the confidence interval, are explained in Chapter 5. In this section, we focus on the
mean, standard deviation, and variance; although we suspect that they are familiar from
an introductory course, it may be useful to review their definitions and properties. We also
discuss the standard errors (labeled Std. Error in Table 2.2) of Table 2.1. Later in this chapter,
we consider skewness and kurtosis, statistics that summarize aspects of the shape of the data
distribution. In presenting some statistical formulas, we occasionally use a summation sign,
a capital Greek sigma, £. The use of notation provides a shorthand; if a picture is worth a
thousand words, a Greek symbol, or an equation, is often worth at least a dozen. Although
our use of notation is intended to be easily understood, readers may find it helpful to refer
to Appendix A, which reviews the algebra of summation and derives several properties of
statistics that we state in this chapter.
2.4.1 The Arithmetic Mean
The familiar arithmetic mean, symbolized by Y (Y bar), is just the sum of all scores divided
by the number of scores. Expressing this as an equation, we have
where N represents the number of scores in a sample. For example, the mean of Y = 1, 2,
3, 5, 9, 10, 12 is Y = 42/7 = 6. The widespread use of the mean reflects two advantages it
OTHER MEASURES OF LOCATION AND SPREAD: THE MEAN AND STANDARD DEVIATION 21
TABLE 2.2 SUMMARY STATISTICS FOR THE DATA OF TABLE 2.1
Score Set Descriptives Statistic Std. Error
Royer Mean 84.61 2.89
95% confidence Lower Bound 78.68
Interval for Mean Upper Bound 90.54
5% Trimmed Mean 85.79
Variance 234.025
Std. Deviation 15.30
Minimum 47
Maximum 100
Interquartile Range 17.25
Skewness -1.326 .441
Kurtosis 1.124 .858
Y Mean 84.61 2.92
Maximum 95
Interquartile Range 6.00
Note. Table is output from SPSS's Explore module.
TABLE 2.3 EXAMPLE OF MEANS BASED ON DIFFERENT SAMPLE SIZES
Parameter A B C D
Mean 17.5 18.3 19.2 22.6
Sample size (n) 26 17 31 24
has over other measures of location. First, we can manipulate it algebraically, and, second,
it has certain properties that are desirable when we estimate the mean of the population
from which the sample was drawn. The role of the sample mean as an estimator of the
population mean is discussed in Chapter 5. In this chapter, we summarize several useful
algebraic properties. These properties can be proven by using the rules of summation in
Appendix A. They can be demonstrated by using the preceding set of numbers, or any other
numbers you choose.
1. Adding a constant, k, to every score in the sample results in the mean being increased
by the amount k; that is, E(Y + k}/N = Y + k. For example, if we add 10 to each
of the values in the preceding set of 7 numbers, the mean increases from 6 to 16.
2. Multiplying every score in the sample by a constant, k, results in the mean being
multiplied by k; that is, E(kY)/N = kY. For example, multiplying each of the
scores in the example by 2 increases the mean from 6 to 12.
3. Perhaps most importantly, means can be combined. For example, given the means
of depression scores and the sample sizes from several clinics, we are able to
calculate the mean based on the combined data sets.
The means and sample sizes (n) for each of four clinics are presented in Table 2.3. It is
tempting to add the four means and divide by 4 to get the mean of the combined data sets.
However, because the four ns vary, this will not do. The mean for Clinic C should carry
more weight and that for Clinic B less weight in combining the means because of their
relative sample sizes. The correct approach requires us to obtain the sum of all the scores
in all four data sets and then divide by N, the sum of the ns. We obtain the sum of scores for
each clinic by multiplying the clinic mean by the number of scores for that clinic. Summing
these four sums, and dividing the grand total by N, the total number of scores, we have the
grand mean of all the scores:
We might have rewritten this slightly:
Equation 2.5 suggests that the mean can be represented as a sum of weighted values
where the weights are proportions or probabilities. The weight for Clinic A is 26/98 because
26 of the 98 depression scores come from Clinic A. To take a somewhat different example,
consider a student who has spent two semesters in College A and compiles a 3.2 grade
point average (GPA). She then transfers to College B, where she earns a 3.8 GPA for the
next three semesters. The student's overall GPA for the five semesters is calculated as in
the preceding example of the clinic means. The overall GPA is a weighted mean in which
the weights are 2/5 and 3/5:
In general, the preceding calculations may be represented by
Equation 2.6 is the formula for a weighted mean. It indicates that each distinct value of
Y is to be multiplied by its weight, p(y), the proportion of all scores with that value. All
of these products are then added together. Note that the usual expression for the arithmetic
mean Y = E Y/N is a special case of the preceding formula for the weighted mean; here
each of the N scores in the sample is given a weight of 1 / N .
Two other properties of the mean, proven in Appendix A at the end of the book, may
help convey the sense in which it reflects the central tendency of a data set:
4. The mean is the balance point of the data in the sense that the sum of the deviations
about the mean is zero; that is, E(y — Y) = 0.
5. The sum of squared deviations of scores from the mean is smaller than the sum
of squared differences taken from any point other than the mean; that is, E(Y —
(Y + k)]2 has its smallest value when k = 0.
Every value in a set of scores is incorporated into the calculation of the mean. One
benefit of this is that, as in our example of the four clinics, means can be combined without
access to the individual values on which they are based. This strength is, however, also a
potential weakness. A few extreme scores can bias the value of the mean, and they may
result in the mean's taking on a value that does not typify the distribution of scores. As we
discussed earlier in this chapter, the median provides an alternative measure of location that
is resistant to the influence of extreme scores.
2.4.2 The Standard Deviation and the Variance
We group these statistics together because the variance is the square of the standard devia-
tion. The sample variance, S2, is the average squared deviation of scores about their mean;
that is, S2 = ^2(Y — Y)2/N. However, as most statistical packages do, we will divide
by N — 1, rather than N. The divisor N — 1 is used, rather than N, because it results in a
better estimate of the population variance. We delay further consideration of this point to
the discussion of estimation in Chapter 5. We denote this revised definition by s2, rather
than S2, to indicate that we are dividing by N — 1. Thus, our formula for s2 is
The calculation of 52 is illustrated with the following set of seven scores:
The sum of the scores is 42, and, therefore, Y = 42/7 = 6. Finding the deviations from the
mean, we have
squaring these deviations, we have
and summing the squared deviations, we have E(Y — Y)2 = 112. Then, s2 = 112/6 =
18.667. Because the variance is based on the squared deviations of the scores about the
mean, the standard deviation, the square root of the variance, is usually preferred as a
measure of spread. In the preceding example, s = \/18.667 = 4.320.
Two properties of the standard deviation should be noted:
1. When a constant is added to all scores in a distribution, the standard deviation is
unchanged. That is, if Y' = Y + k, where k is a constant, then sy = SY . Intuitively,
each score is increased (or decreased) by the same amount so the spread of scores
is unchanged. The range and the H spread also are unchanged when a constant is
added, and it is a desirable property of any measure of variability.
2. When each score is multiplied by a positive constant, k, the standard deviation of
the new scores is k times the old standard deviation. If k is negative, multiplying
each score by k is equivalent to multiplying the standard deviation by — k; the
reason for this is that the standard deviation must always be a positive number.
You can verify these multiplication properties by multiplying the preceding values
of Y by 10, and then by —10, and recalculating both s and s2 in each case. The
new standard deviation is 43.2, 10 times the original value, and the new variance
is 1866.7, 100 times the value on the original scale.
We can summarize the properties of the standard deviation as follows:
1. l f Y ' = Y +£, then SY> =SY.
2. If Y' = kY, then sy = ksy when k > 0 and sy = —ksY when k < 0.
These properties are proven in Appendix A at the back of the book.
Although the standard deviation is less intuitive than other measures of variability, it has
two important advantages. First, the standard deviation is important in drawing inferences
about populations from samples. It is a component of formulas for many significance tests,
for procedures for estimating population parameters, and for measures of relations among
variables. Second, it (and its square, the variance) can be manipulated arithmetically in
ways that other measures cannot. For example, knowing the standard deviations, means, and
sample sizes of two sets of scores, we can calculate the standard deviation of the combined
data set without access to the individual scores. This relation between the variability within
groups of scores and the variability of the total set plays an important role in data analysis.
Both of the properties just noted will prove important throughout this book.
The main drawback of the standard deviation is that, like the mean, it can be greatly
influenced by a single outlying score. Recall that for Y = 1, 2, 3, 5, 9, 10, and 12, Y = 6
and s = 4.320. Suppose we add one more score. If that score is 8, a value within the range
of the scores, then the new mean and standard deviation are 6.25 and 4.062, a fairly small
change. However, if the added score is 20, then we now have Y =7.75 and s = 6.364. The
standard deviation has increased by almost 50% with the addition of one extreme score.
The H spread (or its fraternal twin, the IQR) is resistant to extreme scores and is often a
more useful measure for describing the variability in a data set. We again emphasize that
there is no one best measure of variability (or for that matter, of location or shape), but that
there is a choice, and that different measures may prove useful for different purposes, or
may sometimes supplement each other.
2.4.3 The Standard Error of the Mean
Among the many statistics commonly available from statistical packages is one labeled the
standard error ("Std. Error" in the SPSS output of Table 2.2), or standard error of the mean
(SEM). The SEM is a simple function of the standard deviation:
To understand the SEM, assume that many random samples of size N are drawn from the
same population, and that the mean is calculated each time. The distribution of these means
is the sampling distribution of the mean for samples of size N. The SEM that is calculated
from a single sample is an estimate of the standard deviation of the sampling distribution of
the mean. In other words, it is an estimate of how much the mean would vary over samples.
If the SEM is small, the one sample mean we have is likely to be a good estimate of the
population mean because the small SEM suggests that the mean will not vary greatly across
samples, and therefore any one sample mean will be close to the population mean. We have
considerably more to say about the SEM and its role in drawing inferences in later chapters.
At this point, we introduced it because of its close relation to the standard deviation, and
because it provides an index of the variability of the sample mean.
2.4.4 The 5% Trimmed Mean
The SPSS output of Table 2.2 includes the value of the 5% trimmed mean. This is calculated
by rank ordering the scores, dropping the highest and lowest 5%, and recalculating the mean.
The potential advantage of trimming is that the SEM will be smaller than for the untrimmed
mean in distributions that have long straggling tails, or have so-called "heavy" tails that
contain more scores than in the normal distribution. In view of the preceding discussion
of the SEM, this suggests that in some circumstances the trimmed mean will be a better
estimator of the population mean. However, decisions about when to trim and how much to
trim are not simple. Rosenberger and Gasko (1983) and Wilcox (1997) have written good
discussions on this topic.
2.4.5 Displaying Means and Standard Errors
A graph of the means for various conditions often provides a quick comparison of those
conditions. When accompanied by a visual representation of variability, such as s, or the
SEM, the graph is still more useful. How best to graph the data should depend on the nature of
the independent variable. Although graphics programs will provide a choice, the American
Psychological Association's Publication Manual (2001) recommends that "Bar graphs are
used when the independent variable is categorical" and "Line graphs are used to show the
relation between two quantitative variables" (p. 178). We believe this is good advice. When
the independent variable consists of categories that differ in type, rather than in amount, we
should make it clear that the shape of a function relating the independent and dependent
variables is not a meaningful concept. Figure 2.7 presents mean depression scores4 from
the Seasons data set as a function of marital status and sex; the numbers on the x axis are
the researchers' codes: 1 = single; 2 = married; 3 = living with partner; 4 = separated;
5 — divorced; 6 = widowed. At least in this sample, depression means are highest for
single men and women, and for divorced women, and the means are low for those living
with a partner. Without a more careful statistical analysis, and without considering the
size of these samples, we hesitate to recommend living with a partner without marriage,
but we merely note that the bar graph presents a starting point for comparing the groups.
The vertical lines at the top of each bar represent the SEMs. Note that the SEM bars
Fig. 2.7 Bar graph of mean depression
scores as a function of marital status and
Fig. 2.8 Line graph of subtraction response times as a function
of grade and gender.
indicate that the male data are more variable than the female data, particularly in categories
3 and 6.
Figure 2.8 presents mean times in seconds to do subtraction problems (labeled "Subrt")
from the Royer data set as a function of grade and gender. Because grade level is quantitative,
it is useful to plot the data as line curves, providing a sense of the shape of the functional
relation. It appears that response times for both genders decrease as a function of grade and
seem to level off near the sixth grade. It also appears that variability decreases with grade, as
indicated by the general decrease in the length of the SEM bars. Comparing across panels,
we see that the main difference between boys' and girls' times appears to be in the early
grades. We plotted these data in two panels because the two curves are close together and
the error bars are difficult to disentangle if presented within a single panel.
Software capable of bar and line plots usually offer several options such as the choice
of placing different plots in one panel (as in Fig. 2.7) or in separate panels (as in Fig. 2.8),
or choosing the error bars to represent standard deviations, standard errors, or confidence
intervals. The best advice is to become thoroughly familiar with the software being used,
STANDARDIZED (z) SCORES 27
and then to think carefully about which options will enable you to best communicate the
points you believe are important.
2.5 STANDARDIZED (z) SCORES
An individual's score, by itself, is not very informative. If a student earns a score of 70
on a math test, is that good or bad? On one hand, if told that a score of 70 was at the
90th percentile—that it had been exceeded by only 10% of all scores—we would probably
feel that the performance was quite good. On the other hand, we would be considerably
less impressed if the same raw score fell at the 40th percentile. Although information
about percentile values tells us where a score ranks relative to other scores, it provides no
information about distances among scores. For example, a score at the 90th percentile could
be many points higher than the average or only a little higher; it depends on how variable
the scores are. Standardized, or z, scores tell us more about where the score is located
within the distribution—specifically, how many standard deviation units the score is above
or below the mean. Given a distribution of scores with mean, Y, and standard deviation, s,
the z score corresponding to a score Y is calculated as
For example, if the mean is 75 and the standard deviation is 15, for a score of 90, we would
have ZQO = (90 — 75)/15 — 1; thus, this score is one standard deviation above the mean.
Statistical packages generally include an option such as SPSS's Descriptive Statistics (in
the "Analyze" menu) for calculating z scores.
Standardizing a group of scores changes the scale to one of standard deviation units, thus
permitting comparisons with scores that were originally on a different scale. Nevertheless,
there are aspects of the original distribution that remain unchanged. The following are two
things that remain constant:
1. An individual's z score has the same percentile rank as did that individual's original
score. This is because subtracting a constant, 7, from every score does not change
the rank order of the scores; nor is the order changed by dividing all scores by a
constant, s.
2. The shape of the distribution of z scores is the same as that of the original data.
Subtraction of Y shifts the original distribution and division by s squeezes the
points closer together, but shape information is preserved. If the original dis-
tribution was symmetric, the distribution of z scores will be also. However, if
the original distribution was skewed, this will also be true of the distribution of
z scores.
As we see in Chapter 5, z scores are used in drawing inferences when scores can
reasonably be assumed to be normally distributed. However, the preceding point should
make clear that z scores are not necessarily (or even usually) normally distributed. Their
distribution depends on the distribution of the scores prior to the z transformation.
Two other characteristics of z scores should be noted:
3. The mean (and therefore also the sum) of a set of z scores is zero. We stated earlier
that when a constant is subtracted from every score, the mean is also changed
by that constant. In the case of z scores, Y is subtracted from each of the Y values.
Therefore the mean of the z scores is 7 — 7, or 0.
4. The variance of a group of z scores is 1.0, and, therefore, so is the standard deviation.
Because the average z score is zero, we need only square each member of the group
of z scores, sum the squared values, and divide by N — 1 to obtain the variance.
Doing so yields
Standardized scores can be useful in our understanding of a data set because they
provide a way of comparing performances on different scales. For example, the mean and
standard deviation of the subtraction accuracy scores in the Royer data set were 88.840
and 11.457, respectively; the corresponding values for multiplication accuracy scores were
87.437 and 13.996. Even though a subtraction score of 70 is higher than a multiplication
score of 65, it is actually worse, relative to the other scores in the distribution. You may
verify that, for subtraction, Z70 is — 1.64 (that is, the score of 70 is 1.64 standard deviations
below the mean of the subtraction scores), whereas for multiplication, z65 is —1.60.
Standardized scores play other roles as well. In Chapter 5, we discuss how z scores pro-
vide percentile information when the population distribution has an approximate bell shape.
More immediately, in Chapter 3, we consider the correlation coefficient as an average prod-
uct of z scores. The underlying reason that z scores play a role in deriving a statistic reflecting
the relation between two variables is that such a statistic should not be influenced by differ-
ences in the scales of the two variables. This criterion can be met by converting the original
values to z scores. We show in Appendix 2.1 that the magnitudes of z scores are unchanged
if we change scales by adding a constant to each score and/or by multiplying each score by
a constant. For example, if we measure the weights of each student in a class, the z score of
an individual student will be the same whether we measure weights in pounds or kilograms.
2.6 MEASURES OF THE SHAPE OF A DISTRIBUTION
Early in this chapter, we displayed addition accuracy scores for second graders in a study
by Royer, together with an artificial data set that had the same mean and median as the
Royer data (Table 2.1 and Fig. 2.1). Table 2.2 presented those statistics, as well as several
others, for the Royer data and for the artificial set, Y. Note that not only are the means and
medians of the Y and Royer data identical, but the standard deviations are very similar.
Nevertheless, a look back at Fig. 2.1 suggests that there are differences in the shapes of
the two distributions. These differences are reflected in the skewness and kurtosis values
in Table 2.2. We will consider those values shortly, but first we consider why measures of
shape should interest us.
MEASURES OF THE SHAPE OF A DISTRIBUTION 29
As Fig. 2.1 and the statistics of Table 2.2 indicate, measures of location and spread
may fail to fully characterize a distribution, or differences between distributions. It is useful
to have summary numbers that reflect aspects of the shape of the distribution. Although,
as with measures of location and spread, there are several measures of shape and none are
perfect, they do extend our ability to describe the distribution of data.
Improved description is not the only reason for being interested in numerical indices
of aspects of shape. Many of the most widely used inferential procedures rest on the as-
sumption that the sampled population of scores can be described by a particular equation
that describes a family of bell-shaped, or normal, curves. If the population distribution from
which a researcher samples can be adequately described by this equation, there are many
useful inferential procedures that require only information about sample means and standard
deviations. This assumption of a normally distributed population accounts for the emphasis
on the mean and standard deviation, and on the procedures that use them, in statistical texts
and research reports. In recent years, however, it has become clear that real data are rarely
normally distributed; as we noted, the Royer data and the Beck depression data are not
unusual in their departures from normality. In cases in which there are marked departures
from normality, the reliance on the mean and standard deviation may be inappropriate.
For example, although the mean and median of a symmetric population distribution are
identical, the sample mean is usually considered the better estimate of the location of the
population distribution. However, if the population distribution has long straggling tails,
the sample median, or a trimmed mean, is more likely to have a value close to that of the
center of the population distribution (Rosenberger & Gasko, 1983), and therefore it would
be a better estimate of the location of the population.
The preceding discussion points to two reasons for calculating measures of shape: we
may wish to describe the data we have collected more precisely, or to assess the validity of
the assumption of normality underlying inferential procedures under consideration. There
is a third reason for our interest in measures of shape. An important stage in understanding
the processes that underlie behavior is the construction of mathematical or computer mod-
els, models precise enough to predict the behavior in question. Comparing predicted and
observed measures of the shape of the data distribution provides additional tests of such
Two aspects of shape have received the most attention from statisticians: the degree of
skewness, or departure from symmetry; and tail weight, or the proportion of data in the
extreme tails of the distribution. Indices of these two attributes of shape can be obtained
from various computer packages; those in Table 2.2 were generated by SPSS 10, but most
packages will provide the same results. We next consider the skewness and kurtosis values
in Table 2.2.
2.6.1 Skewness
Skewness statistics are designed to reflect departures from symmetry. The standard defini-
tion of skewness, generally denoted by ^/b\, is the average cubed deviation of scores from
the mean divided by the cubed standard deviation: For a sample, the formula is
Table 2.2 reports Gl, a modified version of this formula that provides a better estimate
of the population skewness parameter. There are several points to consider. First, both the
Y and Royer skewness values are negative, indicating that the distributions are skewed to the
left. This reflects the fact that the left tails in Fig. 2.1 are considerably longer than the right
tails of their respective distributions. If the distributions had been perfectly symmetric, the
skewness values would have been zero. Second, the artificial Y data have a more negative
value than the actual Royer data. Recall that the Y data contained two scores that were
much lower than any in the Royer data; these increased the absolute value of the skewness
statistic. Third, note how much larger the absolute skewness values are than their standard
errors, the SE skewness values. The standard error of skewness, like the SE of the mean
encountered earlier, is based on the idea of drawing many random samples of size N from
the same population, and then calculating Gl for each sample. The SE is an estimate of
the standard deviation of those Gl values. A ratio of skewness (ignoring the sign) to its SE
greater than 2 suggests that asymmetry is present in the population that we sampled from,
and did not occur just by chance in the sample.
2.6.2 Midsummary Scores Based on Different Tail Cutoffs
A problem with Equation 2.10 (or the closely related G1) is that, like the mean and standard
deviation, its value may be distorted by a single outlying point. Several more intuitively
appealing alternatives have been suggested (Hill & Dixon, 1982; Elashoff & Elashoff, 1978;
Hogg, 1974). These typically involve ratios of distances among points in the left and right
tails.
Our own preference involves somewhat more (but simple) calculations and does not
provide a single number, but it does yield a comprehensive picture of the distribution that
can be easily understood (see Hoaglin, 1983, for a more extensive presentation). The basic
idea is to find midsummary scores; these are scores midway between values that cut off
different fractions of the data in the upper and lower tails of the distribution. For example,
one midsummary score might be midway between values cutting off the upper and lower
eighths of the distribution; another might be midway between values cutting off the upper
and lower fourths. If all of these midsummary scores equal the median, it indicates that the
distribution is symmetrical. If they are not all the same, the way they vary provides us with
information about how the distribution departs from symmetry.
We begin by recalling the definition of the median, presented earlier. For the Royer and
Y data, the depth of the median is
Thus, the median is the mean of the 14th and 15th scores when the scores are rank ordered.
Also, recall that the depth of the hinges was
where the brackets imply dropping any fraction. Therefore, when N = 28,
The hinge is sometimes referred to as the "fourth," because roughly one fourth of the data
will always lie below the lower fourth and one fourth will lie above the upper fourth. For
example, if dfourth is 7.5, the lower fourth value for the Royer data is the mean of the 7th and
TABLE 2.4 LETTER VALUES AND MIDSUMMARIES FOR TABLE 2. 1 ROYER DATA
Letter Letter Depth Lower Value Midsummary Upper Value
Median 14.5 89
F (4th) 7.5 79 86.75 94.5
E (8th) 4 69 84.5 100
D(16th) 2.5 50 75 100
8th scores from the bottom, or 79, and the upper fourth value is the mean of the 21st and
22nd scores (i.e., the 7th and 8th scores from the top), or 94.5. The midsummary score is
the mean of these upper and lower values; here it is 86.75.
We can define other, more extreme fractions of the distribution in the same way. We
can talk about upper and lower eighths or sixteenths, where, for example,
The general formula for the depth of the next fraction is
Table 2.4 presents the depths for fourths (F), eighths (£), and sixteenths (D), the upper
and lower values, and the midsummaries for the Royer data. Note the use of letter values
to stand for the various positions in the distribution.
If a distribution is perfectly symmetric, all of its midsummaries should equal the median
of the distribution. The midsummaries for the Royer data decline in an orderly fashion as
we move further from the median, indicating that the distribution is not symmetric, but is
skewed to the left.
2.6.3 Kurtosis
Kurtosis values reflect departures from the normal distribution, and they are generally
sensitive to the height of the peak and to the tail weight of a distribution. The standard
kurtosis statistic, b2, is defined as
The values reported in Table 2.2 are designated G2 and are modifications of b2 derived to
improve the estimate of the population kurtosis value. Turning back once again to Fig. 2.1,
note that the Y data have a very pronounced peak in the interval 86-90, as well as several
scores in the extreme left tail. These two characteristics contribute to the G2 value for Y
that is much higher than that for the Royer data, and much larger than its own SE. G2 has a
value of zero for the normal distribution (b2 is 3 in that case, and the reported value is often
b2 — 3 to provide a simpler comparison with the normal distribution). Distributions with
high peaks and heavy tails (relative to the normal distribution) will have positive G2 values,
whereas the G2 values will be negative for flatter, shorter-tailed distributions. Heavy-tailed
distributions are of particular interest because inferences based on the assumption of
normality have often been shown to be affected by this departure from normality. With
such data, increasing the sample size or removing extreme scores (trimming) may improve
the quality of our inferences. We elaborate on this point in future chapters. For now, we
note that high kurtosis values may signal the need for remedial action. However, kurtosis
is sensitive to more than just the tails of the distribution, and therefore interpretation is
often difficult. Good discussions of kurtosis, together with figures illustrating various
shapes and the accompanying effect on bi, are provided in several sources (e.g., Balanda &
MacGillivray, 1988; DeCarlo, 1997). These and other articles and chapters (e.g., Hogg,
1974; Rosenberger & Gasko, 1983) also suggest alternative measures of tail weight that
are often more resistant to outlying points and that are more readily interpreted.
2.6.4 A Graphic Check on Normality
Because so many commonly used statistical procedures are based on the assumption that the
data were sampled from a normally distributed population, it is helpful to have several ways
of looking at possible violations of this assumption. Skewness and heavy tails in stem-and-
leaf and box plots indicate nonnormality. However, a more direct indication is available in
various computing packages. Basically, those programs rank order the scores and then plot
their expected z scores (assuming normality) against the actual scores. Figure 2.9 presents
two such plots. The left panel presents the plot for multiplication response times for students
in the fifth through eighth grades; the right panel presents the plot for response speeds that
are obtained by taking the reciprocals of the response times. If the data are sampled from a
normally distributed population, the data points fall reasonably close to a straight line. This
is clearly not the case in the left panel. The points in the right panel, except for the lowest
speeds, do fall quite close to the line, indicating that if the population of response speeds
is not normal, it is more nearly so than the population of response times. In Chapter 8, we
present other evidence that data analyses might better be based on the speed than on the
time measure.
Investigators who have precise enough theoretical models to predict distribution func-
tions other than the normal, or who seek to fit the distribution as part of an effort to better
Fig. 2.9 Normal probability (Q-Q) plots of multiplication response
times (MULTRT) and speeds (SPEEDMRT).
KEY CONCEPTS 33
understand their data, will find the Q-Q plot options helpful. There are many options other
than the normal distribution. For each, the expected value, assuming that distribution, is
plotted against the observed values.
The first step in any data analysis should be an examination of the data. The point seems
obvious, but our experience suggests that many, perhaps most, researchers note only means
and (sometimes) standard deviations, and, when a measure of relation between two variables
is of interest, correlation coefficients. Our view of this first step is that it should provide more
information. Researchers should plot their data, perhaps using a variety of graphic devices.
Such plots should not only involve means but also provide a sense of the distribution of data.
Alternative indices of location, spread, and shape should be obtained. This wealth of infor-
mation is available at the click of a mouse, and the payoff can be large. Multiple modes in
a histogram may suggest subpopulations differing in the effect of the independent variable.
Outliers in a box plot may suggest individuals whose performance should be examined
further, and perhaps excluded from further analyses. Differences in shape statistics for two
different experimental conditions may suggest different processes underlying behavior in
these conditions. Marked departures from the theoretical normal distribution may indicate
either that more data should be collected or that the researcher should consider alternatives
to the planned analysis that are less dependent on the assumption of normality. Knowledge
of the shape of the distribution can inform the development of theoretical models. All these,
and more, are possible consequences of exploring data sets. The present chapter introduced
some basic ways in which such exploration can be carried out. More extensive discussions
and other types of graphs may be found in many sources. In particular, we recommend the
three volumes edited by Hoaglin, Mosteller, and Tukey (1983, 1985, 1991). They provide
a clear presentation of many topics beyond the scope of this book, as well as further
discussion and examples of topics we have introduced. Other suggestions of possible ways
of plotting data, and references to useful sources, may be found in the report of the American
Psychological Association's Task Force on Statistical Inference (Wilkinson, 1999).
distribution central tendency (location)
arithmetic mean median
histogram stem-and-leaf display
minimum maximum
5-point summary median
du hinges
range interquartile range (IQR)
percentile H spread
resistant statistics outliers
inner fences outer fences
extreme outliers box plot
weighted mean standard deviation
sampling distribution standard error of the
trimmed means mean (SEM)
line graphs bar graphs
skewness standardized scores (z scores)
midsummary scores tail weight
kurtosis letter values
2.1 We have scores for 16 individuals on a measure of problem-solving ability: Y =21,
40, 34, 34, 16, 37, 21, 38, 32, 11, 34, 38, 26, 27, 33, 47. Without using statistical
software, find (a) the mean; (b) the median; (c) (]T\ y i ) 2 ; (d) ^. Y2; (e) the standard
deviation; and (f) the upper and lower hinges for these data. Then check your results,
using any software of your choice.
2.2 (a) Transform the scores in Exercise 2.1 to a new scale so that they have a mean of
100 and a standard deviation of 15.
(b) What will the new values of the median and hinges be?
2.3 Following are several sets of scores in ranked order. For each data set, is there any
indication that it does not come from a normal distribution? Explain, presenting
descriptive statistics and graphs to support your conclusion.
(a) X = 10 16 50 50 50 55 55 55 57 61 61 62 63 72 73 75 83 85 107 114
(b) X = 15 25 26 37 37 39 45 45 48 49 49 52 53 61 61 63 68 70 72 76
(c) X = 9 9 10 12 14 14 15 16 16 16 17 18 24 28 31 32 32 35 47 59
2.4 For Exercise 2.3 (c), find the median, fourths, and eighths (E and F letter values),
and midsummary values. Are these consistent with the conclusion you drew about
normality? Explain.
2.5 Given the five scores 37, 53, 77, 30, and 28,
(a) what sixth score must be added so that all six scores together have a mean of 47?
(b) What sixth score should be added so that the set of six scores has the smallest
possible variance?
In order to do the following two problems, you may wish to review Appendix A at
the back of the book.
2.6 Given: Y1 = 7 , Y 2 = 1 1 , Y 3 = 14, Y4 = 21, Y5 = 9, X1 = 6, X2 = 5, X3 = 7,
X4 = 1, X5 = 11, a = 3, and b = 9, find (a) £*=1 (Xi + Yi); (b) E,Li Xi2;
(c) (£?=1 X,)2; (d) £?=i X,Y,- and (e) £?=1 (X,- + aY2 + ab\
2.7 We have the following three groups of scores:
Cond 1 Cond 2 Cond 3
EXERCISES 35
Given that Yij is the ith score in the j'th column, find (a) Y,1; (b) Y2.', (c) Y ; (d)
ELiE^and^E^F 2 ,.
2.8 This problem uses the Ex2_8 data set on the CD accompanying this book. Using
any statistical package that you like, explore the distributions and their relationships
by displaying (a) histograms, (b) box plots, (c) stem-and-leaf plots, (d) descrip-
tive statistics (include the median, as well as measures of skewness and kurtosis),
and (e) probability plots. Summarize what you have learned about the distributions
of X, Y, and Z. Be sure to refer to such concepts as the location, spread, and
shapes of the distributions, in particular whether or not they seem to be normally
distributed.
2.9 Suppose we standardized X, Y, and Z (i.e., converted to standard or z
scores) in the Ex2_8 data set. How would this affect the characteristics of the
distributions?
2.10 Find the Royer multiplication accuracy and response time data (Royer Mult Data
in the Royer folder) on your CD. Using whichever descriptive statistics and graphs
you find useful, describe any differences in location, variability, and shape of male
and female distributions in the third and fourth grades. In particular, comment on
whether gender differences change from third to fourth grade. Also comment on
whether the patterns you note are the same or different for accuracy and for reaction
time (RT). Finally, consider whether outlying scores may have an influence on any
of these comparisons.
2.11 The Seasons study was carried out to see if there was variation in the physical or
psychological characteristics of the sampled population over seasons.
(a) Using the data in the Seasons file (Seasons folder in the CD), plot Beck_A (anx-
iety) means as a function of seasons for each age group (Agegrp). Use either a
bar or line graph. Which do you think is preferable? Why?
(b) Discuss the graph in part (a), noting any effects of age, seasons, or both.
(c) Box or stem-and-leaf plots reveal a skewed distribution with many out-
liers. Do you think the pattern of outliers contributed to your conclusions
in part (b)?
2.12 On the CD, in the Seasons folder, find the Sayhlth data set. This includes self-ratings
from 1 (excellent health) to 4 (fair); only three participants rated themselves in
poor health and they were excluded from this file. The file also includes Beck_D
(depression) scores for each season. It is reasonable to suspect that individuals who
feel less healthy will be more depressed.
(a) Selecting any statistics and graphs you believe may help, evaluate this hypothesis.
(b) Discuss any trends over seasons. Are there differences in trends as a function of
self-ratings of health?
2.13 Scores for 30 students on two tests may be found in the Ex2_l 3 file.
One student received a score of 41 on Test 1 and a score of 51 on Test 2. She was
delighted with the improvement.
(a) Should she have been? Explain.
(b) What score on Test 2 would be an improvement for this student, given your
answer to (a)?
(c) Graph the data for each test and describe the distributions.
APPENDIX 2.1
Additional Properties of z Scores
Suppose we have a set of scores Y\, ¥2, ^ 3 , . . . , Y^. The mean of these scores is Y =
]T, Yi/N and the variance is Sy = ^ (7, - Y}2/(N — 1). The z score corresponding to the
score Yf is zy, = (Yi — Y)/sy, where sy =^/sl is the standard deviation of the Y scores.
If we multiply each Y score by the constant b and add the constant a to each of the
resulting products, then the transformed scores can be expressed as Y- = a + bYj. From
the properties of the mean stated in Subsection 2.4.1, we know that adding a constant to
each score increases the mean by that constant, and multiplying each score by a constant
results in multiplication of the mean by that constant. Therefore, Y' = a + bY = a +bY.
From the properties of the standard deviation stated in Subsection 2.4.2, we know that the
addition of a constant to each score does not change the standard deviation. Subsection
2.4.2 also demonstrated that if Y' = kY, then sy> = ksY when k > 0 and sy — —ksy when
k < 0. Therefore, sa+hy = +b sy when b is positive and sa+by = —b SY when b is negative.
Putting this all together, we find that the z score of a transformed score a + bYi is given
Thus the z score of the transformed score is identical to that of the untransformed score if
b is positive, and it is identical in magnitude, but opposite in sign, if b is negative.
Looking at Data: Relations
Between Quantitative Variables
In Chapter 2 we considered how to graph and summarize distributions of single variables.
However, we rarely study individual variables in isolation; rather, we usually are interested
in how variables are related to one another. For example, we may wish to know if depression
varies with the season of the year, how cholesterol level changes with age, or whether math
skills are related to verbal skills in children. Because variability is always present, it is
important to emphasize that, when variables are related, they are usually not perfectly
related. Tall fathers tend to have tall sons, but because of a host of factors, the tallest fathers
do not always have the tallest sons. There is a relation between educational level and income,
but some uneducated people are wealthy and many educated people are not. Variables may
be related in ways that vary in type and degree. Therefore, the major goals of this chapter are
to discuss how to graph the data in ways that help us see how, and how strongly, the variables
are related, and to present statistics that summarize important aspects of the relation.
Also, if two variables are related, it should be possible to use information about one
variable to predict the other. For example, knowing a father's height will be useful in
predicting the height of his son. If we make lots of predictions, even though any single
prediction may not be very accurate, on the average we can predict a son's height more
accurately if we use information about the father's height than if we do not. Clearly, the
stronger the relation, the better our ability to predict. But no matter how strongly the variables
are related, we wish to make the best predictions that are possible with the information that
we have available. Equations that use information about one variable to make predictions
about a second variable are referred to as bivariate regression equations.
3.2 SOME EXAMPLES
Let's first consider two variables from the Royer data set that is on the accompanying CD—
subtraction and multiplication accuracy scores (percentage of correct answers) for third
38 3 / RELATIONS BETWEEN QUANTITATIVE VARIABLES
TABLE 3. 1 DESCRIPTIVE STATISTICS FOR THIRD-GRADE
SUBTRACTION AND MULTIPLICATION ACCURACY
Subtraction Multiplication
No. of cases 32 28
Median 89.182 79.447
Mean 87.014 78.469
SE 2.081 3.426
Std. dev. 11.770 18.127
Skewness (Gl) -1.430 -0.835
SE skewness 0.414 0.441
Kurtosis (G2) 3.078 0.813
SE kurtosis 0.809 0.858
Fig. 3.1 Histograms for subtraction and multiplication accuracy for third graders.
graders. The basic statistics for each variable are given in Table 3.1, and their histograms
are presented in Fig. 3.1. For subtraction accuracy, the median and mean are both high,
89.182 and 87.014, respectively. The distribution of scores is negatively skewed, Gl =
— 1.430; most scores are high, but there is a "tail" of lower scores. For multiplication,
performance is less good, with both the median and mean approximately 10 points lower
than in the subtraction data. The distribution is again negatively skewed, but less so than for
subtraction (Gl = -0.835). The distribution is flatter; G2 = 0.813 as compared with 3.078
for subtraction. There is less of a pileup of scores toward the high end for multiplication,
and, because the scores are more spread out, measures of variability have larger values. The
standard deviation is 18.127 for multiplication, compared with 11.770 for subtraction. The
multiplication scores not only tend to be smaller than subtraction scores, but also exhibit
greater variability.
3.2.1 Scdtterplofrs
But how are subtraction and multiplication accuracy related in this sample of third graders?
We might expect that children with larger subtraction scores will also have larger multipli-
cation scores, because we know that some children are better at arithmetic than others. The
most common way of displaying the relation between two variables is to use a scatterplot,
SOME EXAMPLES 39
Fig. 3.2 Scatterplot for subtraction and
multiplication accuracy for the 28 third-grade
children having both scores; box plots for each
variable appear on the borders.
Fig. 3.3 Scatterplot for response time and
multiplication accuracy for third grade.
which is simply a plot in which each point has coordinates corresponding to paired values
on each of the variables. The Scatterplot for the 28 third graders for whom we have both
multiplication accuracy and subtraction accuracy scores is presented in Fig. 3.2. Note that
some statistical packages, in this case SYSTAT, allow us to present the univariate distribu-
tions such as histograms or box plots along the borders of the scatterplot, so we can see
information about both the univariate and joint distributions in the same display.
What we see in the scatterplot is a tendency for larger multiplication scores to go
together with larger subtraction scores; when this happens we say there is a positive relation
or association between the two variables. If larger scores on one variable tend to go together
with smaller scores on the other, we have a negative relation. The scatterplot in Fig. 3.3,
Fig. 3.4 Scatterplot for TC score and age.
in which mean time to answer multiplication problems is plotted against multiplication
accuracy, shows a strong negative relation, reflecting the tendency for children who are
more accurate to take less time to answer. A notable exception is the child represented by
the data point in the lower left of the display, who took the least time to answer but also had
the lowest accuracy. We can also see from the box plots on the axes that this data point is
labeled as an outlier with respect to the multiplication accuracy distribution.
We conclude this section with two additional examples of scatterplots to which we
refer later in the chapter. To obtain the first, presented in Fig. 3.4, we found the mean of
the four seasonal total cholesterol (TC) scores for each of the 431 individuals who had
scores in all four seasons in the Seasons study conducted by researchers at the University
of Massachusetts Medical School; we then plotted TC against age. Although there is a
great deal of variability in the cholesterol scores, there seems to be a positive relation
between cholesterol level and age; that is, there is a tendency for older people to have
higher cholesterol scores.
The second example uses data obtained from an elementary statistics class. Table 3.2
contains two scores for each of 18 students—the score on a math-skills pretest taken during
the first week of class and the score on the final exam. The scatterplot for the 18 data points
is presented in Fig. 3.5. Not surprisingly, the pretest and final scores covary. We would like
to be able to develop ways to summarize how the two variables are related and to use the
pretest scores to predict final exam performance. If we could find the equation that did the
best job in predicting the 18 final exam scores from the pretest scores, it could be useful in
predicting final exam performance for students who take the pretest in other classes.
3.2.2 Extracting the Systematic Relation Between
Two Variables
Scatterplots can be very useful in helping us understand how, and to what extent, the variables
are related. For example, it is quite clear in Fig. 3.3 that there is a strong tendency for the
time taken to perform multiplication to decrease as accuracy improves. However, real data
are often extremely messy. Any systematic relation that exists between the variables may be
TABLE 3.2 STATISTICS CLASS EXAMPLE DATA
Pretest Score Final Exam Score
(X) (Y)
obscured by variability caused by factors such as individual differences and measurement
error. In such cases we try to see through the "noise" in order to extract the "signal," that is,
the underlying systematic relation, if one exists. This can be difficult to do by eye, especially
when there are many data points and a great deal of variability, as in the plot of cholesterol
level against age in Fig. 3.4.
Fig. 3.5 Scatterplot for pretest and final exam
scores in a statistics class.
Fig. 3.6 Scatterplot for TC and age with
LOWESS curve.
One way of trying to get at the underlying relation is to use some type of averaging
to reduce the complexity of the display. In Chapter 2 we used the average to represent a
univariate distribution of scores. Here we can find the average cholesterol score for each age
and plot it against age. Some statistical packages assist us in understanding the relation by
fitting curves called smoothers to the data points in the scatterplot. We can use these curves
to smooth out some of the random variability by plotting a "moving average" of cholesterol
scores against age, such that the height of the curve for a particular age represents the
mean or the median of the cholesterol scores corresponding to a range of neighboring ages.
There are different types of smoothing functions available in both SAS and SYSTAT, some
of which have the desirable property of being resistant in the sense that they give less
weight to outlying data points than to those near the average. We can choose not to put any
preconditions on the curve, or we can plot the best-fitting curve of a particular type, such
as the best-fitting straight line or logarithmic function.
An example of one kind of smoothing is provided by Fig. 3.6, which displays the
scatterplot for cholesterol and age with local weighted scatterplot smoothing (LOWESS;
Cleveland, 1979) by using SYSTAT 10. For each value of X, LOWESS smoothing plots
the Y score predicted by a procedure that gives more weight to data points near the value of
X than to data points that are further away (for details, see Cook & Weisberg, 1999). The
resulting curve indicates a positive relation between cholesterol and age that approximates
a straight line.
In practice, we usually first look to see whether there is a systematic tendency for the
variables to have a straight line relation, because this is the simplest and most common
way that two variables can be related. We then look further to see whether there are also
systematic departures from linearity. The most common numerical measures that are used
to summarize the relation between two quantitative variables are those that (a) indicate how
well a straight line captures the scatterplot, and (b) describe the straight line that gives the
best fit. In Section 3.4 we introduce the first type of measure, the Pearson correlation
coefficient, which is a measure of the extent to which two variables are linearly related.
In Section 3.5 we introduce the idea of linear regression, which provides a way to find
LINEAR RELATIONS 43
the straight line equation that fits the data points best, in the sense that it allows the best
prediction of one of the variables from the other. We first take a closer look at what it means
for two variables to be linearly related.
3.3 LINEAR RELATIONS
Each of the scatterplots in Fig. 3.7 contains a number of (X,Y) data points. If all of the data
points fall exactly on a straight line, we say there is a perfect linear relation between X
Fig. 3.7 Examples of scatterplots.
and Y. This linear relation is said to be positive if the slope of the line is positive; that is, if
Y increases as X increases as in panel (a). The linear relation is negative if Y decreases as
X increases as in panel (b). In panel (c), there is a systematic increase in 7 as X increases.
However, not all the data points fall on the straight line that seems to best capture this
systematic increase, although they cluster closely around it. In this case, we say that there
is a strong positive linear relation between X and Y but not a perfect one. In panel (d) there
is less clustering around the line, indicating a weaker linear relation. In panel (e) there is a
linear component to the relation between X and Y; that is, the best-fitting straight line seems
to capture part of how Y and X are related. However, not only do the points fail to cluster
closely around the line, but also there seems to be a systematic nonlinear component to the
relation. In panels (f) and (g), there is no overall linear relation between X and Y; no straight
line passing through the center of either "cloud" of data points is better at characterizing
the overall relation between X and Y than a line parallel to the X axis. In (g), however, X
and Y are positively related for small values of X but negatively related for large values,
whereas in (f) there does not seem to be any indication of a linear relation for any part of
the X distribution.
A straight line can be represented by an equation of the form
where bo and b\ are constants, because all points (X, Y) that satisfy this linear equation
fall on a straight line. The constant b\ is called the slope of the line and indicates the rate
of change of Y with X. We can see in Equation 3.1 that, for every one-unit change in X, Y
changes by b\ units. The constant b0 is the Y intercept, the value of Y when X is equal to
zero. We show later that the regression line for Fig. 3.4 that best predicts cholesterol level
from age is
cholesterol level = 171.702 + 0.945 x age
For this line, the slope is 0.945; that is, the predicted cholesterol level increases by 0.945
units for each additional year of age.
3.4 THE PEARSON PRODUCT-MOMENT CORRELATION
3.4.1 Obtaining the Correlation Coefficient
The Pearson correlation is a measure of the extent to which two quantitative variables are
linearly related. We indicated in the previous section that the more tightly the data points
are clustered about the best-fitting straight line, the stronger the degree of linear relation.
The notion of clustering around a straight line leads directly to a useful measure of linear
relation. However, in developing this idea further, we consider standardized or z scores
instead of raw scores. When raw scores are used, the appearance of the scatterplot and the
apparent degree of clustering around the best-fitting straight line depends on the units in
which X and Y are measured. This is not true for z scores.
THE PEARSON PRODUCT-MOMENT CORRELATION COEFFICIENT 45
In Section 2.5 we indicated that each member of a set of scores X1, X2 , X3 , .... XN
can be converted to a z score by using
where X is the mean of the set of scores and sx is the standard deviation. The z score that
corresponds to a raw score just tells us the number of standard deviations the raw score is
above or below the mean of the distribution. In Section 2.5 we showed that the mean of a
complete set of z scores is zero, and that the standard deviation and variance both have a
value of 1. An equation that will be useful in understanding the correlation coefficient is
This equation follows from the fact that the variance of a set of z scores is 1 and the
expression for this variance is
because zx, the mean of a set of z scores, is equal to 0.
If X and Y are positively related, larger scores in the Y distribution will tend to be
paired with larger scores in the X distribution and smaller scores in the Y distribution
will tend to be paired with the smaller Xs. This means that large positive values of zy
will tend to be paired with large positive values of zx, small values of zy will tend to be
paired with small values of zx, and large negative values of ZY will tend to be paired with
large negative values of zx. It can be shown (see Appendix 3.1) that, if there is a perfect
positive linear relation between X and Y, ZY is exactly equal to zx, and if there is a perfect
negative linear relation, zy is exactly equal to —z x . If there is no linear relation between
X and Y, there is no overall tendency for larger zy scores to be paired with either larger
or smaller zx scores, or for positive zy scores to be paired with either positive or negative
zx scores.
The Pearson product-moment correlation coefficient for two variables, X and Y, is
defined as
The letter r is used to denote the Pearson correlation coefficient in a sample, and p (the Greek
letter rho) denotes the correlation in a population. The correlation coefficient is basically
the average of the cross products of the corresponding z scores (it would be exactly the
average if we divided by N instead of N — 1 when we obtained the standard deviations of
X and Y). We can think of rxy as a measure of how similar, on the average, ZYi, is to zxi,.. If
there is a perfect positive relation between X and 7, then for each data point (X,Y), ZY = zx .
so that the correlation is
from Equation 3.3. If there is a perfect negative relation, ZY = -zx, so that rXY = — 1. If
there is no linear relation between Y and X, there will be no tendency for Zyi, and zXi to
have the same or different signs. Their cross products zxi, ZYi, should be equally likely to be
positive (when zXi and zyi have the same sign) or negative (when zXi and zyi have opposite
signs). Therefore we would expect the cross products to sum to 0, so that rXY = 0.
In summary, rXY provides a numerical measure of the degree to which X and Y are
linearly related. It takes on a value of +1 when there is a perfect positive linear relation,
a value of — 1 when there is a perfect negative linear relation, and a value of 0 when there
is no overall linear relation between X and 7. Intermediate values provide measures of
the "strength" of the linear relation. Going back to the examples we introduced earlier
in the chapter, for multiplication and subtraction accuracy for third graders (Fig. 3.2),
r = .594; for multiplication accuracy and the time taken to answer (Fig. 3.3), r = — .487; for
cholesterol level and age (Fig. 3.4), r = .286; and for final exam and pretest score (Fig. 3.5),
r = .725.
3.4.2 Interpreting the Correlation Coefficient
Although it is easy to calculate a correlation coefficient, we must be cautious in how we
interpret it. Rodgers and Nicewander (1988) discussed 13 different ways to look at the
correlation coefficient, and others (e.g., Falk & Well, 1996, 1998; Rovine & Von Eye, 1997)
have considered additional interpretations. Although we have a good deal more to say about
the correlation coefficient in Chapter 18, here we list some basic things to keep in mind
when interpreting a correlation coefficient.
First, how large must a correlation coefficient be in order to indicate that there is a
"meaningful" linear relation? Cohen (1977, 1988) discussed guidelines according to which
rs of .10, .30, and .50 correspond to small, medium, and large effects. Cohen arrived at
these values by noting the sizes of correlations encountered in the behavioral sciences and
by considering how strong a correlation would have to be before the relation could be
perceived by an observer. These values should be considered only as loose guidelines and
not as criteria for importance. As we discuss later, in some contexts, even small correlations
might be of great practical significance. We should also emphasize that unless the sample
is large, the correlation may be quite different in the sample than in the population from
which the sample was selected. Later we discuss what the sample allows us to say about
the population.
Second, we must always keep in mind the fact that the Pearson correlation coefficient
is a measure of strength of the linear relation between X and Y. The correlation coefficient
is not a measure of relation in general, because it provides no information about whether
or not there is a systematic nonlinear relation between the two variables. As can be seen in
panels (e) and (g) of Fig. 3.7, two variables can have a systematic curvilinear component
to their relation in addition to, or instead of, a linear one. Therefore, finding a correlation
of 0 does not necessarily mean that the variables are independent. The data points in all
four panels of Fig. 3.8 (see Table 3.3) have identical correlations and best-fitting straight
lines. However, whereas panel (a) displays a moderate linear relation with no curvilinear
component, panel (b) displays a strong curvilinear relation that has a linear component. It
cannot be emphasized strongly enough that, to understand how variables are related, one
must plot them and not simply rely on statistics such as the correlation coefficient or the
slope of the best-fitting straight line.
TABLE 3.3 FOUR HYPOTHETICAL DATA SETS
a-c a b c d d
Case No. X Y Y Y X Y
1 10.0 8.04 9.14 7.46 8.0 6.58
2 8.0 6.95 8.14 6.77 8.0 5.76
3 13.0 7.58 8.74 12.74 8.0 7.71
8 4.0 4.26 3.10 5.39 19.0 12.50
9 12.0 10.84 9.13 8.15 8.0 5.56
10 7.0 4.82 7.26 6.42 8.0 7.91
Note. From "Graphs in Statistical Analysis," by F. J. Anscambe, 1973, American Statis-
tician, 27, pp. 17-21. Copyright 1973 by The American Statistical Association.
Fig. 3.8 Scatterplots for the data sets in Table 3.3.
Third, expressions for the correlation coefficient are symmetric in X and Y. The cor-
relation of X with Y is the same as the correlation of Y with X. The correlation between
cholesterol level and age is the same as the correlation between age and cholesterol level.
Fourth, although a nonzero correlation coefficient indicates that there is a linear com-
ponent to the relation between two variables, it generally doesn't describe the best-fitting
straight line. Figure 3.9 displays several scatterplots that all have the same correlation even
though the slopes of the best-fitting straight lines vary considerably. We show later on that
the correlation between X and Y depends not only on the slope but also on the standard
deviations of X and Y.
Fifth, the Pearson correlation coefficient is not a resistant statistic. It can often be
changed considerably by the presence of just a few extreme data points. In Fig. 3.10 we
used SYSTAT to display the influence plot for multiplication and subtraction accuracy
for the sample of third graders. The influence plot is just a scatterplot in which each case
is plotted as an open or filled circle that can vary in size. The size of the circle indicates
how much the correlation would change if that point was omitted, and whether the circle is
filled or open indicates whether omitting the data point would make the correlation larger or
smaller. The very large open circle in the left of the plot indicates that the corresponding data
point has a large effect on the correlation. If we omit this one data point, the correlation drops
from .594 to .388. It is important to identify these influential data points because we would
have less confidence in the value of a correlation coefficient if it was strongly influenced by
a few extreme data points. There are measures of correlation that are more resistant than the
Pearson coefficient because they diminish the importance of extreme scores. An example
is the Spearman rho coefficient, for which the X and Y scores are first ranked, and then
the ranks are correlated. We have more to say about such measures in Chapter 18.
Sixth, because the correlation coefficient is defined in terms of z scores, the size of the
correlation coefficient does not change if we change the units in which we measure either
of the variables by a linear transformation (i.e., if we multiply each score by a constant or
add a constant to each score, as when we change units from ounces to kilograms, or from
degrees Fahrenheit to degrees Celsius - see Appendix 2.1, in which we showed that such
transformations do not change the sizes of the z scores). It follows from this that knowing
the correlation between two variables tells us nothing about the mean or variance of either
variable.
Seventh, because correlation is a measure of strength of relation, it is tempting to
consider the correlation coefficient as a measure of the extent to which changes in X cause
changes in Y. However, correlation does not imply causation—in fact, no statistic implies
causation. Just because two variables are correlated does not necessarily mean that they
are causally related. For example, the fact that in elementary school there is a positive
correlation between shoe size and verbal ability does not mean that foot growth causes
enhanced verbal ability or vice versa. Rather, the correlation follows from the fact that both
physical and mental growth occur as children get older.
3.4.3 Some Other Ways of Expressing the Pearson
Correlation Coefficient
The value of the Pearson correlation coefficient can always be obtained from Equation 3.4.
However, other expressions are often encountered. If we substitute the expressions for
Fig. 3.9 Scatterplots with the same r values but different values of sx and SY .
Fig. 3.10 Influence plot for third-grade multiplication and subtraction
accuracy.
z scores (Equation 3.2) into Equation 3.4, we get
is the sample covariance of X and Y.
Although the covariance, SXY (i-e., the amount of variance shared by X and 7), plays
an important role in statistics, it is not usually employed as a measure of relation because
it changes value when we change the units of measurement. For example, if we measured
the heights and weights of a number of people and then found the covariance of height and
weight, the covariance would be 12 times larger if we measured height in inches than if
we measured it in feet. The correlation coefficient would be the same in either case. The
correlation can be thought of as the standardized covariance; that is, the covariance of the
Another expression that is commonly encountered in elementary textbooks is the so-
called computational formula for the Pearson correlation coefficient:
The computational formula gives the same result as Equation 3.4, within rounding error,
but it is less "transparent" than Equation 3.4; that is, the interpretation of the expression
is less readily apparent. However, it has the advantage of allowing simpler calculations
LINEAR REGRESSION 51
and less rounding error if it is used to calculate the correlation by hand or with a simple
calculator. Although computational ease may have been an important factor in the past when
researchers used simple calculators, it is not today, when almost all serious computation is
performed by computer, and even inexpensive hand calculators can calculate the correlation
at the press of a button. Therefore, we generally do not deal with computational formulas
in this book, but instead concentrate on conceptual (or definitional) formulas that are
expressed in ways that make their meanings most apparent.
3.5 LINEAR REGRESSION
3.5.1 Predicting Y From X With the Least-Squares
How do we find the best-fitting straight line for a set of data points (X, Y) represented in
a scatterplot? What we usually mean by the "best-fitting straight line" is the line that best
predicts the value of Y corresponding to each value of X, the linear regression equation.
The linear regression equation that predicts Y from X has the form
where Yi, is the predicted value of Y when X — Xi, and bo and b\ are constants chosen in a
way that results in the smallest amount of prediction error. Before we can find bo and b\ we
must decide what to use as the measure of error. If, on a given trial, we predict that Y has
a value of Yi and it actually has a value of Yi, then the error in prediction is ei = Yi — Yi.
The mean of these errors for a set of N predictions is not a good index of error because
positive and negative errors cancel, so the mean error could be small even if there were large
positive and negative errors. An index of error that is often used is the mean of the squared
prediction error. This measure is equal to zero only if prediction is perfect for the entire
set of data points and it is also easy to work with mathematically.1 Regression equations
that minimize the mean of the squared errors are said to be optimal or best according to the
least-squares criterion.
Thus, to find the best linear regression equation according to the least-squares criterion,
we must find values of b0 and b1 that minimize the mean of the squared errors, MSE, where
It can be shown (see Appendix 3.2) that these values are given by
Applying Equations 3.8 and 3.9 to the statistics class data in Table 3.2 allows us to find
the linear equation that best predicts final exam performance from pretest score. Substituting
into the expressions for b1 and b0, or obtaining the output from a computer package, when
we regress Y on X (i.e., when we predict Y from X), we find that the regression equation is
A difference of 1 point on the pretest translates into a predicted difference of about 3.6
points on the final exam. Our prediction for the final exam score of a student who scored
30 on the pretest would be -36.08 + (3.55)(30) = 70.42, or 70, rounding to the nearest
integer.
Note that in the general case for which sx = sy, the regression equation and the corre-
lation coefficient tell us different things about the linear relation. In Equation 3.6, X and Y
play different roles: X is the predictor variable and Y is the criterion variable, the vari-
able that is predicted. In contrast, the correlation coefficient is symmetric in X and Y; both
variables are treated in the same way. The regression equation describes the straight line
that is best for predicting Y from X, whereas the correlation coefficient serves as a measure
of the extent to which Y and X are linearly related. If we solve for r in Equation 3.8, we get
From this equation we can see that the same correlation may arise from different com-
binations of the slope and the standard deviations of X and Y. For example, both of the
combinations b1 = 1, sx = 3, sy = 5 and b1= .5, sx = 6, sy = 5 will correspond to rs
of .6. Because of this, we have to be extremely cautious if we wish to compare the relation
between X and Y in different groups. Two groups that have the same slope may have differ-
ent correlations, and two groups that have the same correlation may have different slopes.
If we are primarily concerned with the rate of change of Y with X, we should compare the
slopes, not the correlation coefficients.
We conclude this section by pointing out that there are several additional ways of
writing the regression equation that can be useful. Substituting the expression for bo in
Equation 3.9 into Equation 3.6 yields
Note that, in Equation 3.11, if Xi = X, then Y = Y. This tells us that the least-squares
regression line always must pass through (X, Y). In addition, if we subtract Y from both
sides of Equation 3.11 and divide both sides by sy , we get
the z score form of the regression equation. Note that the regression line that predicts zy
has slope r and passes through the origin.
3.5.2 Predicting X From Y
So far, we have discussed the regression equation for predicting Y from X that is optimal
in the sense that it minimizes £](Y — Y) 2 /N; see panel (a) of Fig. 3.11. Exactly the same
reasoning can be used to find the regression equation for predicting X from Y. In this case,
the index of error that is minimized is E (Xi — X i ) 2 /N, the mean of the squared prediction
errors when Y is used to predict X. These prediction errors are indicated in panel (b) of
Fig. 3.11.
Fig. 3.11 Graphical representation of (a) the regression of Y on X and (b) the regression
of X on Y.
The expressions that have been developed for predicting Y from X can be transformed
into expressions for predicting X from Y by simply interchanging X and Y. For example,
Of course, whether it makes any sense to predict X from Y depends on the nature of the
variables. It is unlikely that we would want to predict pretest scores from final exam scores
because pretest scores are available first.
Figure 3.12 indicates how the regression lines that best predict ZY from zx and zx from
zy differ from one another. Imagine that the elliptical "envelope" that has been drawn in
the figure to represent an imperfect linear relation contains a large number of data points.
Imagine further that the ellipse is divided into a number of narrow vertical strips. Notice
that even though the envelope is symmetrical about a straight line with a slope of 1 drawn
through the origin (i.e., ZY = zx), the mean value of ZY associated with any given value
of zx is closer to 0 than zx is. The line that best fits the points representing the mean
values of ZY in the vertical strips will approximate ZY = rzx, the regression equation2 for
predicting ZY from zx.
In contrast, if we divide the ellipse into a large number of narrow horizontal strips,
the line that best fits the mean values of zx in the strips will approximate the regression
equation for predicting zx from zy, zx = rzY. It should be apparent from Fig. 3.12 that
these two regression lines are not the same.
Fig. 3.12 Regression lines for predicting (a) ZY from zx and (b) zx
from ZY when there is an imperfect relation between X and Y.
3.6 THE COEFFICIENT OF DETERMINATION, r2
The square of the correlation coefficient, r2, called the coefficient of determination, is
another commonly encountered measure of strength of linear relation. The r2 measure is
usually defined as "the proportion of the variance in Y accounted for by X.,.3 What this
actually means is that r2 is the proportion by which prediction error is reduced if the
regression equation is used to predict the Y scores instead of using Y to predict each of the
Ys. The specific interpretation is as follows:
1. If we do not use any information about X in predicting the corresponding value of
Y, the best prediction for each Y can be shown to be Y, the mean of the Y scores.
INFLUENTIAL DATA POINTS AND RESISTANT MEASURES OF REGRESSION 55
In this case, the sum of the squared prediction errors for the set of NY scores is the
total variability in the Y scores, the sum of squares of Y, SSy = Ei (Yi —Y)2.
2. If we use the regression equation to predict the Ys, the sum of the squared prediction
errors is SSerror = E (Yi - Yi)2, where the Yis are obtained by using the regression
equation. Substituting the expression for Yi, from Equation 3.11 and simplifying,
we can show the sum of the squared prediction errors to be SSerror = (1 — r2 )SSy.
3. The amount by which prediction error is reduced when the regression equation is
used is, therefore, SSregression = SSY - SSerror = SSY - (1 - r2 ) SSY = r2 SS Y .
Therefore, the proportion by which prediction error is reduced (or the proportion
of the variability in Y accounted for by the regression on X) is
Therefore, r2 is simply a measure of how well the linear regression equation fits
the data. According to the Cohen (1977, 1988) guidelines introduced in Subsection
3.4.2, r2 values of .01, .09, and .25 correspond to small, medium, and large linear
relations, respectively.
For the statistics class data, the correlation between the pretest and the final exam score
is .725, so the coefficient of determination is (.725)2 = .53. This tells us that the variability
of the Y scores about the regression line is (1 —.53) = .47 of their variability about their
mean. Therefore, if we use the regression equation to predict Y instead of using Y, we will
reduce the squared prediction error by approximately one half. However, measures of the
actual variability about the regression line such as the variance of estimate (basically, the
mean of the squared prediction errors) or its square root, the standard error of estimate,
provide more useful information than either r or r2 about the accuracy of the predictions
of Y based on X. Such measures are available in most statistical software and are discussed
further in Chapter 19.
We conclude this section by noting that r2 has frequently been misinterpreted, and
that some of these misinterpretations have resulted in inappropriate claims being made. For
example, the statement has been made in a number of psychology textbooks that children
achieve about 50% of their adult intelligence by the age of 4 years. The origin of this
statement can be traced to a misinterpretation of the data from a longitudinal study that
found IQ scores at age 17 to have a correlation of about .7 with IQ at age 4. The resulting
r2 of about .5 (or 50%) provides an indication of how predictable adult IQ is from IQ at age
4, using a linear equation. However, it says nothing about the relative levels of intelligence
at age 4 and age 17, and therefore provides no evidence for the statement.
3.7 INFLUENTIAL DATA POINTS AND RESISTANT MEASURES
OF REGRESSION
Although the least-squares procedures described here are easy to work with, they produce
best-fitting straight lines that are sensitive to the effects of extreme scores. All of the major
statistics packages provide diagnostics that allow us to identify unduly influential data points
and assess how much the regression statistics would change if the extreme data points were
not present. We describe these diagnostics in Chapter 19. There are also "robust" regression
procedures that provide better fits to the majority of data points than ordinary least-squares
procedures because they are resistant to the effects of extreme data points. They are resistant
because they somehow reduce the influence of extreme scores by trimming them or giving
them less weight, by using ranks, or by using more resistant measures of prediction error
such as the median of the squared errors or the mean of the absolute errors. Some of
these procedures are discussed by Hoaglin et al. (1985), Neter, Kutner, Nachtscheim, and
Wasserman (1996), and Rousseeuw and Leroy (1987).
3.8 DESCRIBING NONLINEAR RELATIONS
In the past few sections, we focused on describing the linear component of the relation
between two variables, and on measures of its strength. Indeed, correlation coefficients
and regression slopes are by far the mostly commonly reported measures of relation for
quantitative variables. This is reasonable, given that an approximately linear relation is
the simplest and most common way that two variables can be related. However, there will
certainly be situations in which it is apparent from the scatterplot and the smoothers that the
relation has a nonlinear component. How are we to describe and measure the strength of
this component or to describe the overall function that best seems to fit the data? We address
this question when we discuss trend analysis in Chapter 10, and again after we extend our
knowledge of regression in Chapter 20.
When we explore how two quantitative variables are related in a sample of data, the first
step is to plot the data and look at both the scatterplot and the univariate distributions.
Inspecting the scatterplot by eye and using smoothers, we can try to extract, and describe,
any underlying systematic relation. We can use the Pearson correlation as a measure of the
strength of linear relation and the regression equation as a description of the straight line
that allows the best prediction of one variable from the other. We can then try to determine
if there are any systematic departures from linearity, and, if so, we can describe them. Later
we discuss how to assess the fit of various kinds of functions to the scatterplot.
We must also consider the variability in the distribution of data points. The processes
that determine how the variables are related may not be the same for all cases. Separate
clusters of points may suggest the presence of subpopulations for which the variables
are related differently or have different means (more about this later). Outliers or "ex-
treme" data should be examined closely because they may have a very large influence on
statistics such as the correlation or regression slope. Extreme data points may come from
subjects who perform in ways that are qualitatively different from the majority of sub-
jects. Extreme data points may also arise because of errors in data collection or copying.
If there are a few extreme outliers, we should examine our data records for errors, and
we may wish to describe the data both when the extreme scores are included and when
they are not. Statistical packages make it easy to identify outliers and to perform these
analyses.
As Wilkinson (1999) states in his guidelines for statistical methods in psychology
As soon as you have collected your data, before you compute any statistics, look at your data. Data
screening is not data snooping. It is not an opportunity to discard data or change values to favor
your hypotheses. However, if you assess hypotheses without examining your data, you run the risk
of publishing nonsense Computer malfunctions tend to be catastrophic: A system crashes; a file
fails to import; data are lost. Less well-known are the more subtle bugs that can be more catastrophic
in the long run. For example, a single value in a file may be corrupted in reading or writing (often
the first or last record). This circumstance usually produces a major value error, the kind of singleton
that can make large correlations change sign and small correlations become large. (p. 597)
Finally, it should be noted that, in Chapters 2 and 3, we have been concerned with
describing samples of data. We have yet to address the issue of what we can infer about the
populations from which these samples were selected. In Chapter 4, we begin to develop a
framework for statistical inference that will allow us to do this.
scatterplot positive relation
negative relation smoothers
resistant measures Pearson correlation coefficient
linear relation linear equation
slope Y intercept
influence plot covariance of X and Y
conceptual formula computational formula
linear regression equation least-squares criterion
predictor variable criterion variable
coefficient of determination the sum of squares of Y, SSy
variance of estimate standard error of estimate
3.1 Given the following data,
(a) Draw a scatterplot.
(b) What is the correlation between Y and X?
(c) What is the least-squares equation for the regression of Y on X?
(d) What is the proportion of variance in Y accounted for by X?
(e) Find the equation for the regression of X on Y.
(f) What is the proportion of the variance in X accounted for by 7?
3.2 Given the following data for three variables X, Y, and W,
W X Y
and, using a statistical package, find the correlations among W, X, and Y. Standardize
the variables and recompute the correlations. They should be identical. Why?
3.3 (a) Using the Royer data set on the CD, find the correlation between multiplication
accuracy (MULTACC) and the time taken to solve multiplication problems (MUL-
TRT) for third graders.
(b) Generate the influence plot for these variables.
(c) What is the correlation coefficient if the single most influential point is removed?
3.4 (a) A psychologist is interested in predicting Y from X in two distinct situations and
finds the following results:
Sit. 1 Sit. 2
b1 = 38.41 b1 = 0.25
SY =512.31 SY = 8.44
sx = 2.00 sx = 23.17
In which situation is the correlation between X and Y higher?
(b) You are given a large number of data points (X,Y) and find that the correlation
between X and Y is rxy = 0.70. You now add 10 to each of the X scores. What
happens to the correlation coefficient (i.e., what is the new correlation between Y
and the transformed X)?
(c) You have the same situation as in (b)—except instead of adding 10 to each of the
X scores, you multiply each of the Y scores by 3. Now what is the value of the
correlation coefficient?
(d) Now perform both operations: multiply each Y score by 3 and add 10 to the product.
What happens to the correlation coefficient?
3.5 For parts (a)—(c), indicate whether the use of the correlation coefficient is reasonable.
If it is not, indicate why not.
(a) A social critic has long held the view that providing enriched programs for disad-
vantaged students is a waste of money. As evidence to support this position, the
critic describes the following study:
Two thousand 8-year-old children were selected from economically deprived homes, given
a battery of mental tests, and then randomly assigned to either Group 1 or Group 2. The
1,000 children in Group 1 entered special enriched programs, whereas the 1,000 children
in Group 2 continued in their regular classes. After 3 years, another battery of mental tests
was given to all the children. It was found that the correlations between children's IQ scores
at age 8 and their IQ scores at age 11 was just about the same in Group 1 as it was in
Group 2.
Our critic claims that finding very similar correlations in the enriched and regular
groups proves that the enriched classes are ineffective in improving IQ.
(b) The research division of the Old Southern Casket and Tobacco Corporation has
just released the results of a study that they argue is inconsistent with the nega-
tive health claims made against cigarette smoking. For a large sample of heavy
smokers, a substantial positive correlation was found between the total number of
cigarettes smoked during a lifetime and length of life, a result they claim leads to
the conclusion that cigarette smoking is beneficial to health.
(c) It is found that, for eighth-grade children, there is a fairly strong negative correlation
between the amount of television watched and school performance as measured
by grades. It is claimed that this finding constitutes proof that watching television
interferes with intellectual ability and has a negative effect on the ability to focus
attention. Does this argument seem valid?
3.6 In a large study of income (Y) as a function of years on job (X), the data for 2,000
men and 2,000 women in a certain profession are
Income (Y) Years (X) Income Years
Mean 80 15 76 10
s2 324 100 289 25
rxY .333 .235
Note that income is recorded in thousands of dollars.
(a) Find byx (i-e., bIncome,Years, the regression coefficient for the regression of Income
on Years of Service) for men and for women. What is your best estimate of the
amount by which salary increases per year for males and females? Is this result
consistent with differences in the correlations? Explain.
(b) Using separate regression equations for men and women, what salary would you
predict for men and women with 10 years of experience? With 20 years of expe-
rience?
3.7 Using the Seasons data file on the CD, correlate height with weight, and then correlate
height with weight separately for men and women. How might you account for the
discrepencies among the three correlations?
3.8 SSerror = E (Yi - Yi )2 is the sum of the squared errors in prediction for a set of N data
points. Starting with Equations 3.10 and 3.11, show that SSemT = (1 — r2)SSy =
SSy — b2SSx, where r is the correlation of X and Y, b\ is the slope of the regres-
sion of Y on X, SSY = E (Yi - Y)2, and SSX = £ (Xi– X)2.
3.9 Given that rXY = .60, and starting with Equations 3.10 and 3.11, find the correlations
between (a) Y and Y, (b) Y and Y - Y, and (c) Y and Y - Y.
Proof that zY = ±zx when Y= b0 + b1 X
We want to show that if X and Y have a perfect linear relation, ZY = zX when the relation
is positive and zY = –zx when the relation is negative.
For any data point (X, 7) that falls on a straight line, we have Y = b0 + b1 X. Substitut-
ing into the usual expressions for the mean and standard deviation and simplifying, we have
Y = b0 + b\X and SY = ±b 1 sx, with sy = +b1sx when b\ is positive and sy = –b1sx
when b\ is negative (see Appendix 2.1). Therefore, if there is a perfect linear relation
between X and Y,
Where Do the Expressions for bo and b1 Come From?
Here we sketch out the procedures that produce Equations 3.8 and 3.9, although knowing the
details is not necessary for understanding any of the rest of the chapter. Finding expressions
for b0 and b1 , the Y intercept and slope of the regression line, is just a minimization problem
in calculus—we want to find the equation of the straight line that minimizes prediction error.
We first take partial derivatives of the error measure (the mean of the squared prediction
errors, MSE) with respect to both bo and b1; that is, we find
Setting these partial derivatives equal to zero and simplifying, we obtain a set of what are
called normal equations:
Solving the normal equations for b1 and b0 yields Equations 3.8 and 3.9. Note that b1 can
be expressed in any of a number of equivalent ways, including
where SXY is the covariance of X and Y, SSY = ENi=1 (Yi - Y)2, SSX = ENi=1 (Xi–X)2
and sy and sx are the standard deviations of Y and X, respectively.
Probability and the
In a study of long-term memory for childhood events, N. A. Myers and Chen (1996)
tested 20 teenagers who had previously participated in an experiment at the University of
Massachusetts as 3- to 5-year-old children 12 years earlier. In the earlier experiment, the
children had been exposed to a number of objects. The teenagers were presented with four
objects, only one of which they had seen as children. They were asked to decide which of
the four objects they had seen 12 years earlier.1 At one level, the question is whether the
teenagers remember the earlier event. But just what does this mean? We need to frame this
question more precisely if we are to use the data to answer it. A more precise statement
of the question requires us to place the experiment in a clear conceptual framework, one
which we introduced in Chapter 1.
We begin by noting that the 20 teenagers may be viewed as a sample from a population
of participants of similar age and experience. Accordingly, the responses in the experiment
are viewed as a sample of responses in the population. The researchers are interested in
whether the responses of the 20 participants provide evidence of memory in the population.
We restate this question as, Is the proportion of correct responses in the population greater
than we would expect by chance? We are closer to the goal of stating the question precisely
enough that we may use the data to answer it, but now we need a definition of "chance."
We can think of chance performance as that which would occur if none of the teenagers
retained any memory of the objects they were exposed to 12 years earlier. When they were
required to choose one of the objects, we assume they would simply "guess" in such a way
that each of the four objects would be equally likely to be chosen. Therefore, assuming
chance performance, the probability of a correct choice is 1/4, or .25. If the experiment was
performed many times, each time employing a new random sample of 20 teenagers from
a population that had experienced the correct object in childhood, we would expect that,
"on the average," 5 of the 20 would choose correctly. Note that this assumption of chance
62 4/ PROBABILITY AND BINOMIAL DISTRIBUTION
responding does not mean that in any one replication of the experiment there will be exactly
five correct responses, but that the average number of correct responses will be five if the
experiment were carried out many times.
However, the experiment is performed only once. Given that the probability of a correct
choice is .25, finding exactly 5 correct responses would be consistent with the assumption
of chance responding. But suppose 6 of the 20 teenagers responded correctly. Would this
demonstrate that the probability of a correct response in the population was greater than .25?
Not necessarily. It is at best weak evidence in favor of memory because 6 correct responses
out of 20 could easily occur if the probability of a correct choice was .25. Suppose, however,
that 18 of the 20 choices were correct. This would seem to be strong evidence that more
than chance is involved, because it seems very unlikely that this result could occur if the
probability of a correct choice was .25. How much evidence does it take to convince us that
the probability of a correct response is greater than .25—8 correct? 10 correct? In order to
make a reasonable decision about whether or not there is evidence of long-term memory
for the childhood experience, we need two things:
1. We need to know the probability distribution, assuming only chance responding.
If the members of the population had no memory for the childhood event, and if
the experiment was performed many times, each time employing a new random
sample of teenagers from a population that had experienced the correct object in
childhood, what proportion of such experiments would yield 11 correct? Or 12
correct? Or any other number of correct responses from 0 to 20?
2. We need a decision rule for deciding whether the observed number of correct
responses is so much greater than 5, the number expected by chance, that we are
willing to reject the hypothesis that only chance factors (and not memory) are
In summary, our inferences are fallible because performances vary as a function of
many factors beyond the researcher's control. Statistics such as the proportion of correct
responses in a sample, or the sample mean and variance, will rarely, if ever, exactly match
the population parameters they estimate. However, despite this uncertainty, inferences about
population parameters can be made. The data from the sample, together with certain as-
sumptions about the population of scores, provide a basis for such inferences. Understanding
the process by which inferences are drawn requires understanding random variables, their
distributions, and probability. In the next part of this chapter, we present these topics. Given
that conceptual foundation, we can then return to the question of whether the participants in
the Chen and Myers experiment exhibited better than chance memory. To develop a statisti-
cal test to address this question, we will make use of a particular probability distribution, the
binomial. Although the applicability of the statistical test we present is somewhat limited,
it should be easy to understand. Furthermore, the issues raised, and the concepts defined,
are basic to inferential processes in general.
4.2 DISCRETE RANDOM VARIABLES
Continuing with our example of the memory experiment, we state that the number of
participants (out of 20) who correctly choose the object seen in the earlier experiment
might be symbolized by Y. Y is referred to as a random variable; in this example, the
PROBABILITY DISTRIBUTIONS 63
variable Y can take on any of the 21 integer values in the range from 0 to 20. 7 is a discrete
random variable because there are values within the range of Y that cannot occur. Although
only whole numbers can be observed in the preceding example, the potential values of a
variable need not be integer or even equally spaced for the variable to be discrete. We might
discuss the results of the experiment in terms of the proportion of participants who were
correct, in which case the random variable would take on the fractional values 0, 1/20,
2/20,..., 20/20.
Variables that can take on any value within their range are called continuous random
variables. Consider the time it takes a participant to make a response in some experimental
task. Typically, in this era of high-speed microcomputers, response time can be measured
to the nearest millisecond. Although we may not be able to observe response times more
accurately, response time is a continuous random variable. The clock may not be capable of
recording it, but any time can occur; the limitation is in the measuring instrument and not
in the variable itself. Considerably more will be said about continuous random variables
in Chapter 5 and, in fact, throughout this book. In this chapter, the focus is on discrete
random variables simply because the ideas we wish to develop about inference are more
easily understood in this context.
4.3 PROBABILITY DISTRIBUTIONS
As we indicated in Section 4.1, we are frequently interested in whether scores that are
actually observed, or statistics based on these scores, differ very much from what would
be expected by chance. In this section, we begin to clarify what we mean by the expres-
sion "expected by chance." Consider the participants in a memory experiment similar to
the one just described, but, to simplify things at this stage, assume that each participant
is presented with only two objects, one of which is the correct (i.e., previously presented)
one. In this simplified memory experiment, if a participant were to make a response by
guessing, the probability of being correct on any problem would be .5. For now we also
restrict our discussion to only 4 participants in this memory experiment. An appropri-
ate random variable reflecting the performance of this group of participants would be 7,
the number of participants responding correctly. Given that each response must either be
correct (C) or in error (E), and assuming 4 participants, there are only 24 (or 16) pos-
sible patterns of correct and error responses; each of these is associated with a value
of 7. These patterns are presented in Table 4.1, together with the corresponding values
of 7.
If the participants are guessing, any of the possible values of 7 may occur, although
some values are more likely to occur than others. The set of probabilities corresponding to
each possible value of 7 is called the probability distribution of 7. The column labeled
p ( y ) in Table 4.1 contains the probabilities associated with guessing, assuming that there
are only two choices. These probabilities are also plotted in Fig. 4.1.
Where do these probabilities come from? How can they be used in answering questions
that may be of interest to us? We will consider each of these issues in turn.
The probability distribution is derived by beginning with a statistical model, a set of
assumptions about how responses are generated that is explicit enough for probabilities
to be calculated. Different sets of assumptions will lead to different values of p(y). In the
current situation, a desirable model would be one that allows the calculation of probabilities
64 4 / PROBABILITY AND BINOMIAL DISTRIBUTION
TABLE 4. 1 POSSIBLE PATTERNS OF C AND E RESPONSES
FOR 4 PARTICIPANTS
Pattern No. Correct (y) p(y)
(E 1 E 2 E 3 E 4 ) 0 1/16 = .0625
( E1E2 E3 C4 ) 1
(E E C E } 1
( E1 1 C2 2 E33 E 44 ) 1 4/16 = .25
(C1E2E3E4) 1
(E 1 E 2 C 3 C 4 ) 2
(E1C2E3C4) 2
(E1C2C3E4) 2
(C 1 E 2 E,C 4 ) 2 6/16 = .375
(C 1 E 2 C 3 E 4 ) 2
(C 1 C 2 E 3 E 4 ) 2
(C 1 C 2 C 3 E 4 ) 3
(C 1 C 2 E 3 C 4 ) 3
(C 1 E 2 C 3 C 4 ) 3 4/16 = .25
(E 1 C 2 C 3 C4) 3
(C 1 C 2 C 3 C 4 ) 4 1/16 = .0625
EP(y) = 1
Note. The subscripts denote the 4 different individuals.
Fig. 4.1 Theoretical distribution of the number of correct
responses when p(C) = .5.
to be relatively simple, yet captures the essential features of what we mean by guessing.
We employ the following model:
1. On each trial (in the example, for each participant) the probability of a correct
response is .5.
2. The responses are independent of one another; that is, the probability of a correct
response on any trial does not depend on the outcomes of any other trials. In our
example, the probability that any one participant makes a particular choice does
not depend on the choice of any other participant.
The first assumption seems reasonable given that each participant has only two possible
responses. It is equivalent to assuming that participants perform as though they are randomly
selecting responses from a box containing two slips of paper, one marked with the word
"correct" and the other with "error." Each participant shakes the box vigorously and one
of the slips of paper is selected. The second assumption requires that after selecting a slip
of paper, the participant replaces it so that the next participant is not influenced by the
preceding participant's response.
Given these assumptions, it makes sense that all 16 patterns in Table 4.1 are equally
likely to occur. Therefore, the probability of 0 correct responses is 1/16 or .0625; we
usually write p(Y = 0) = .0625. Similarly, because we know there are four equally likely
sequences with one correct response, we know that p(Y = 1) = 4/16 or .25. We have
just assigned numbers called probabilities to the events Y = 0 and Y = 1. In general, we
calculate p(Y = y), where Y is the random variable and y is a specific value that Y can
take. As we noted earlier, these values, which we collectively refer to as p(y), are presented
in Table 4.1 and are also graphed in Fig. 4.1, in which the height of each bar indicates the
probability of the corresponding value of Y.
It should be emphasized that different assumptions lead to different probability dis-
tributions. If, as in the Chen and Myers memory experiment, there were four alternatives
rather than two, assumption 1 would have to be modified to state that the probability of a
correct response on each problem was .25 (as though there were now four slips of paper in
the box, one marked "correct" and three marked "error"). All the response patterns and the
corresponding values of Y listed in Table 4.1 would still be possible, although the proba-
bilities associated with them would change. As you might guess, the probability of getting
three or four problems correct would now be much less than indicated in Table 4.1. If the
participants were not guessing and we could assume that the probability of a correct re-
sponse was, for example, .8, still another probability distribution would be indicated. Later
in the chapter, we develop a general formula for calculating values of p(Y = y).
Keep in mind why we generated the probability distribution of Table 4.1: to answer
questions about whether or not the participants were guessing, we had to get some idea of
what kind of data to expect if they were, in fact, guessing. The probability distribution we
generated is called a theoretical probability distribution because it was generated on the
basis of a statistical model. In this case, our statistical model was a theory we had about
how people would respond if they were guessing. If the assumptions we made were valid,
and if we performed many random replications of the memory experiment (assuming four
participants and two choices), in 1/16 of the experiments no one would be correct; in 1/4
of the experiments one participant would be correct; and so on. In short, the proportions of
experiments yielding various values of Y would match the theoretical values in Table 4.1.
We can see this, and get a better sense of what we mean by "many random replications
TABLE 4.2 PROPORTION OF SIMULATED EXPERIMENTS IN WHICH
THERE WERE Y CORRECT RESPONSES
No. of Experiments
No. Correct (y) 10 100 1,000 10,000 1TheoreticalProb.
0 .1 .09 .058 .0603 .0625
4 0 .04 .060 .0580 .0625
Note. p(c) = .5 and n = 4.
of the experiment," by considering the results of computer simulations of the experiments.
We simulated participants performing according to our model in either 10, 100, or 10,000
"experiments," each of which had four participants for whom the probability of a correct
response was .5. We then recorded a value of Y for each simulated experiment; recall that Y
is the number of correct responders out of four participants. The observed probability dis-
tributions of Y are presented in Table 4.2, together with the theoretical values of p(y). The
numbers in each column are the proportions of experiments in which there were y correct
responses; note that each column sums to 1.0. When there are only 10 simulated experi-
ments, the probabilities clearly differ from those for the theoretical distribution, though the
distribution shapes have some similarity. The observed proportions more closely approach
the theoretical probabilities as the sample size increases. Thus, the theoretical probabilities
may be viewed as the proportions of an infinitely large set of experiments having a particular
value of Y, assuming the statistical model is correct.
The idea of repeating an experiment many times and obtaining the value of some
statistic (here Y, the number of correct responses in the example) from each experiment
is basic to the inferential procedures described throughout this book and used by most
researchers. The idea is important enough for us to summarize the general steps:
1. A statistical model is formulated.
2. On the basis of this model, a theoretical distribution of a statistic of the experiment is
derived; this distribution is called a sampling distribution. This is the distribution
we would obtain if our model is correct and we were to repeat the experiment
many times, plotting the distribution of the statistic (Y in the example) over the
many replications of the experiment.
3. The sampling distribution is then employed, together with the data from our exper-
iment, to draw inferences about the population.
We have our statistical model and a theoretical sampling distribution (Table 4.1) for four
trials (participants) and p = .5. Now we can use this theory together with observed data to
investigate whether people in the memory experiment are simply guessing. To accomplish
this, we have to be clearer about the question. We can formulate it this way: Is the probability
of a correct response in the sampled population .5 or does our experimental evidence indicate
it is higher than .5? One possible decision rule is to conclude that performance is better
than chance if 3 or 4 responders—more than 50% of the sample of responders—are correct.
SOME ELEMENTARY PROBABILITY 67
However, the values in Table 4.1 suggest that 3 or 4 correct is not strong evidence that
performance is better than we would expect by chance. As we can see from that table,
assuming our model of chance guessing, the chance a sample would have 3 or 4 correct
responses is .3125 (that is, .25 + .0625). In other words, even if participants were guessing,
there is almost one chance in three of getting at least 3 correct responses. Stronger evidence
of memory would be provided if we require all 4 participants to be correct. According to
our model, if people are guessing, only 1/16 of similar experiments (.0625) would yield 4
correct responses. Therefore, if all 4 participants are correct, either they are guessing and
a relatively unlikely event has occurred, or performance in the population is better than
chance.
We can now outline the rationale underlying statistical testing. If in the context of
a particular experiment we wish to examine the hypothesis that only chance is involved,
a model of chance performance is used to generate a probability distribution. There are
certain outcomes, consistent with an alternative hypothesis (e.g., that the probability of a
correct response is above the chance level), that will be very unlikely if the model is valid.
If one of this set of outcomes is obtained in the experiment, we will conclude that the model
is not valid and that something other than chance is involved. Although these basic ideas
are involved in a variety of statistical tests, the advantage of first developing ideas about
inference by using a discrete random variable is that the relevant probabilities are easier to
understand and calculate. Although we will not have to deal with very complicated aspects
of probability, a thorough understanding of a few basic concepts will be required. The next
section provides an introduction to some of these concepts. Following that, we present a
class of probability distributions of which the one in Fig. 4.1 is a member. These are then
used to demonstrate further how inferences can be drawn from data.
4.4 SOME ELEMENTARY PROBABILITY
Suppose we have a class of 100 students, of whom 60 are men and 40 are women. The
instructor, a kindly statistician, gives no grades lower than C. The number of male and
female students receiving each grade is presented in Table 4.3. Suppose further that the
sex of each student, along with his or her grade, is written on a separate slip of paper. The
100 slips are placed in a box. If a slip of paper is randomly selected from the box, we can
determine the probability that the slip drawn belongs to a particular sex and has a particular
grade written on it. We can use this very simple "probability experiment" to introduce some
basic ideas about probability.
TABLE 4.3 DISTRIBUTION OF GRADES FOR MEN
AND WOMEN
Sex A B C Total
Female 12 24 4 40
Male 15 36 9 60
Total 27 60 13 100
First, by random selection, we mean selecting a slip of paper in such a way that each
slip is equally likely to be chosen. We might achieve this by vigorously shaking the box
before selecting the slip. There are 100 slips of paper and one of them will be selected.
There are only 100 possible outcomes, each corresponding to a value of sex and grade.
The possible outcomes of a probability experiment are called elementary events, and the
complete set of elementary events is called the sample space for the experiment. Here, if
we assume random selection, each of the 100 elementary events has an equal probability
of occurring.
We are frequently not so much interested in the probability of a particular elementary
event as we are in the probability of meaningful collections of elementary events, which
are usually called event classes or simply events. We might be interested in the probability
of the event "getting a grade of A," which we can denote by p(A). When the probabilities
of elementary events are equal, the probability of an event is easily computed. In this case,
where n(A) is the number of elementary events in A and n(S) is the number of elementary
events in the entire sample space. For our probability experiment,
because there are only 100 elementary events and 27 of them belong to the event A. It should
be clear that p(A) cannot take on any values greater than one or less than zero. Similarly, if
the event of interest is M, "being male," p(M) = 60/100 = .60. Because events like A and
M contain all the elementary events in a row or a column of the table, their probabilities
are often referred to as marginal probabilities.
4.4.1 Joint Probabilities
The probability of obtaining a particular combination of events is referred to as a joint
probability. For example, p(A and M), which is read as "the probability of A and M," is
the probability of the joint event (A, M); that is, it is the probability of selecting a slip of
paper with both "A" and "male" written on it. If the probabilities of the elementary events
are equal, p(A and M) can be obtained by using
where n(A and M) is the number of elementary events that belong to both events A and M.
For the data of Table 4.3, p(A and M) = . 15, because 15 of the 100 slips of paper correspond
to grades of A obtained by male students. Similarly, if the events B and F correspond to
"getting a grade of B" and "being female," respectively, p(B and F) = 24/100 = .24. Note
that p(A) must always be at least as large as p(A and M) because event A will always
contain at least as many elementary events as joint event (A, M). These ideas may be
clarified by reconsidering Table 4.3. Each column represents the event of a letter grade and
has two nonoverlapping parts. For example, the column representing event A consists of
joint events (A, M) and (A, F). Note that n(A) = n(A and M) + n(A and F), and it follows
from Equations 4.1 and 4.2 that p(A) = p(A and M) + p(A and F). An additional fact to
note is that, because of the way in which it is defined, p(A and M) = p(M and A).
4.4.2 Probabilities of Unions of Events
The union of two elementary events consists of all the elementary events belonging to either
of them. The elementary events forming the union of events A and M are the following cells
of Table 4.3: (A, F), (A, M), (B, M), and (C, M). The expression p(A U M), or p(A or
M) refers to the probability of obtaining an elementary event belonging to either A or M,
that is, falling into any of the four cells just noted. Therefore,
because 72 of the 100 elementary events belong either to A or to M. Note that n(A or M)
does not equal the sum of n(A) and n(M). As should be clear from Table 4.3, this sum
counts twice the 15 elementary events that belong to both A and M. Verify for yourself that
p(A or M) = p(A) + p(M) - p(A and M). Also verify that p(A or F) = 55/100 = .55.
In general, if E1 and E2 are two events of interest,
4.4.3 Conditional Probabilities
We may be interested in the probability of obtaining a grade of A when only the male students
in the class are considered. This probability is called a conditional probability because it
is the probability of A given the condition that M occurs. It is denoted by p ( A | M ) , and it
is read as "the probability of A given M." There are 60 slips of paper labeled "male" and
15 of them correspond to grades of A. Therefore, p ( A | M ) = 15/60 = .25. More generally,
p ( A | M ) is the proportion of all elementary events in M that also belong to A, or
Verify that, for the current example, p(B|M) = 36/60 = .60;/?(M|A) = 15/27 = .56; and
p(A|fl) = 0/60 = 0.
Two important ideas about conditional probabilities should be noted. First, people
have a tendency to confuse conditional probabilities with joint probabilities. Look care-
fully at Equations 4.2 and 4.5. The conditional probability p(A\M) is the probability of
selecting a slip of paper that is labeled "A" if a selection is made from only the 60 slips
labeled "male." The joint probability p(A and M) is the probability of selecting a slip
labeled both "A" and "male" if selection is randomly made from all 100 slips of paper.
A conditional probability has to be at least as large as, and generally is larger than, the
corresponding joint probability because the set from which we sample is a subset of the en-
tire sample. For example, when we calculate p ( A | M ) , we are dividing by only the number
of male students, a number less than the total sample. Bear in mind, however, that al-
though joint and conditional probabilities are not the same, they are related, as Equation 4.5
demonstrates.
The second idea is that for any two events A and M, there are two conditional probabil-
ities, p ( A | M ) and p(M|A). These two conditional probabilities will generally not have the
same values; in our current example, p( A \ Af) = 15/60 = .25andjp(M|A) = 15/27 = .56.
As this example illustrates, the denominators are based on different subsets of the entire
sample, and these often will have different numbers of elementary events.
An important extension of the ideas about joint and conditional probability is incorpo-
rated into Bayes' Rule, a mathematical formulation that has implications for understanding
the relations among conditional and joint probabilities, and that has also provided the foun-
dation for an entirely different approach to statistics, Bayesian analysis. An introduction to
Bayes' Rule and an example of its application may be found in Appendix 4.3 at the end of
this chapter.
4.4.4 Mutually Exclusive/ Exhaustive/
and Independent Events
Two events E\ and £2 are mutually exclusive if they are incompatible; that is, if an
elementary event belongs to E\, it cannot belong to E2. It follows that if E\ and £2 are
mutually exclusive, p(E\ and £2) = 0, p ( E 1 | E 2 ) = 0, and p(E 2 |E 1 ) = 0. In our current
example (Table 4.3), p(A and B) = 0, because if a student received a grade of A in the
course, he or she did not receive a B; p(A and M) is not equal to 0 because some of the
men in the course did receive As.
A set of events is exhaustive if it accounts for all of the elementary events in the sample
space. In our example, the events A, B, and C collectively account for all the students in
the class and are also mutually exclusive. Therefore, p(A or B or C) = 1.00 and we can say
that A, B, and C constitute an exhaustive and mutually exclusive set of events.
Two events E1 and £2 are independent if p(E\ | E2) = p(E\)', that is, if the probability
of event E1 is the same whether or not event £2 occurs. We may wish to ask questions such
as, Is getting a grade of A independent of sex? This is another way of asking whether the
probability of getting an A is the same for male and female students. If there is independence,
p ( A | M ) = p ( A | F ) = p ( A ) Returning to Table 4.3, we see that p ( A | M ) = 15/60 = .25;
p ( A | F ) - 12/40 - .30; and p(A) = .27. Clearly, for these data, getting an A is not
independent of being a male or female student, so (A, M) and (A, F) are pairs of events
that are not independent. In contrast, p ( B | M } = p ( B | F ) = p ( B ) so getting a grade of B
is independent of the student's sex. For both male and female students, the probability of
getting a B is .60.
We may also wish to ask more general questions, such as Are the variables grade and
sex independent of each other? For the answer to be yes, each of the six pairs of events
formed by combining levels of sex and grade, specifically, (A, M), (A, F), (B, M}, (B, F),
(C, M), and (C, F), would have to be independent. The variables, sex and grade, are not
independent of each other in this example.
Several important concepts concerning independence should be noted. First, if E1 and
E2 are two independent events, p ( E 1 and E2) = p ( E 1 ) x p(E2). To see why this is so,
consider the definition of conditional probability, given by Equation 4.5:
p ( E 1 | E 2 ) = p ( E 1 and E 2 )/p(E 2 )
Multiplying both sides of this equation by p ( E 2 ) yields
p(E\\E2) x p(E2) = p(E\ and£ 2 )
However, we know that if E1 and E2 are independent, p(E1|£2) = p ( E 1 ) . Replacing
p ( E 1 \ E 2 ) by p ( E 1 ) in the last equation, we have, if E1 and £2 are independent events,
It is important to understand that if events E1 and £2 are mutually exclusive, they cannot
be independent. If E1 and £2 are mutually exclusive, E1 cannot occur if £2 does. There-
fore, if E1 and £2 are mutually exclusive, then their joint probability and both conditional
probabilities must be zero; that is,
However, p ( E 1 ) and p(E2 )may be greater than zero, so the basic condition for indepen-
dence—that p(E 1 |£2) = P(E 1 ) or p(E 2 \E 1 ) = p(E2)—is not met.
4.4.5 Rules of Probability
We can now summarize the basic rules of elementary probability.
The Multiplication Rule. If £1 and £2 are two independent events,
In Table 4.3, the events B and M are independent, so p(B and M) = p(B)p(M) =
(.60)(.60) = .36. Note that Equation 4.6 does not hold if the events are not independent;
for example, A and M are not independent and p(A and M) = .15, but p(A)p(M) =
(.27)(.60) = .162. As indicated in the last section, Equation 4.6 follows directly from the
definitions of independence and conditional probability. The rule can be extended to any
number of independent events; for example, if three events E1, E2, and £3 are independent
of one another,
Although in this chapter we are concerned with independent events, the multiplication
rule can be extended to events that are not independent. In this case,
Equation 4.8 follows directly from the definition of conditional probability (Equation 4.5):
Multiplying both sides of this last equation by p(E2) yields
For example, applying Equation 4.8 to the data of Table 4.3, we can see that p(A and M) =
p(M) p(A\M) = (.60)(15/60) = .15.
The Addition Rule. If E1 and £2 are two mutually exclusive events,
TABLE 4.4 SOME PROBABILITY DEFINITIONS AND RULES
Definition or Rule Formula
Some probability definitions
Probability of event A p(A) = n(A)/n(S)
Probability of joint event A and B p(A and 5) = n(A and B)/n(S)
Probability of the union of events A and B p(A or B) = n(A or B)/n(S)
Conditional probability of A given B p(A\B) = p(A and B)/p(B)
= n(A and B)/n(B)
Some probability rules
The addition rule for unions of events p(A or B) = p(A) + p(B)-p(A and B)
Special case of the addition rule if the
events are mutually exclusive p(A or B) = p(A) + p(B)
The multiplication rule for joint events p(A and B and C)
= p(A)p(B\A) p(C\A and B)
Special case of the multiplication rule p(A and B and C) = p(A)p(B)p(C)
for independent events
This can be extended to any number of mutually exclusive events; for example, if E1, £2,
and £3 are mutually exclusive events,
As we explained in Subsection 4.4.2 (and see Table 4.3), if events E1 and £2 are not mutually
exclusive,
For example, in Table 4.3, p(A or M) = p(A) + p ( M ) - p(A and M) = .27 + .60 -
.15 = .72.
Table 4.4 summarizes much of what has been presented in Section 4.4 to this point. It
includes important definitions and the rules embodied in Equations 4.6-4.10.
Although the multiplication and addition rules are very simple, people often mix them
up. It should be emphasized that the multiplication rule tells us how to calculate p ( E 1 and
£2), the probability of the joint occurrence of E1 and E2. The addition rule tells us how to
calculate p ( E 1 or £2), the probability that E1 or £2 occurs. This union of E1 and £2 (E1 or
£2) will include the joint event (E1 and £ 2 ), but it also includes occurrences of E\ without
£2 and of £2 without E1.
4.4.6 The Sample Space for an Experiment
In the previous few sections, we discussed a sample space in which the elementary events
were the 100 combinations of sex and grade that could be sampled from a class. We now
apply some of the ideas that we have developed about probability to the memory experiment
that was introduced earlier. In the terminology we have developed, each of the possible
patterns of 4 correct and error responses presented earlier in Table 4.1 may be viewed as an
elementary event. These elementary events are mutually exclusive and exhaust the sample
space for an experiment with four participants, each making either a correct or incorrect
response. Figure 4.2 represents the events of Table 4.1 in a tree diagram.
Fig. 4.2 Tree diagram for four trials (C = correct, E = error).
In Fig. 4.2, the top branch corresponds to the pattern ( C 1 , C2, C3, C 4 ) , the next is ( C 1 ,
C2, C3, E 4 ), and so on. The subscripts indicate the 4 individuals in the experiment. The
16 patterns are mutually exclusive and exhaust the possibilities that can occur. If we assume
independent outcomes for the 4 participants, we can find the probability of each pattern by
using the multiplication rule. If p(C j ) = 1/2 for j = 1,2,3, and 4, then
and the probabilities for each of the other 15 sequences are also 1/16.
We can now obtain the probability distribution of Y, the number of correct responses,
using the addition rule as necessary; for example,
and similarly for the rest of the possible values of Y. The rules of probability allow us to
generate the theoretical probability distribution that was presented in Table 4.1 and Fig. 4.1.
The outcome probabilities can easily be calculated, no matter what the probability of
a correct answer is for either problem. In the Chen and Myers experiment, each participant
had to select one of four objects. Assuming guessing, p(C), the probability of a correct
response is 1/4 and p ( E ) = 3/4. Then, for example, the probability of exactly 1 correct
response is now
Before ending this introduction to probability, we should emphasize several points.
The sample space, S, consists of all of the elementary events, and the probabilities of the
elementary events must sum to 1. This must be the case because the elementary events are
by definition mutually exclusive and exhaustive. In the memory experiment example, the
events 0, 1,2, 3, and 4 correct responses are also mutually exclusive and exhaustive and so
their probabilities must also sum to 1. In general,
The sample space can always be partitioned into two mutually exclusive and exhaustive
sets of elementary events; call these A and A ("not A" and called the complement of A).
For example, let A be "zero correct responses" in the 4-participant memory experiment.
Then A is "1 or more correct." We could calculate p( 1 or more correct) by using the addition
rule; that is, we could add p(1 correct) + p(2 correct) + p(3 correct) + p(4 correct). It is
simpler to note that, because p(S) = 1, p(1 or more correct) must equal 1— p(0 correct).
In general,
It should be evident from the definition of probability in Equation 4.1 that a probability
must have a value within the range from zero to one. More precisely,
4.4.7 Sampling With and Without Replacement
Suppose we select two individuals from the sample summarized by Table 4.3. What is the
probability that both will be men? When sampling is done with replacement, if a man is
selected on the first draw, he is put back into the sample (i.e., replaced) and is therefore
eligible to be selected again on the second draw. This means that if sampling is performed
with replacement, selecting a man on the first draw (M 1 ) and selecting a man on the second
draw (M2) are independent events, so p ( M 1 and M2) = p(M 1 ) p(M2) = (60/100)(60/100) =
.360. However, when sampling is performed without replacement, if a man is selected on
the first draw, he is not replaced, so there is one less man eligible for selection on the second
draw. Consequently, the events M1 and M2 are not independent. Now, p ( M 1 and M2) =
p ( M 1 ) p ( M 2 \ M 1 ) = (60/100)(59/99) = .358. Note that here, even though the events are
not independent, the probability is similar to that calculated by assuming independence,
because the sample is fairly large (see Appendix 4.2).
This concludes our introduction to probability. It is brief and far from complete. Nev-
ertheless, it provides the basis for discussing a particular type of probability distribution
known as the binomial. In turn, the binomial distribution provides a context within which we
can introduce aspects of inference involved in the use of many other statistical distributions.
So, without further ado, let's consider the binomial distribution.
THE BINOMIAL DISTRIBUTION 75
4.5 THE BINOMIAL DISTRIBUTION
Figure 4.1 and the p(y) column of Table 4.1 present the distribution of one random variable,
the number of correct responses, for the special case in which there are four trials and the
probability of a correct response on each trial is .5. It would be useful to have a general
formula for the probability distribution for any number of trials and for any probability of a
correct response. For example, in the Chen and Myers experiment, 20 teenage participants
were presented with four objects, one of which they had been exposed to in a laboratory
task when they were preschoolers. In order to decide whether memory for the previously
seen object was at a better than chance level, the chance probability distribution had to be
determined. The question becomes, If the participants are guessing, what is the probability
distribution? In this example, p(C) = .25, and n, the number of trials (participants), is 20.
Once we have the chance distribution, we can formulate a decision rule. For example, using
an equation that we develop in the next two sections, we can calculate that if p(C) = .25
andrc = 20, the probability of 9 or more correct responses is less than .05 (.041). Therefore,
if 9 or more participants are correct, either they are guessing and are very lucky, or p is
actually greater than .25. Given that guessing is very unlikely to produce 9 or more correct
responses, we might form the following decision rule: If 9 or more responses are correct,
reject the hypothesis that the true probability of a correct response is .25 in favor of the
hypothesis that it is greater than .25. To form such decision rules for any experiment of
this type, we need to be able to calculate the theoretical probability distribution assuming
chance performance. We now develop a formula to enable us to do just that.
4.5.1 Basic Assumptions
Look again at Figs. 4.1 and 4.2. They represent specific instances of a general experimental
situation that has the following three characteristics:
1. Bernoulli Trials. On a Bernoulli trial, there are exactly two possible outcomes;
examples would be "correct" or "error," "head" or "tail," and "success" or "failure."
The two outcomes possible on each trial will be referred to as A and A ("not A")
and their respective probabilities will be denoted by p and q. Because A and A
exhaust the possible outcomes on a trial, p + q = 1.
2. Stationarity. This is the assumption that p and q stay constant ("stationary") over
trials. Thus, if the probability of a correct response is 1/4 on trial 1, it is 1/4 on all
3. Independence. In the example of the memory experiment, we assumed that the
probability that a participant responded correctly was the same regardless of how
other participants responded. In general, the assumption of independence is that
the probability of an outcome of any trial does not depend on the outcome of any
other trial.
The preceding assumptions justify the probability calculations that yield the binomial
distribution. If that distribution is used to draw inferences about some population, and
the underlying assumptions are not correct, the inferences may not be valid. We will have
more to say about the consequences of violating assumptions after illustrating the use of
the binomial distribution in testing hypotheses about the population.
4.5.2 The Binomial Function
Consider an experiment with n Bernoulli trials, each with possible outcomes A and A. The
outcome probabilities, p andq, are stationary over trials and do not depend on the outcomes
of other trials. We want a formula for p(Y= y), where Y is a discrete random variable and
the values it can take on are the possible number of A responses that can occur in n trials.
In an n-trial experiment, Y can take on the values 0, 1,2, ..., n. Suppose that n and p are
specified and we calculate p(Y = y) for each possible value of Y in turn. The result is a
probability distribution; the distribution in Fig. 4.1 is a specific example obtained by setting
n at 4 and p at .5. An infinite number of distributions that belong to this general family can
be generated by using different combinations of p and n. Any distribution in this family is
referred to as a binomial distribution. In this section, we develop a formula for calculating
the probability of y responses as a function of n and p. This binomial probability function
will be denoted by p(y; n, p) to indicate that it is the probability of y responses of type A
when there are n trials with p(A) = p on each trial.
Table 4.1 and Fig. 4.2 present the 16 possible sequences for a four-trial experiment.
Note that the trial outcomes, A and A, need not come from different individuals as they
did in the example of the memory experiment. For example, A and A could represent
correct and error responses by an individual on four multiple-choice questions. Then each
pattern would represent a sequence of such responses for the four questions and p would
be the proportion of correct responses in a population of such items. From now on, we
will use the more general term combination to refer to a sequence or pattern of A and
A responses.
Suppose we wish to find the probability of obtaining exactly three A responses in
four trials. Assuming that the responses A and A are independently distributed over tri-
als, we can use the multiplication rule developed earlier to calculate the probability for
each combination of A and A responses. For example, the probability of the combination
(A, A, A, A) would be ( p ) ( p ) ( p ) ( q ) or p3q. What we want, however, is p(3, 4, p), the
probability of exactly 3 A responses in four trials. That is, p((A, A, A, A) or (A, A, A. A)
or (A, A, A, A) or (A, A, A, A)). These four combinations are mutually exclusive (i.e.,
exactly one will occur in a four-trial experiment in which there are three A responses and
one A response). Therefore, the probability of three A and one A response in any order
is the sum of the probabilities of the combinations, or 4p3q. In general, we calculate the
probability of a combination having y A responses; we then multiply by the number of such
combinations.
The approach can be generalized to any value of n. The probability of any one specific
combination of y A responses and n — y A responses is p y q n - y . The probability of exactly
y A responses and n — y A responses is
where k is the number of combinations consisting of y A and n — y A responses. We just
about have our binomial function; all we still need is a formula for k, the number of ways
in which y A and (n — y) A responses can be combined. This number of combinations is
This is referred to as the binomial coefficient. The derivation of this coefficient is presented
in Appendix 4.1 at the end of this chapter. Its formula is
where n! = (n)(n - 1)(n - 2) ... (3)(2)(1) and 0! = 1. Also, note that
have the same value. Substituting y = 0, 1,2,3, and 4 in turn into Equation 4.15, verify that
the formula yields the values 1, 4, 6, 4, and 1, respectively; these are the numbers of com-
binations that appear in Table 4.1. Of course, writing out and counting all the combinations
becomes rather tedious when there are many trials. For large ns, Equation 4.14 becomes
quite useful.
Replacing k in Equation 4.14 with the formula for k in Equation 4.15 yields the binomial
probability function:
Numerical values of this probability for various values of n, p, and y are contained in
Appendix C at the back of the book (Table C.I). You may use the values contained there
to verify our statement that the probability of 9 or more correct in 20 trials is .041 if
p = .25.
Figure 4.3 presents several binomial distributions for various values of n and p. For
easier comparison across different values of n, the outcome probability is plotted as a
function of the proportion of A responses, Y/n. For example, when n= 10 and p = .5,
we expect to observe 40% correct responding (4 A responses in 10 trials) with probability
.2051. In the long run (i.e., if the experiment were repeated many times), the proportion of
experiments with 4 A and 6 A responses should equal .2051, if the binomial model is correct.
Several points should be noted about these distributions. First, when p = .5, the distributions
are symmetric. Second, when p = .25 and skewness (asymmetry) is present, as n increases
the distribution becomes more symmetric about the value of Y/n that corresponds to p.
Third, the distributions appear more continuous in form as n increases. The importance
of these observations lies in the fact that if n is sufficiently large, particularly when p is
close to .5, the binomial distribution looks much like the normal distribution, which then
can be used to get binomial probabilities with considerably easier calculations. This point
is developed in Chapter 5.
A fourth detail to note about Fig. 4.3 is that the probability of getting a value of Y/n
close to p increases with n. Consider the probability if p = .5 that Y/ n lies in the range from
.4 to .6; that is, p(A < Y/n < .6). When n is 10, p(.4 <Y/n< .6) is the probability that Y =
4, 5, or 6, which is .2051 + .2461 + .2051, or .6563. When n is 20, p(.4 < Y/n < .6) is the
probability that Y = 8, 9, 10, 11, or 12, which is .1201 + .1602 + .1762 + .1602 + .1201,
or .7368. When n is 40, the probability is .8461. This point is very important; it means that
as n grows larger, the proportion of A responses observed in a single experiment is more
likely to be close to the population parameter. We do not prefer larger data sets to smaller
Fig. 4.3 Binomial distributions as a function of sample size and
probability.
ones because of some deeply ingrained work ethic; we do so because it is more likely to
result in a sample statistic that is closer to the population parameter of interest. Statistics that
have a higher probability of being within any fixed range of the population parameter as n
increases are called consistent estimators of the parameter. We have more to say about this
desirable property of sample statistics, and about other properties important to estimating
population parameters, in Chapter 5.
MEANS AND VARIANCES OF DISCRETE DISTRIBUTIONS 79
4.6 MEANS AND VARIANCES OF DISCRETE DISTRIBUTIONS
In drawing statistical inferences, we are concerned with the parameters of a theoretical pop-
ulation of scores; quite often the population mean and variance will be the parameters of
interest. Equations 2.4 and 2.7 for the sample mean and variance are not directly applicable to
the calculation of the population mean and variance, because the individual scores in the pop-
ulation usually are not available for summing and the number of such scores is often infinite.
However, when the random variable is discretely distributed, we usually know its possible
values, and, assuming some theoretical distribution such as the binomial, we also know the
probability of each value. This provides us with a way of defining the mean and variance for
discrete distributions. We present the necessary equations in the following section, using the
binomial distribution to illustrate. In Chapter 5, we present analogs to these equations, suit-
able for dealing with the mean and variance of continuously distributed random variables.
4.6.1 The Population Mean
Now that we have developed the probability distribution of a discrete random variable,
we can take the opportunity to demonstrate how to find the mean of the distribution. The
concept of a weighted average provides a good way to think about population means. For
example, reconsider Table 4.1, which presents the probability distribution of the number of
correct responses (Y) when there are four independent trials and p is .5 on each trial. We can
conceive of doing this four-trial experiment many times, each time recording the number
of successes. Thus, we have a hypothetical population of numbers from 0 to 4 and we wish
to know what the mean of that population is: Over an infinite number of replications of the
experiment, what will the average number of successes be? This population mean will be
referred to as the expected value of Y and denoted as E(Y), and often by the Greek letter,
u (mu). In our example, Y can take on the values 0, 1,2, 3, and 4, and we can calculate the
proportion of each of these values in the population by using the binomial formula.
The equation for E(Y) is the same as Equation 2.5 for the weighted mean of an observed
set of numbers; the proportions are replaced byp(y), the theoretical probabilities of each
value of Y. Then we define the expected value as
Appendix B at the back of the book presents some useful information about expectations.
If n = 4 and p = .5, Equation 4.17 yields the following result for the mean of a binomial
In words, take each possible value of Y (zero through four in the example of the four-trial
experiment), multiply it by its probability of occurrence, and sum the products. The prob-
abilities in this example are given by the binomial equation, 4.16, and appear in Table 4.2.
It is as if we had a very large number of people, each tossing four fair coins. We note the
number of heads obtained by each person. The average number should be two. In Appendix
B, which presents the algebra of expectations, we demonstrate that the mean number of
successes in n independent Bernoulli trials is pn, where p is the probability of a success on
a single trial. In the preceding example, p = .5 and n = 4 and (.5)(4) = 2.
4.6.2 The Population Variance
Analogous to the sample variance, the population variance is the average squared deviation
of scores about their mean, or E(Y — u) 2 . We denote this quantity by a2 (the Greek letter,
sigma).We can calculate this population variance by
Applying this equation to our example of the binomial distribution with n = 4 and
p = .5, and recalling that u = 2 in this case, we have
p2 = (0 - 2)2(.0625) + (1 - 2)2(.25) + (2 - 2)2(.375) + (3 - 2)2(.25) + (4 - 2)2(.0625)
In the case of the binomial distribution, this result can be obtained more simply; p2 =
p(1 — p)n, as we demonstrate in Appendix B.
4.7 HYPOTHESIS TESTING
4.7.1 A Significance Test
Now that we have a probability distribution, the binomial, we can apply it to the example
presented at the beginning of this chapter. Recall that there were 20 participants, each of
whom had been exposed to a set of objects in the University of Massachusetts laboratory
12 years earlier, and who were now required to choose the one object from a set of four that
they had seen in the earlier experiment. The sample statistic of interest will be the number
of participants who make a correct response. In the language developed earlier, we have
20 independent Bernoulli trials; n = 20 and Y is the number of correct responses.
Before proceeding with the actual significance test, a review of the conceptual frame-
work is helpful. Imagine a population of 15- to 17-year-olds who had been in the experiment
12 years previously. Further imagine that a lottery has been held such that each individual in
the population had an equal chance to appear in our study. In this sense, the 20 students who
participated in our study can be viewed as a random sample from a hypothetical population
of 15- to 17-year-olds. The 20 responses actually obtained in the study can be regarded as
having been sampled from the hypothetical population of responses that could be obtained
from this population. Our estimate of the probability of a correct response in the population
will be p, the proportion of correct responders in the sample. We denote the population
probability by the Greek letter TT (pi). From now on we will use Greek letters to stand for
population parameters and the more common Latin letters to stand for sample statistics in
order to lessen confusion between the two.
We now must decide between two hypotheses about IT: the null hypothesis, H0, and
the alternative hypothesis, H1. Here, the null hypothesis states that the members of the
HYPOTHESIS TESTING 81
population are guessing; that is, the probability of a correct response is .25:
The alternative hypothesis states that the probability of a correct response is higher than the
chance level; that is, p is greater than .25:
If we assume that the null hypothesis is true, we can specify the probability distribution of
the random variable, Y, the number of correct responses out of 20 responses. We use the
theoretical distribution that Y should have if the null hypothesis is true to assess whether
we have enough evidence to reject the null hypothesis. Letting n be 20 and replacing p by
p (.25 if H0 is true) in Equation 4.16, we can generate the values of p ( y ) found in the
p = .25 column of Table 4.5.
The next step is to determine those values of Y that, if obtained in the study, would lead
to rejection of H0 in favor of H1. Such values constitute a rejection region. This is a set of
possible values of Y that are consistent with H\ and very improbable if H0 is assumed to be
true. Indeed, these values are so unlikely if H0 is assumed that their occurrence leads us to
reject H0. An arbitrarily chosen value, a (alpha), defines exactly how unlikely "so unlikely"
is. Traditionally, researchers have set a at .05. We want very strong evidence against H0
before we reject it.
TABLE 4.5 THE BINOMIAL DISTRIBUTION
No. Correct P(y)
y n = .25 n = .35 7T = .50
0 .0032 .0002 .0000
10 .0099 .0686 .1762
Note. The binomial distribution is shown for n = 20 and
TT = .25, .35, and .50.
We will now establish a rejection region for the Chen and Myers experiment. Turn
to the p = .25 column of Table 4.5 and sum the probabilities, beginning at the bottom of
the column. We begin at the bottom (larger values of Y) because these are the values of Y
consistent with H1 ; values of Y at the top of the table are consistent with HO being true. If
HO is true, the probability that Y has a value between 9 and 20 is
Note that if we included Y = 8 in the rejection region, the probability of obtaining a value
of Y in that region (assuming H0 to be true) would be .1019, which is greater than the value
of a we had set. Therefore, our decision rule for this experiment is to reject H0 if 9 or more
subjects in the experiment make the correct response.
The logic of our approach underlies the application of many other statistical tests.
Therefore, it is worth reviewing the basic steps in hypothesis testing:
1. State a null and alternative hypothesis.
2. Obtain the distribution of the test statistic assuming HO to be true. In our example,
HO implies p = .25. The test statistic is 7, the number of correct responses. In
general, the test statistic (a) is a quantity calculated from the data that is sensitive
to the truth or falsity of the null hypothesis; and (b) has a known probability
distribution when HO is assumed to be true.
3. Decide on a value of a and establish a rejection region. If H0 is true, the probability
that the experiment yields a value of the test statistic within the rejection region
should be less than or equal to a. Typically, this significance level is .05. Because
we are dealing with a discrete distribution, that value of a. was not available to us,
so in our example a was .041.
4. Run the experiment and calculate the value of the test statistic. If it lies within the
rejection region, HO is rejected in favor of H\. Otherwise, fail to reject H1.
A statistically significant result means that a value of the test statistic has occurred that
is unlikely if HO is true. Of course, "unlikely" is not the same as "impossible." We may be
rejecting a true null hypothesis. Such incorrect rejections of the null hypothesis are called
Type 1 errors. Alpha (a) is the probability of such errors; that is, a =p(reject H0|H0 true).
Note that a is a conditional probability, the probability of rejecting HO given that HO is true.
A useful way to conceptualize this is that if the individuals in the population are actually
guessing (Ho is true), and if we were to replicate the experiment many times, we can expect
to obtain a value of y in the rejection region in .041 of these experiments. By setting a at
this level, we express a level of risk of a Type 1 error that we are willing to tolerate.
Statistical packages usually report an exact p value, that is, the probability that a result
at least as "extreme" as that obtained in the experiment would occur, if the null hypothesis
was true. A result is statistically significant if p is less than the value of a that has been
chosen. For example, if Y was 12, p = p(Y > 12|p = .25); that is,
The result is statistically significant because p is less than the alpha value of .05. Researchers
have tended to misinterpret these p values in at least two ways. First, they often view the
reported p value as the probability that the null hypothesis is true. This amounts to viewing
p as p ( H 0 true | data). But, in fact, p is the probability of the observed data given that the null
hypothesis is true, or p(data|H0 true). It would be nice if we could calculate the probability
of the status of H0 from our analysis of the data, but we cannot. What we can do is calculate
the probability of the data under the assumption that the null hypothesis is true, and that is
a very different thing. Second, there is a tendency to compare p values across experiments
or experimental conditions, concluding that if one p value is smaller than another, it must
represent a larger, or more important, effect. However, p values depend on sample size and
variability as well as effect size, and direct comparisons are rarely valid. This is one reason
why we present ways to estimate effect sizes in the following chapters, and emphasize the
importance of such methods.
4.7.2 One- and Two-Tailed Tests
The test we just described for the Chen and Myers study is referred to as a one-tailed test,
or directional test. Because we were interested in whether the sampled population performs
better than chance, the rejection region consisted of only the largest values of Y. On one
hand, in the context of this research example, that makes sense. On the other hand, one
can conceive of many situations in which a departure from the null hypothesis in either
direction would be of interest. For example, in the case of Royer's data on arithmetic skills
(see Chapter 2), we might wish to know if there is a significant difference in performance on
arithmetic and subtraction; if there was, it might influence the way in which these skills were
taught. We might assign a plus to each student who had a higher addition than subtraction
score, and we might assign a minus if the subtraction score was higher. Then we ask if
the probability of a plus (or, equivalently, a minus) was significantly different from .5. As
another example, as described in Chapter 2, University of Massachusetts medical school
researchers collected data on seasonal variation in clinical states such as depression and
anxiety. Comparing depression scores in winter and summer, we might assign a plus if the
winter score was higher, or a minus if it was lower. In both of these examples, H0 would be
The alternative hypothesis would be
Suppose n is again 20. Turning to the column labeled p = .50 in Table 4.5, and assuming
that equal weight is given to both directions and a is close to .05, we would reject H0 if
Y <5orY > 15. This is usually referred to as a two-tailed or nondirectional test. Note that,
if we use these rejection regions, the actual probability of a Type 1 error is .021 + .021 =
.042. A larger, but still symmetric, rejection region would include the next value of Y in
each tail; then a would be .042 + .037 + .037, or .116.
4.7.3 The Power of a Statistical Test
In deciding whether or not to reject a null hypothesis, a researcher can make two types of
errors. If the null hypothesis is true, rejecting it is called a Type 1 error. The probability
of such an error is alpha and it is determined by the experimenter in the way we have
illustrated. Suppose the null hypothesis is false. Failure to reject a false null hypothesis is
called a Type 2 error, and its probability is referred to as p (the Greek letter beta). The
probability of rejecting a false null hypothesis, that is, p(reject H 0 \H 0 is false), is called the
power of the test. The sum of power and B is one.
The following table may help to clarify the meanings of a, B, and power:
H0 Reject Fail to Reject
True a 1-a
False power = 1 — B P
The rows represent two mutually exclusive events: H0 is either true or false. Given either
of these two events, the researcher may make one of two mutually exclusive decisions:
reject or do not reject H0. The cell probabilities are conditional probabilities representing
the probability of the decision given the event. Because one of the two decisions must be
made, the probabilities in each row sum to one.
The Chen and Myers experiment on memory for a childhood event will serve to illustrate
the general principles involved in computing power and should clarify the relation between
power and other quantities such as a and n. We begin by noting that the power of a statistical
test depends on how false H0 is. If the probability of a correct response in the sampled
population is .9, it is very likely that the sample of 20 subjects will yield a value of Y in the
rejection region; therefore, the statistical test has high power. In contrast, if the true value
of p is only .6, the probability of a large value of Y is much less and so, therefore, is power.
Of course, the true value of the population parameter is never known. However, if
some value of the parameter is assumed, the power of the test against that alternative can
be calculated. For example, suppose we wish to test H0: p = .25 against the alternative
hypothesis, H1: p > .25. We can calculate the power of the test for different assumed values
of p. If we assume a specific alternative hypothesis, say,
we can determine the power of the statistical test of H0 against HA . In other words, we can
calculate the probability of rejecting H0 if the probability of a correct response is .35 in the
sampled population.
Table 4.5 can be used to obtain the value of power for this example. On the basis of
H0, H1, and a, the rejection region was determined to be Y >9. Power is the probability of
obtaining a value in this region when HA is true. The actual steps in this calculation are as
1. Calculate the probability distribution of Y, assuming H0 to be true. In this example,
the distribution is presented in the p = .25 column of Table 4.5.
2. Determine the rejection region. In this example (n = 20,a = .05, H1: p >.25), the
rejection region is Y >9.
3. Calculate the probability distribution of Y, assuming HA is true. In this example,
p in Equation 4.16 is replaced by .35. The results are presented in the p = .35
column of Table 4.5.
4. Sum the probabilities for Y >9 (the rejection region) in the .35 column. This
sum is p(Y > 9\HA = .35), the power of the test of H0 against HA. In this case,
power = .1158 + .0686 + .0336 + .0136 + .0045 + .0012 + .0003 = .2376.
Fig. 4.4 Power functions based on the binomial distribution.
The Type 2 error probability, B, is 1 — .238, or .762. These calculations mean that a
test of H0: p = .25 has .238 probability of yielding a significant result if p is actually .35.
If the true value of p is even greater than .35, the power of the test will be greater than .238.
Using the p = .5 column, verify that power against this alternative to the null hypothesis is
The approach just illustrated underlies power calculations for all statistical tests. The
possible values of the test statistic are divided into rejection and nonrejection regions; this
division is determined by the value of the parameter under H0 and the nature of H1 (one
or two tailed). Then the probability of obtaining a value of the test statistic that falls in the
rejection region is obtained, with the assumption that a specific alternative distribution is
the true one.
It is important to be clear about the distinction between H1 and HA. H1 is the class
of alternative hypotheses that determines where the rejection region is placed (right or left
tail, or in both tails). HA is a specific alternative; power is calculated for the test of H0 by
assuming HA to be true.
Figure 4.4 presents the power of the binomial test of H0: p= .50 against several
alternatives. The power functions have been plotted for three conditions: (1)n = 20, a =. 15;
(2) n = 20, a = .06; and (3) n = 15, a — .06; HA represents specific alternative values
of p. The left panel presents power for a one-tailed alternative and the right panel presents
power for a two-tailed alternative. Three points should be noted that are typical for power
functions for all statistical tests. First, power increases as a increases. This is because the
increase in a requires an increase in the rejection region. For example, when n = 20 and the
alternative is one tailed, the rejection region increases from Y > 14 to Y > 13 as a increases
from .06 to .15. Because power is also calculated for this larger set of Y values, it too is
increased. Second, power is affected by the nature of H0 and H1. In the left panel of Fig.
4.4, the alternative hypothesis is H1: p >.5. This one-tailed test has more power than the
two-tailed test in the right panel whenever p is greater than .5. This is because the rejection
region is concentrated in that tail of the distribution in the case of the one-tailed test, whereas
4/ PROBABILITY AND BINOMIAL DISTRIBUTION
it is divided in half and distributed over the two tails in the case of the two-tailed test. In
contrast, the one-tailed test has virtually no power against specific alternatives of the form
TT <.5, whereas the two-tailed test does have power to reject H0 against these alternatives.
The third aspect of power illustrated in Fig. 4.4 is that increased sample size, n, results
in increased power. This follows from the discussion of Fig. 4.3 in which it was noted that
Y/n is a consistent estimator of IT. This means that as n increases, Y/n is more likely to
be close to the true value of the parameter. Therefore, if H0 is false, a larger n increases
the probability of getting values of Y consistent with the alternative hypothesis; therefore,
power is increased.
The study of power functions has other implications for researchers. First, power func-
tions for different statistical tests of the same H0 can be compared. Assuming that the choice
among tests is not dictated by some other factor (such as validity of assumptions, ease of cal-
culations, or availability of tables), the test with the higher power function should be chosen.
Second, the effects of violations of assumptions on power functions can be assessed. For
example, when the population of scores is normally distributed, the t test is more powerful
than other tests that can be used to compare two experimental conditions. However, when
the population of scores is not normally distributed, other tests may achieve markedly more
power (Blair & Higgins, 1980, 1985). Finally, and most important, the relation between
power and sample size can be used to decide how much data should be collected. Suppose
we want power of at least .90 to reject H0 if TT is at least .75. We can derive power functions
for various values of n similar to those depicted in Fig. 4.4. The n we want for our study is
the one that gives rise to a power function such that there is an ordinate value (power) of at
least .90 when the abscissa value (TT) is .75.
The last point deserves further comment. The null hypothesis is almost always false.
If we collect enough data, we are likely to obtain statistically significant results. Whether
the results will be of practical importance or theoretical significance is another matter. The
effect may be trivially small, or in a direction that makes no sense in terms of any theory,
or practical concern. Therefore, it makes good sense before we collect data to ask these
questions: What is the smallest size effect that would be of interest? and What power do we
want to detect such an effect? The answer to these two questions will be major factors in
determining the sample size for our research. Sometimes the required n will be impractically
large and we will have to compromise, have less power, or target a larger effect; or we may
be able to redesign the research so that a smaller sample will achieve the desired power
against the specific alternative hypothesis we had in mind. We return to these issues in
discussions of power in subsequent chapters.
4.8 INDEPENDENCE AND THE SIGN TEST
Throughout the preceding sections on hypothesis testing and power, we have assumed that
Y has a binomial distribution. All the computed probabilities have been based on Equation
4.16. The derivation of that equation rests on the assumption that p is constant across trials
(stationarity) and that the probability of an outcome on any one trial is independent of the
outcome on any other trial. The assumption of independence is of particular importance for
several reasons. First, it is frequently violated in psychological research. Several measures
taken from the same participant will usually be correlated. In addition, whenever responses
INDEPENDENCE AND THE SIGN TEST 87
are obtained from members of the same discussion group, school class, or litter of animals,
the responses obtained are likely to be correlated. Social, environmental, and biological
factors will tend to affect the members of such units in a similar way. Second, violation
of the independence assumption frequently will result in a Type 1 error rate very different
from the alpha assumed by the experimenter. Some assumptions can be violated with
minor consequences, but the independence assumption is often quite critical. Third, the
independence assumption plays some role in all statistical test procedures. The binomial
test is used to illustrate the consequences of its violation, but the implications are much
Consider a study in which 10 pairs of participants discuss a topic. After the discussion,
each of the 20 participants casts a "yes" or "no" vote on the issue under consideration.
Previous research has established that votes are evenly divided between the two positions
when there is no discussion. However, theoretical principles lead the researcher to believe
that "yes" votes will be more frequent than "no" votes following discussion. Thus, the null
hypothesis is H0: p(yes) = .5 and the alternative hypothesis is H1: p(yes) >.5. If alpha
is set equal to .06, the binomial table indicates that H0 should be rejected if the observed
number of "yes" responses is 14 or more.
There is a problem with this procedure: the two individuals in each discussion pair may
have influenced each other and their responses may not be independent. Let's see what this
means and then attempt to understand the implications for our testing procedure. We begin
with a case in which the independence assumption is valid. Suppose we had a population
of such discussion pairs. Randomly label one member of each pair M1 and the other M2.
Over all pairs in the population, the joint probabilities of "yes" and "no" responses might
look like this:
yes .49 .21 .70
no .21 .09 .30
.70 .30
In .49 of the pairs, both members voted "yes," in .21 of the pairs, M\ voted "yes"
and M2 voted "no," and so on. Comparing the products of the marginal probabilities with
the joint probabilities (e.g., .7 x .7 = .49), we should find it apparent that the response of
each member in a pair is independent of that made by the other member. We can verify
this by calculating conditional probabilities; the probability of one member's response is .7
regardless of whether the other member responded "yes" or "no."
Unfortunately, this independence result is not the usual outcome in studies of social
interaction. The joint probabilities in the population are more likely to look something like
A check of the products of the marginal probabilities against the joint probabilities reveals
that the independence assumption no longer holds. Calculate conditional probabilities and
verify that the probability that a pair member votes "yes" is higher when the partner also
votes "yes" than when the partner votes "no."
In order to make clear what the consequences of this dependency within pairs is for the
binomial test, we consider an extreme example. Suppose the joint probabilities of votes were
yes .50 0 .50
no 0 .50 .50
In this case, the dependence within pairs is complete: The conditional probability of a "yes"
vote is 1 when the partner votes "yes" and 0 when the partner votes "no." Note, however,
that the null hypothesis [p(yes) = .5] is true.
Recall that the researcher had sampled 10 pairs from this population and, on the basis
of the binomial distribution table, had decided to reject H0 if there were 14 or more "yes"
votes from the 20 individuals; the researcher assumed a .06 significance level. Unknown to
the researcher, the two members of each pair vote the same way. Therefore, the probability
of 14 or more "yes" votes is really the probability that 7, 8, 9, or 10 pairs vote "yes." There
are only 10 independent events; they are the pair (not the individual) votes. If this violation
of the independence assumption occurs, the probability of a Type 1 error is not the .058
assumed by the researcher; rather it is the probability that Y = 7, 8, 9, or 10 when n= 10
and p(yes) = .5. Using Equation 4.15, we can show that probability to be .172. The Type 1
error rate is much higher than the researcher believed. Most researchers would feel that it
is an unacceptably high Type 1 error rate.
Although complete dependence between the members of the pair is improbable in a real
experiment, some dependence is often likely. Consequently, the distortion in Type 1 error
rate will be smaller than in our example, but there will be distortion. Frequently, the true
error rate will be intolerably high. The opposite result occurs when responses are negatively
related. For example, suppose that the null hypothesis is false, but in a high proportion of
pairs, the partners agree to split their votes. In cases such as this, power will be greatly
reduced. Thus, depending on the nature of the dependency, either Type 1 or Type 2 error
rates will be increased. Positive dependencies are far more likely, and, therefore, the greatest
danger is an increased rate of rejection of true null hypotheses.
CONCLUDING REMARKS 89
4.9 MORE ABOUT ASSUMPTIONS AND STATISTICAL TESTS
Independence is only one assumption that plays a role in many statistical tests. In general,
the consequences of failures of assumptions are not simple and have to be thought through in
each research situation. Many factors affect error rates. The example in the preceding section
illustrates two of these—the magnitude and direction of the failure of the assumption. A
third factor is which assumption is violated. Some assumptions, despite being used in the
derivation of the test statistic, are less critical; their violation has little effect on error rates.
A fourth factor is sample size; certain assumptions (but not all) are less critical when there
are many observations. Appendix 4.2 provides an example of the interaction of assumptions
and sample size.
In summary, every inferential procedure involves some statistical distribution, and the
derivation of that distribution rests on certain assumptions. The consequences of violat-
ing these assumptions will vary depending on the factors noted herein. Throughout this
book, we emphasize the statistical model underlying each inferential procedure, detailing
the conditions that cause assumptions to be violated, the results of such violations, and al-
ternative analyses that remedy the situation when the violations are severe enough to make
the proposed analysis untrustworthy.
4.10 CONCLUDING REMARKS
The only thing certain about the inferences we draw from samples of data is that there is
no certainty. As a consequence, one cornerstone of inferential statistics is probability. Ac-
cordingly, this chapter provided a brief review of elementary probability. Inferences from
a sample to a population require a statistical model, a set of assumptions that underlie the
probabilities associated with our conclusions. Chapter 4 has illustrated this, developing
the relation between assumptions of independence and stationarity and an important the-
oretical distribution, the binomial. Finally, we used the binomial distribution to illustrate
how a theoretical distribution can be used in one kind of inferential process, significance
testing. In doing so, we introduced many of the concepts and much of the machinery of
hypothesis testing in general. Throughout this chapter, we focused on discrete random
variables because the relationship between the assumptions and the resulting theoretical
probability distribution is quite transparent. In subsequent chapters, continuous random vari-
ables are introduced, and the role of their probability distributions in making inferences is
A limitation of this chapter has been its focus on hypothesis testing. In the example
of the memory experiment, we asked whether the true probability of a correct response
was greater than .25. We might have asked a related but different question: What is the
actual value of p, the population probability of a correct response? Our best estimate
of that parameter is the sample proportion of correct responses, p. However, such a point
estimate of a population parameter is of limited value. It is a best guess about the population
parameter, but we do not know how good a guess it is. If we ran the experiment several
times, we would have as many estimates as experimental replications. If the estimates were
close to each other, we might feel confident that any one estimate was close to the population
parameter. Unfortunately, like most researchers, we will run our study only once. From that
one data set, we would like an idea of how good our estimate is, how close it is likely to
be to the parameter being estimated. The statistic we calculate to accomplish this is called
a confidence interval. In future chapters, we develop formulas for confidence intervals
for various population parameters, illustrate their interpretation and use, and discuss their
relation to significance tests.
probability distribution continuous random variable
discrete random variable statistical model
theoretical probability distribution sampling distribution
random selection elementary event
sample space event classes (events)
marginal probability joint probability
union of events conditional probability
mutually exclusive events exhaustive set of events
independent events tree diagram
complement of an event sampling with replacement
sampling without replacement Bernoulli trial
stationarity independence
binomial distribution combination of events
binomial probability function significance test
null hypothesis alternative hypothesis
rejection region test statistic
significance level statistically significant result
Type 1 error p value
one- and two-tailed tests Type 2 error
power of a test specific alternative hypothesis
sign test point estimate
decision rule
4.1 Suppose an experiment is designed to test for the existence of ESP (extrasensory
perception—the supposed ability to be aware of events in the environment through
means that do not use the normal sensory channels). An experimenter is seated in
a room with a deck of five different cards, which we can refer to as 1, 2, 3, 4, and
5. On each trial of the experiment, the experimenter shuffles the cards well and
then randomly selects one of them. A participant, P, who is seated in a room in a
different building, knows when each trial of the experiment is to occur and tries to
"perceive" and then record each card that was chosen by the experimenter. Evidence
that P does better than would be expected by chance will be taken as support for
ESP. Therefore, it is important to be able to calculate what we could expect as a
result of pure chance. Suppose P has no ESP and simply picks one of the five
cards on each trial. Assuming independence, what is the probability that P is (a)
correct on the first trial? (b) correct on each of the first three trials? (c) correct
on the second trial but wrong on the first and third? (d) correct on exactly one of
the first three trials? (e) correct on at least one of the first three trials? (f) correct
on exactly two of the first three trials? (g) correct for the first time on the fifth
trial?
4.2 Suppose a certain trait is associated with eye color. Three hundred randomly selected
individuals are studied with the following results:
Trait Blue Brown Other
Yes 70 30 20
No 20 110 50
Suppose a person is chosen at random from the 300 in the study.
(a) For each of the following pairs of events, indicate whether or not they are ex-
haustive, whether or not they are mutually exclusive, and whether or not they
are independent: (i) "yes" and "no," (ii) "blue" and "brown," and (iii) "yes" and
"brown."
(b) Find: (i) p(blue|yes); (ii) p(yes)blue); (iii) p(yes or blue); and (iv) p(yes and
blue).
Suppose two people are chosen at random from the 300.
(c) What is the probability that the first person has the trait and has brown eyes?
(d) What is the probability that both people have the trait and have brown eyes if
they are selected with replacement?
(e) What is the probability that both people have the trait and have brown eyes if
they are selected without replacement?
4.3 The following demonstrates why it is hard to screen populations for the presence
of low-incidence diseases: enzyme-linked immunosorbent assay (ELISA) tests are
used to screen donated blood for the presence of the HIV virus. The test actually
detects antibodies, substances that the body produces when the virus is present.
However, the test is not completely accurate. It can be wrong in two ways: first, by
giving a positive result when there are no antibodies (false positive), and second, by
giving a negative result when there actually are antibodies (false negative).
When antibodies are present, ELISA gives a positive result with a probabil-
ity of about .997 and a negative result (false negative) with a probability of about
.003. When antibodies are not present, ELISA gives a positive result (false positive)
with a probability of about .015 and a negative result with a probability of .985.
That is,
p (correct positive) = p (positive | HIV) = .997
p(false negative) = p(negative|HIV) = -003
p(false positive) = p(positive|no HIV) = .015
p(correct negative) = p (negative | no HIV) = .985
Suppose 100,000 blood samples are obtained from a population for which the inci-
dence of HIV infection is 1.0%; that is, p(HIV) = .01.
(a) Using the information given here, fill in the cells in the following 2x2
Test Results HIV No HIV Total
(b) Given that a randomly chosen sample tests positive, what is the probability that
the donor is infected?
(c) Given that a randomly chosen sample tests negative, what is the probability that
the donor is not infected?
4.4 We are often able to use key words such as "and," "or," and "given" to decide among
probability rules. In the following problem, we use more everyday language, so read
carefully and for each part think about whether the wording dictates marginal, joint,
or conditional probability.
Suppose that a survey of 200 people in a college town has yielded the following
data on attitudes toward liberalizing rules on the sale of liquor:
Attitude Student Nonstudent Student Nonstudent Row Total
For 70 10 40 0 120
Against 5 30 10 20 65
No opin. 5 0 10 0 15
Col. total 80 40 60 20 200
(a) What is the probability that someone is for if that person is male?
(b) What is the probability that a randomly selected individual is a female who has
no opinion?
(c) What is the probability that a female student would have no opinion?
(d) What is the probability that a student would have no opinion?
(e) What is the probability that someone with no opinion is male?
4.5 Assume that, in a particular research area, .30 of the null hypotheses tested are true.
Suppose a very large number of experiments are conducted, each with a = .05 and
power = .80.
(a) What proportion of true null hypotheses will be rejected?
(b) What proportion of false null hypotheses will not be rejected?
(c) What proportion of nonrejected null hypotheses will actually be true?
(d) What proportion of all null hypotheses will be rejected?
4.6 A study reported in the local newspapers indicated that a psychological test has been
developed with the goal of predicting whether elderly people are at high risk for
developing dementia in the near future. For healthy people at age 79, the probability
of developing dementia within the next 4 years is approximately .20. In the study, a
group of healthy 79-year-olds was given the test. For those who went on to develop
dementia within the next 4 years, the probability of a positive test at age 79 was
found to be .17; that is, p(positive|dementia) = .17. For those who did not develop
dementia within the next 4 years, the probability of a positive test was .008; that is,
p (positive | no dementia) = .008.
(a) What is p (negative | dementia)?
(b) What is p(negative|no dementia )?
From the data given here, find the predictive accuracy of the test. That is, find the
probability that a 79-year-old who takes the test will develop dementia within the
next 4 years (c) if the test result is positive and (d) if the test result is negative. Bayes'
rule (Appendix 4.3) can be used to answer parts (c) and (d); alternatively, see our
answer to Exercise 4.5.
4.7 For each of the following, state the null and alternative hypotheses.
(a) The recovery rate for a disease is known to be .25. A new drug is tried with a
sample of people who have the disease in order to determine if the probability of
recovering is increased.
(b) An experiment such as that described in Exercise 4.1 is conducted to provide
evidence for the existence of ESP.
(c) In the ESP experiment of Exercise 4.1, a proponent of ESP (Claire Voyant?)
claims that she will be successful on more than 60% of the trials.
4.8 Use the binomial table (Appendix Table C.I) to find the rejection region in each of
the following cases (IT is the population probability):
Case Ho HI n a
(a) IT = .25 p > .25 20 .01
(b) IT = .25 TT > .25 5 .01
(c) TT = .25 7T < .25 20 .05
(d) TT = .5 TT #.5 20 .01
4.9 In an experiment, data are collected such that, when a hypothesis test is conducted,
the null hypothesis is rejected with p = .003.
(a) Can you conclude that H0 is true with probability .003? Why or why not?
(b) Can you conclude that H\ is true with probability .997? Why or why not?
4.10 In each of the following, (i) State the null and alternative hypotheses; (ii) state n; and
(iii) state the appropriate rejection region assuming a = .05.
(a) An important quality in clinical psychologists is empathy, the ability to perceive
others as they perceive themselves. In a simplified version of one investigation
of empathy, 5 first-year graduate students were asked to rate a target individual
on a particular trait as they believed the individual would rate himself or herself
A 4-point scale was used. The question of interest was whether the raters would
do better than chance.
(b) In a study of group problem solving, the investigator uses the solution rate for
individuals in a previous study to predict that 40% of 3-person groups will reach
the correct solution. Fifteen groups are run in the study. The question of interest
is whether the theory is correct.
4.11 Suppose a sign test is to be done with H0: p = .50, H1: p <.50, n = 20, and a =
(a) What is the rejection region?
(b) What is the power of the test if p is actually .35?
(c) What would the rejection region be if the alternative hypothesis was nondirec-
tional, that is, if H1 : p# .50 ?
(d) What is the power with an assumption of a two-tailed rejection region and an
alternative of p= .35?
4.12 Ten students took a course to improve reasoning skills. Before the course they took a
pretest designed to measure reasoning ability, and after the course they took a posttest
of equal difficulty. The results for the 10 students are as follows:
Student 1 2 3 4 5 6 7 8 9 10
Pretest score 25 27 28 31 29 30 32 21 25 20
Posttest score 28 29 33 36 32 34 31 18 32 25
The instructors of the course try to decide whether performance on the posttest is
significantly different from performance on the pretest by looking at the signs of the
difference scores, on the reasoning that if the course had no effect whatsoever, each
student would be equally likely to get a plus or minus.
(a) State H0 and H1 .
(b) Perform a sign test on these data (a = .06) and report your conclusion.
(c) The researchers believe that if at least 75% of the population sampled improves
on the posttest, the reasoning course is worth using more widely. They redo the
analysis, testing the null hypothesis that TT = .75 against the alternative that it is
less than .75. Do you see a problem with this approach? Explain.
4.13 A researcher studying memory performs an experiment that compares two strategies
for remembering pairs of words. Twelve students are each given a number of sets of
word pairs to learn. They learn half the sets by using rote memorization and the other
half by using imagery. The order of conditions is counterbalanced appropriately. It
is found that 9 students do better with the imagery strategy and 3 do better with rote
memorization.
(a) Using the binomial distribution, test the null hypothesis that both strategies are
equally effective using a = .05. Write down the appropriate null and alternative
hypothesis and describe the steps you take in testing the null hypothesis. What is
the result of the significance test?
(b) What is the power of this test if the probability of doing better using the imagery
strategy is actually .9 in the population (so that the probability of doing better
using the rote strategy would be .1)?
APPENDIX 4.1 95
4.14 Reconsider the study of empathy described in Exercise 4.10, part (a).
(a) If the true probability of an empathetic response is .5, what is the power of the
significance test in your answer to the earlier question?
(b) What is meant by "true probability"?
4.15 Consider the hypothetical population that corresponds to a random variable Y where
Y takes on each of the values 2, 4, 6, and 8 with probability .25.
(a) What are the values of E(Y) and var(F)?
(b) Samples consisting of two scores are drawn with replacement from this popula-
tion, and the mean of each sample, Y, is obtained. Complete the following table,
generating the sampling distribution of 7:
Y = 23 4 5 6 7 8
P(Y) = ?
(c) Find E(Y) and var(Y) for the distribution in (b). How do these values relate to
your answer to part (a)?
4.16 Given a parent population that consists of just five scores, 2, 4, 6, 8, and 10:
(a) What is the mean, u, and the variance, a2, of the population?
(b) Consider all the samples of two scores that can be selected with replacement from
the population. Generate the sampling distribution. That is, state each possible
value of the sample mean and its probability of occurrence.
(c) Find the mean and variance of this sampling distribution.
4.17 It is known that, in a school with several thousand students, the mean IQ is 100.
You select a random sample of 5 students. The first student you select has an IQ of
150. Given the above information, answer the following questions and justify your
(a) What is your best estimate of the mean IQ of the next 4 students you select?
(b) What is your best estimate of the mean IQ of all 5 students in the sample?
(c) Do either of your answers to (a) and (b) change if the sample size is increased to
10? If so, what is the nature of the change?
4.18 Suppose we draw all samples of size 2 without replacement from the population of
five scores in Exercise 4.16.
(a) Generate the sampling distribution of the mean.
(b) What is the mean and variance of this sampling distribution?
Understanding the Combinatorial Formula (Equation 4.15)
Consider five individuals who are running for positions on the city council; the two top vote
getters will be elected. First consider all the possible assignments of individuals to ranks
where the ranks are the position in the final vote. There are five possibilities for the first
position in the vote count, and four possibilities for the second position (e.g., A could be
followed by B, C, D, or E). The total number of sequences is (5)(4)(3)(2)( 1) or 5! In general,
there are n\ sequences of n objects.
Suppose the question is, How many outcomes can this election have? Here, by "out-
come" we mean patterns of election and nonelection. For example, A and B might be
elected and C, D, and E fail to be elected. Notice that the order of finish within each
of the two classes (elected and nonelected) is irrelevant. The following sequences are all
equivalent in that they constitute the same outcome: A and B elected, and C, D, and E not
A,B/C,D,E B,A/C,D,E
A,B/C,E,D B,A/C,E,D
A,B/D,C,E B,A/D,C,E
A,B/D,E,C B,A/D,E,C
A,B/E,C,D B,A/E,C,D
A,B/E,D,C B,A/E,D,C
Note that the two (2!) possible sequences of A and B, paired with the six (3!) possible
combinations of C, D, and E, correspond to one combination (A and B elected; C, D, and
E not elected). In general, r!(n — r)! sequences will correspond to a single combination
when n items are split into one class with r items and one with n — r items. Therefore, the
number of combinations is n!/r!(n — r}! In our example, the number of ways the election
can turn out is
In general, the number of different ways of selecting r items from n items is
Sample Size and Violations of the Independence
Although violations of assumptions can often lead to erroneous inferences, the consequences
can sometimes (though not always) be minimized by using large samples. The violation
of the independence assumption in calculating probabilities provides a nice illustration of
this point. Consider an urn containing five red and five black balls. We draw a marble three
times from the urn. If we assume that the marble is replaced and the urn is thoroughly
shaken after each draw, so that we have independence, according to the multiplication rule,
the probability of drawing three red marbles is p ( R 1 and R2 and R3) = p(R)p(R)p(R) =
(5/10)3 = . 125. However, suppose that the drawn marble has not been replaced each time.
This violates our assumption of independence. We can see this by the following analysis:
If a red ball is drawn on trial 1, the probability of drawing a second red ball is now 4/9,
whereas if a black ball is drawn on the first draw, the probability of a red on the second
draw is 5/9. In fact, the probability of drawing a red (or black) ball on any trial depends on
the sequence of preceding draws. So, although our assumption of independence leads us to
conclude that p ( R 1 and R2 and R3) = .125, the true probability is p ( R 1 and R2 and R3) =
p(R 1 )p(R 2 \R 1 )p(R 3 \R 1 and R2) = (5/10)(4/9)(3/8) = .063, roughly half the inferred
Suppose the urn consists of 50 red and 50 black balls. Our assumption of independence
leads us to the same probability, .125. This time, however, the true probability if we select
without replacement is p ( R 1 and R2 and R3) = (50/100)(49/99)(48/98) = .121, and the
true and inferred probabilities are quite close; that is, the violation of the independence
assumption did not lead to a very large error.
There are two implications of our examples that extend beyond simple probability
calculations and violations of the independence assumption. First, violations may lead to
very wrong conclusions, as the urn with 10 marbles attests. Second, the consequences of
violations of assumptions may be less damaging when sample size is large. Neither of these
statements will be true for every inferential procedure, but they are often true and therefore
worth bearing in mind.
Bayes' Rule
We defined conditional probability in Subsection 4.4.3. For any two events, X and Y, there
are two conditional probabilities: the probability of event X given event F,
and the probability of Y given X,
We also mentioned that people tend to confuse the opposite conditional probabilities,
p(X\Y) and p(Y\X), and to confuse both of them with the joint probability p(X and Y).
Bayes' rule provides a way of expressing one conditional probability in terms of the other.
Because p(Y and X) = p(X and Y), from Equations 4.19 and 4.20 we have
Also, because when X occurs, Y either occurs or does not occur, we can write
where Y represents "not K" Combining Equations 4.21 and 4.22, we can write Bayes' rule
These equations tell us a number of useful things. From Equation 4.21, we see that p ( X \ Y } =
p ( Y \ X ) only if p(X) = p(Y). Equation 4.23 provides a way to find p(Y X), given that
we know the opposite conditional probability, p ( X \ Y ) , and have some appropriate addi-
tional information. Equation 4.23 can also be thought of as providing a way of updating
probabilities in the light of additional information. Suppose you know p ( Y ) and are now
given information about X. Equation 4.23 gives you a way of updating your estimate of
the probability of Y given the information X, resulting in p(Y\X). When the equation is
used in this way, p ( Y ) is called the prior probability and p ( Y \ X ) is called the posterior
The importance of distinguishing between opposite conditional probabilities is illus-
trated by the following example. For diagnostic tests, one can distinguish between the
predictive accuracy, p(disease(positive result), and the retrospective accuracy, p(positive
result|disease), of the test. Consider the ELISA test that is used to detect the presence of
HIV antibodies in samples of donated blood. The test is not completely accurate, and it
can be wrong in two ways: it can give a positive result when there are no antibodies (false
positive), and it can give a negative result when there actually are antibodies (false negative).
Although accuracy varies somewhat from laboratory to laboratory, when antibodies
are present, the ELISA test gives a positive result with a probability of about .997 and a
negative result (false negative) with a probability of about .003 (these two numbers have to
add to 1). When antibodies are not present, ELISA gives a positive result (false positive)
with a probability of about .015 and a negative result with a probability of .985. That is,
p (correct positive) = p (positive! HIV) = .997
Xfalse negative) = p (negative! HIV) = .003
p(false positive) = p (positive| no HIV) = .015
p(correct negative) = p(negative|no HIV) = .985
Suppose a blood sample is randomly selected from a population known to have an HIV
infection rate of 2%. Before the blood sample is tested, p(HIV) = .02. Now suppose the
ELISA test is performed on a sample and gives a positive result. What is the probability
that the sample contains HIV antibodies given that we know the test was positive, p(HIV|
positive)?
According to Bayes' rule,
Note that whereas p(positive)HIV) = .997, p(HIV|positive) is only .576 given the infor-
mation presented here. The two conditional probabilities would be equal only if the rate of
HIV infection was 50%; that is, p(HIV) = p(no HIV).
Even though the probability of a positive test is much higher when antibodies are
present than when they are not, with only a 2% infection rate, almost half the positive tests
will come from samples that do not contain antibodies because there are so many more of
them. To see this more clearly, suppose we tested 100,000 blood samples. Of the expected
2,000 samples with antibodies, we would expect (.997)(2,000) = 1994 positive tests. Of
the expected 98,000 samples without antibodies, we would expect (.015)(98,000) = 1,470
positive tests. Therefore, of the 1,994 + 1,470 = 3,464 positive tests, 1,994 would come
from samples with HIV antibodies, so that p(HlV[positive) = 1,994/3,464 = .576. Bayes'
rule is just a way of formalizing this type of reasoning.
Estimation and Hypothesis
Tests: The Normal Distribution
In this chapter, we review many of the concepts introduced in Chapter 4, but with new
procedures. For example, we again consider hypothesis tests and the errors associated with
them. However, whereas we previously employed the binomial distribution as a basis for
our inferences, we now use the normal distribution for that purpose.
The general outline of this chapter is as follows. Because the normal distribution is
continuous, we begin by expanding on the brief statement in Chapter 4 about continuous
distributions. Next, we consider the normal distribution and some reasons for its central
role in much of statistical inference. Following that, we discuss sampling distributions (also
introduced in Chapter 4), and we provide several illustrations. This enables us to address
questions such as, What do the statistics of our data set tell us about the parameters of the
population? What are the characteristics of a "good" estimate of a population parameter?
How do we set limits on our estimate so that we have an interval within which we are rea-
sonably confident the parameter falls? We then analyze several measures from the Seasons
data set (Seasons folder on the CD) collected at the University of Massachusetts Medical
School to illustrate interval estimation and hypothesis testing, basing probabilities on the
normal distribution. In Chapter 6, we consider smaller samples and introduce additional
concepts; our inferences there are based on the t distribution.
5.2 CONTINUOUS RANDOM VARIABLES
5.2.1 The Density Function
A continuous random variable is one that can take any value within a given interval. A
common example in psychological research is response time, which theoretically takes on
any value from zero to infinity. Of course, observed response times usually fall between some
CONTINUOUS RANDOM VARIABLES 101
Fig. 5.1 Example of a continuous distribution.
bounds such as 200 ms and 10 s. Even within such bounds, continuity is more theoretical
than real because the best laboratory timing devices rarely record in units smaller than
thousandths of a second. Not all the possible values of a continuous random variable can
be recorded. Nevertheless, the concept is important because many inferential procedures
assume continuously distributed random variables.
A logical problem arises when we try to deal with the probabilities of values of a
continuous random variable. We can see this by considering a relatively crude clock, one
capable of registering response times within a tenth of a second. Times longer than .95 but
shorter than 1.05 s will be registered as 1 s. Suppose we now substitute a more accurate
clock capable of measuring to the nearest hundredth of a second. Only response times in
the interval from .995 to 1.005 s now will be registered as 1 s. Of course, there will be fewer
times between .995 and 1.005 s than between .95 and 1.05 s; the probability of registering 1 s
is lower with the more accurate clock. Extending the argument, we should find it apparent
that the probability of a response time with a duration of exactly 1 s is essentially zero.
We may understand the issue better by considering Fig. 5.1. The distribution in the figure
is continuous. The area segment between y1 and y2 represents the proportion of observations
that falls between these two values. If y1 and y2 are placed closer together, the probability
of a score between these two values becomes smaller. Theoretically, we could make that
probability as close to zero as we desired by just reducing the separation between y1 and y2.
This line of reasoning suggests that it is not sensible to speak of the probability of some
exact value of Y when Y is a continuous random variable. To distinguish among continuous
random variables having different distributions, we need to represent the distribution by
some function of Y other than its probability. One function we can define is F(y), the
cumulative probability function. For example, F(y1) is the probability of getting a value
less than y1; in terms of Fig. 5.1, it is the area to the left of y1. The area in the segment
between y1 and y2 is F(y2) - F(y1). This is the probability that y takes on a value between
y1 and y2. This probability forms the basis for another function often used to characterize
a distribution: let
when y2 — y1 is very small. This ratio is the probability density function for continuous
distributions. We can view f(y) as the height of the curve at the value y. To see why
this is so, realize that we are dividing an area (between y1 and y2) by its width (y2 — y 1 );
102 5 / NORMAL DISTRIBUTION
area/width = height. Even though the area gets increasingly close to zero as the width gets
smaller, the ratio approaches a constant value that ordinarily is greater than zero. The value
of f(y) will depend on just what the value of y is, but the formula for f(y) will allow us
to calculate the probability density for any y. In other words, if we had a formula for f(y)
for the distribution of Fig. 5.1, we could plug values of y into it and plot the distribution in
that figure.
In contrast to f(y), F(y) is a probability, a proportion of the curve up to the point
y. It will be of primary interest in most inferential procedures; recall, for example, that
the hypothesis test in Chapter 4 involved evaluating the area in the tails of the binomial
distribution. Nevertheless, f(y) is important because continuous probability distributions
are characterized not by p(y), the probability that the variable, Y, has the value y, but by
f(y), the probability density function. This function represents the limiting ratio obtained
when the probability of a very small interval is divided by the width of that interval. The
density function is different for different distributions and provides a way of characterizing
a continuous distribution, just as Equation 4.16 characterizes the binomial distribution.
5.2.2 Expected Values
The expected value of a discrete random variable was defined in Chapter 4 as E(Y) =
E y p ( y ) . Because p(y) is essentially zero when Y is continuously distributed, the definition
for the continuous case involves the calculus. We present that definition, together with rules
for defining expected values, in Appendix B at the back of the book. For now, it is only
important to understand that the expected value of a population distribution is its average.
We use the notation E(Y), or the Greek letter mu (u), to indicate the mean of the population
of Y scores. The variance of the distribution is, as in discrete distributions, the average
squared deviation of scores from the mean:
This can also be written as (see Appendix B for the derivation)
5.3 THE NORMAL DISTRIBUTION
The assumption that scores are normally distributed plays a central role in many inferential
procedures. In large part, this is because the derivations of other distributions such as the
chi-square, t, and F rest on that assumption. There is some justification for the assumption
of normality; many random variables do have at least an approximately normal distribution.
Consideration of an individual's score on a test such as the SAT may clarify why this is so.
The score might be represented as
where Y is the obtained score, u is the mean of the population of test takers, and e (Greek
letter epsilon) is a sum of "errors," positive and negative deviations from the population
mean that are due to many random factors that affect the obtained performance. Such
factors would include test-taking skills, amount of knowledge relevant to the test, amount
THE NORMAL DISTRIBUTION 103
and type of preparation, motivation, and the current state of alertness. There is an important
theorem, the central limit theorem, which says that the sum of many such effects will
be normally distributed. Therefore, if e can be viewed as a sum of many independent
random effects, such as the ones we have indicated, it (and therefore Y) will tend to be
normally distributed. Another consequence of the central limit theorem is that even if Y is
not normally distributed, the distribution of the sample mean will tend toward the normal
as the sample size increases. Because of this, tests on means may be valid even when the
data are not normally distributed.
Although the normal distribution is a reasonable approximation to the distribution of
many variables, many others are not normally distributed. We cited published reviews of data
sets in Chapter 2 and presented examples from the Royer and Seasons data sets that make this
point. With this in mind, in several chapters, we consider the consequences of nonnormality
and the alternatives to those classical statistical procedures that rest on the assumption of
normality. Nevertheless, because of its central role in statistical inference, we devote much
of this chapter to considering the normal distribution. The normal distribution also merits
our consideration because it provides a relatively simple context within which to continue
our presentation of inferential procedures such as interval estimation and hypothesis testing.
The normal distribution is characterized by its density function:
where u and or are the mean and standard deviation of the population and p and e are
mathematical constants. The random variable Y can take on any value between — i and
+i, and the curve is symmetric about its mean, u.
Infinitely many normal distributions are possible, one for each combination of mean
and variance. However, inferences based on these normal distributions are aided by the
fact that all of the possible normal distributions are related to a single distribution. This
standardized normal distribution is obtained by subtracting the distribution mean from
each score and dividing the difference by the distribution standard deviation; specifically,
it is the distribution of the z score:
As we showed in Chapter 2, the mean of the distribution of z scores is zero and its standard
deviation is one. This is true of any complete set of z scores. In addition, if the variable Y
is normally distributed, the corresponding distribution of z scores also will be normal. In
this case, the variable z is often referred to as a standardized normal deviate.
Standardization provides information about the relative position of an individual score,
and it is very helpful. For example, assume a normally distributed population of scores
with u = 500 and p = 15. A value Y of 525 would correspond to a z score of 1.67; z =
(525 - 500)/15 = 1.67. Turning to Appendix Table C.2, we find that F(z) = .9525 when z
is 1.67. F(z) is the proportion of standardized scores less than z in a normally distributed
population of such scores. In this example, we may conclude that the score of 525 exceeds
.9525 of the population. Of course, this conclusion may not be valid if our values of u and
p are incorrect or if Y is not normally distributed.
Equation 5.6 defined a z score as (Y — u)/cr . In fact, this is just a special case of a general
formula for a z score. Instead of Y, we could have any observed quantity; examples would
be the sample mean, the difference between two sample means, or some other statistic. Call
this V for observed variable. To transform V into a z score, subtract its expected value from
it. Then divide the difference by pv. To conceptualize ov, assume that many samples have
been drawn from a population and a value of the statistic V is calculated for each sample.
The standard deviation of these values is (TV , which we refer to as the standard error of
the sampling distribution of V. Thus, a general formula for z is
The variable zv will be normally distributed if (and only if) Vis normally distributed. In
that case, we can assess the probability that V exceeds some specified value by referring to
Appendix Table C.2, which tables probabilities under the normal distribution. We illustrate
the application of the table and Equation 5.7 for drawing inferences about means in Sections
5.5-5.8, but first we consider the general issue of estimating population parameters.
5.4 POINT ESTIMATES OF POPULATION PARAMETERS
The basic problem in using the statistics of a single study to draw inferences about population
parameters is that the values of the statistics are not identical to those of the parameters they
estimate. The sample mean will vary over independent replications of a study, as will all
other statistics we can compute from a sample. This raises many questions, among which
are the following: What does the sampling distribution of a given statistic look like? Can
more than one statistic be used to estimate a particular population parameter? If so, how do
we decide between these possible estimates? These are some of the issues we deal with in
the following subsections.
5.4.1 What Is a Sampling Distribution?
The concept of a sampling distribution, introduced in Chapter 4, is implicit in statistical
inference. For example, consider the following marketing study. Fifty individuals are sam-
pled from some well-defined population and asked to rate a new brand of breakfast cereal.
The ratings range from 1 ("strongly dislike") to 11 ("strongly like") with 6 as the neutral
point. We might wish to test whether the mean of the sampled population is different from
the midpoint of the scale, 6. The mean of the sampled ratings is 8.6. On one hand, if the
sample mean changed little from one sample to another, this value would provide strong
evidence against the hypothesis that u = 6. On the other hand, if the sample mean was
quite variable over samples, then a sample value of 8.6 could well have occurred even
when the population mean was 6. The critical point is that it is useful to picture many
random replications of the 50-subject sampling experiment with each replication giving
rise to a value of Y. This hypothetical probability distribution of Y is called the sampling
distribution of the mean for samples of size 50. As we can see from our example, knowing
the properties of this sampling distribution may help us evaluate inferences made on the
basis of a single sampled value of Y. If we know that the sampling distribution has little
variability, we have considerable confidence that our one estimate is close to the population
parameter; conversely, we are less satisfied with an estimate when the variability of the
sampling distribution is high. Furthermore, as we shall see, if we have knowledge of the
POINT ESTIMATES OF POPULATION PARAMETERS 105
shape of the sampling distribution—for example, that it can be described by the normal
density function—we can draw various inferences about the parameter. Every statistic has
a sampling distribution, because, each time a new sample is drawn from a population, the
sample statistic is based on a new set of values. For now, we focus on the mean and variance
of the sampling distribution of Y. These two properties of the sampling distribution of the
mean will prove useful to know when we study subsequent developments.
5.4.2 The Sampling Distribution of the Mean
We can never observe the sampling distribution of a statistic, because we never take a large
number of samples from the same population. Fortunately, we can derive properties of the
sampling distribution without actually drawing even one sample. This point may be clearer
if we consider some examples.
Sampling from a Population with Equiprobable Values. Assume that we
toss a single die. As usual, the die has six sides, each with a different one of the values from
1 to 6. If this experiment is carried out many times, and the resulting number is recorded
each time, we have the distribution displayed in Fig. 5.2(a); in the long run, each possible
Fig. 5.2 (a) A discrete population distribution and (b) the
sampling distribution of the mean for n = 2.
integer from 1 to 6 will occur on 1/6 of the trials, if we assume that the trial outcomes are
independent and that each value has probability 1/6 of occurring.
Now let's change the experiment slightly so that a trial consists of tossing two dice. If
we record the mean of the two numbers that come up on each of many trials, then—still
assuming independence and equal probability of the six values for each die—the sampling
distribution of the trial mean will be that depicted in Fig. 5.2(b). The distribution now has
a definite peak. For example, the sample mean is more likely to equal 3.5 than 1 or 6. The
reason for this follows from the multiplication rule for independent events. The mean will
equal 1 only if both dice on a trial result in a 1, an event that occurs with probability p =
1/6 x 1/6. In contrast, the mean will equal 3.5 if one die shows a 1 and the other shows a 6,
or if either die has a 2 while the other shows a 5, or if the result is a 3 and a 4. Therefore,
there are six outcomes that can yield a sum of 7, or a mean of 3.5. Each outcome has a
probability of 1/36, so the probability of a mean of 3.5 is 6 x 1/36. If we were to further
increase the number of dice tossed in each replication of the experiment, consistent with
the central limit theorem, the resulting sampling distribution of the mean would be more
closely approximated by the normal density function of Equation 5.5.
Figure 5.2(a) includes values of the population mean and variance, and Fig. 5.2(b)
includes the mean and variance of the sampling distribution when two dice are thrown; that
is, when n = 2. Note that the mean of the sampling distribution of the mean, uy, is identical
to the mean of the population, uY; that is,
In words, the expected, or average, value of the sample means equals the population mean.
In Appendix 5.1, we show that Equation 5.8 must be true for all sampling distributions
regardless of the shape of the population distribution. Also note that, when there are two
scores (e.g., two dice are thrown) in the sample, the variance of the sample means,p2/Y,is 1/2
the population variance. In general, if the scores are independently distributed,p2/Y= p2Y/N,
where N is the sample size (see Appendix 5.1 for examples). For example, in panel (a), the
population variance is 2.917, and in panel (b), which depicts the sampling distribution of
means of samples of size 2, the variance is 2.917/2, or 1.458. If we construct the sampling
distribution of means for samples of size 10, the variance would be .2917.
In summary, (a) the average of many sample means will be the same as the population
mean, and (b) the sample-to-sample variability of the sample mean will be less when N is
large. Therefore, a single sample mean is more likely to be close to the population mean it
estimates when the sample is large than when it is small. This makes sense; the larger the
sample, the more likely it is to resemble other samples from the same population and the
closer its mean will be to those of other samples.
Sampling from a More Representative Population Distribution. As
Micceri (1989) and others have noted, distributions of many, perhaps most, variables mea-
sured by psychologists and educators are not normally distributed; they tend to be skewed,
or have pronounced peaks, or marked gaps among values, or some combination of these.
To consider a distribution with some of these characteristics, we created a distribution with
mean, standard deviation, skewness, and kurtosis similar to that of the sample of Beck de-
pression seasonal change (winter — spring) scores in the Seasons data set. This population
distribution is displayed in Fig. 5.3(a). There is a slight asymmetry and a marked peak
Fig. 5.3 (a) A population distribution, and sampling
distributions of the mean for (b) n = 2 and (c) n = 25.
near the mean, as well as several outliers, all characteristics present in the Seasons data.
Another salient characteristic is the presence of several large gaps among the values. We
drew 2,000 samples of size 2 from this population; the sampling distribution of the 2,000
means is displayed in panel (b). The sampling distribution is still slightly skewed to the right
but there are now fewer gaps, and the peak and outliers are less prominent. Averaging the
infrequent high scores with the more frequently occurring zero values has produced these
effects. Panel (c) displays the sampling distribution of 2,000 means, each based on samples
of 25 scores. Now we can see the central limit theorem at work; the gaps have been filled
in and the distribution is beginning to look more like that described by the normal density
function. Skewness and kurtosis values are quite close to the theoretical value of zero for
the normal density function. This does not mean that a sample of size 25 will always suffice
to yield a sampling distribution approximated by the normal density function. With a very
skewed population distribution, a still larger n would be required.
In summary, the following points should be kept in mind:
1. The sampling distribution of the mean approaches the normal distribution as sample
size increases. This approach will be slower when the population distribution is not
symmetric.
2. The mean of the sampling distribution of the mean equals the mean of the pop-
ulation. Because of this, we say that the sample mean is an unbiased estimate
of the population mean. This and other properties of estimators are discussed in
Subsection 5.4.3.
3. The variance of the sampling distribution of the mean of a sample of size N equals
the population variance divided by N. The square root of this variance is known
as the standard error of the mean (SEM) and plays an important role in many
inferential procedures.
4. Not all sampling distributions are described by the normal density function when n
is large. The central limit theorem applies only to linear combinations1 of variables,
and even then an N so large as to be impractical may be required before the normal
distribution is a good fit to the sampling distribution.
The mean of the sampling distribution and its standard deviation (the SEM) provide
the key to understanding the estimation of population parameters. Having developed some
basic ideas about sampling distributions, we can now consider the properties of estimators.
5.4.3 Some Properties of Estimators
An infinite number of possible estimators of any single population parameter exist. The
population mean might, for example, be estimated by the sample mean, the sample median,
or even the first score drawn from the sample. The choice of an estimator may seem
intuitively obvious. Why not just estimate the population mean by the sample mean, the
population variance by the sample variance, and so on? The answer is that the "obvious"
estimator may not be a very good estimator. For example, suppose we wanted to estimate
the height of the tallest man in a country; call this parameter G. Intuitively, we might use g,
the tallest height in a sample of men, to estimate G. However, it is unlikely that the tallest
man in the country will be included in a random sample of men; g usually will be less than
G. Why use an estimator that, on the average, will give a value that is systematically too
small?
Sometimes intuition suggests competing choices for estimators. Suppose a sample is
taken from a symmetrically distributed population of scores. Then, the population mean
and median are identical. In that case, intuition leaves two choices for an estimator. Do
we take the sample mean as the estimate? The sample median? Does it matter? Clearly,
we need something more than intuition to guide us in estimating (and testing hypotheses
about) population parameters. The decision about which quantity best estimates a particular
population parameter can be made by establishing criteria for good estimators, and then
examining how closely various estimators meet these criteria. The criteria that are generally
agreed on are based on knowledge of the sampling distribution of the estimator. We consider
three important criteria for selecting estimates in turn.
Unbiasedness. Suppose we wish to estimate some population parameter, 9 (theta); 0
might be a mean, a variance, or any other quantity of interest. A statistic, 0, is calculated
from a sampled set of N scores. One desirable quantity for a good estimator is that the mean
of its sampling distribution should equal the parameter being estimated; that is,
If Equation 5.9 holds for an estimator, the average of many independent estimates will
equal the population parameter. Estimators conforming to Equation 5.9 are called unbiased
estimators. One example of biased estimation is the use of the largest score in a sample (g)
to estimate the largest score in a population (G). We noted earlier that E(g) < G. A second
example is S2 as an estimator of a2, where S2 = E(Yi — Y)2/N. In Appendix B, we show
where s2 = £ (Y - Y)2/(N - 1). Therefore, because S2 = [(N - 1 ) / N ) ] s 2 ,
and E(S2) < a2. In other words, s2 is an unbiased estimator of the population variance but
S2 is a biased estimate. Equations 5.10 and 5.11 together convey this message: If we were
to take many samples, and compute S2 and s2 each time, the average value of S2 would be
smaller than the population variance, a2, whereas the average value of s2 would equal p2.
Because S2 tends to underestimate a2, we follow the usual practice of calculating s2 rather
than S2 for the sample variance.
Bias, or the lack of it, cannot be the sole, or even the most important, property of an
estimator. Suppose we took the first score, Y1, in a sample as an estimate of u. If we drew
many samples, discarded all but the first score each time, and then calculated the mean of
the sampling distribution of Y\, that mean would equal u. Thus, the first score (or, for that
matter, any single score) drawn from a sample is an unbiased estimate of u. Nevertheless,
this estimator does not feel right. For one thing, it violates our work ethic; collecting more
data in the sample does not improve our estimate because we are discarding all but one
score. This line of reasoning suggests the next criterion for an estimator.
Consistency. Again, let 6 be some estimator of 0. It is a consistent estimator of
6 if its value is more likely to be close to t as N increases.2 A familiar example of a
consistent estimator is the sample mean; because O2/y = O2y/N, it is evident that the sampling
variability of Y about u decreases as N increases. Because inferences based on consistent
estimators are more likely to be correct as sample size increases, consistency is an important
property of an estimator. Nevertheless, even consistency combined with unbiasedness is not
a sufficient basis for selecting between possible estimators of a parameter. A very important
consideration in selecting an estimator of a parameter is its variance about the parameter
being estimated. The less variable the sampling distribution is, the more likely it is that any
single estimate will have a value close to that of the population parameter. We consider this
criterion next.
Relative Efficiency. Assume that a sample of size n has been drawn from a symmetric
population. In that case, the sample mean and median are both unbiased estimators of the
population mean because the population mean and median have the same value in any
symmetric distribution. Furthermore, both the sample mean and median are consistent
estimates of u. They do differ in one respect, however. For any sample of size n, the
sampling distributions of the median and mean will differ in their variances. Assume that
many samples are drawn from a normally distributed population and the average squared
deviations of the sample means and medians about u are then calculated. For large samples,
the variance of the sample means will be approximately 64% of the variance of the sample
medians when the population is normally distributed. This smaller variance of the mean
relative to that of the median is expressed by saying that the relative efficiency (RE) of the
median to the mean as estimators of the mean of a normally distributed population is .64.
Conversely, the relative efficiency of the mean to the median is 1/.64 or 157%. Because of
its greater efficiency, the mean is preferred to the median as an estimator of the mean of a
normally distributed population.
In general, assume a population parameter, t, which can be estimated by either of two
statistics, t1 or t2. The RE of t1 to t2 is
Thus, RE is the ratio of two averages of squared deviations of estimates about the same
population parameter. Note that this is a measure of the efficiency of the estimator in the
denominator relative to that in the numerator.
5.4.4 Which Estimator?
Most of the estimation and hypothesis testing procedures presented in this and similar books,
and in published journal articles, make use of the sample mean, Y, and the unbiased variance
estimate, s2. If the population from which the data are drawn has a normal distribution, these
statistics will be efficient relative to their competitors. Consequently, estimates based on
them are more likely to be close to the true value of the parameter being estimated, and
hypothesis tests are more likely to lead to correct inferences. But what if the population
distribution is not normal? We address this question by considering the relative efficiencies
of several estimators of u for different population distributions.
To examine the efficiencies of various estimators, we used a computer to draw 2.000
random samples of size 20 from a normally distributed population that had u = 0 and a =
1. Three statistics were calculated for each sample. These were the mean (F), the median
TABLE 5.1 VARIANCES AND REs OF THREE ESTIMATES OF A POPULATION MEAN
Normal Distribution Mixed-Normal Distribution
Statistic Variance RE Variance RE
Y .051 1.000 .260 1.000
Y.10 .054 0.952 .061 4.238
Note. RE for each statistic is its sampling variance relative to that of the sample mean.
(Y), and the 10% trimmed mean (Y.10); this last statistic is obtained by rank ordering the
scores in the sample, discarding the highest and lowest 10% (the top and bottom two scores
for n = 20), and then calculating the arithmetic mean. The variances of the 2,000 values
of these three statistics are presented in the first column of Table 5.1. The column also
contains the efficiencies of Y and Y.10 relative to Y; these are obtained by taking ratios of
the variances, as in Equation 5.12. It appears that, when the population of scores is normal,
the mean is the more efficient statistic and therefore the better estimator of the population
mean. The situation is quite different if we make one change. Suppose 19 of the 20 scores in
each sample were drawn from the population with u = 0 and a = 1; however, one score is
drawn from a population with u = 0 and cr = 3. This second population looks much like the
first except that extreme scores are more likely. Think of the extreme scores as coming from
those rare individuals who come to the experiment hung over from the previous night's party.
Such scores might contribute to the variance, increasing the proportion of very small and
very large scores. Variances of the three statistics and efficiencies relative to the mean are
presented in the second column of Table 5.1. The interesting result here is that the variances
of the sampling distributions of both the trimmed mean and the median are markedly less
than that of the mean, and their relative efficiencies, accordingly, are greater.
Contrary to popular mythology and intuition, the sample mean is not always the best
estimator of the population mean. Other estimators may be more efficient when the popula-
tion is skewed, or is symmetric but with long tails, or when there are a few outlying scores,
as in the example of Table 5.1. This happens because the sampling variance of the mean is
increased much more than that of the trimmed mean or median by the inclusion of even a
few extreme scores. The sampling procedure we used to generate the results in the second
column of Table 5.1 is probably representative of what happens in many studies. The result
of the occasional inclusion of a deviant score is that we have less confidence in our inferences
about population parameters. In many cases, it might be best to use inferential procedures
that do not rest on the sample mean. Several nonparametric, or distribution free, procedures
are presented in this book; these procedures will be particularly useful in fairly simple de-
signs, but less so in more complex designs involving several independent variables. Another
possible approach implicit in the results presented in Table 5.1 is to trim data from the tails
of sample distributions. This involves adjusting estimates of population variances. Hogg,
Fisher, and Randies (1975) describe a t test based on trimming, and they compare it with
several nonparametric tests (as well as with the standard t test) for distributions exhibiting
various degrees of tail weight and skew. They also suggest ways of estimating tail weight
and skew and of using these estimates to select the best hypothesis-testing procedure.
To sum up the developments of this section, we state that unbiasedness, consistency,
and efficiency are desirable properties in the statistics we use in drawing inferences. The
prevalent use of inferential procedures based on Y and pY reflect the fact that these statistics
are known to have these properties under many conditions. However, there will be situations
in which we encounter distributions for which other statistics will be more efficient. The
researcher should be aware of this and, when such situations arise, consider alternative
approaches to inference.
5.5 INFERENCES ABOUT POPULATION MEANS:
THE ONE-SAMPLE CASE
One of the measures available to us in the Seasons data set is the seasonal total cholesterol
score (TCI,..., TC4). We calculated the average over the four seasons (the variable labeled
TC) for those participants who had been measured in all four seasons (some participants
missed at least one of the four sessions); the data are in the TC file in the Seasons folder on
the CD; We decided to use our sample data to estimate the mean TC of a subpopulation—
namely, male participants who were 50 years of age or older (Agegrps 3 and 4 in the file).
Keeping in mind that doctors frequently recommend that TC levels should be at 200 or
less, we also wanted to know if the subpopulation mean differed from this recommended
maximum level. In what follows, we analyze the TC data of 117 male participants in
Agegrps 3 and 4 to estimate the mean of a population of such individuals, and to test the
null hypothesis that the population mean equals 200.
5.5.1 A First Look at the Data
Figure 5.4 presents a plot of the data obtained from SYSTAT's t-test module. Several aspects
of the plot are of interest. First consider the box plot of the TC scores. Because the median
approximately bisects the box, and the whiskers are of about equal length, it appears that the
distribution is symmetric. Note that the median is clearly above the recommended maximum
TC level of 200. Furthermore, the lower hinge is to the right of the dashed line that represents
this level; therefore, at least 75% of the participants have TC scores above the recommended
Fig. 5.4 SYSTAT graphs of the
TC data for 117 male subjects in
Agegrps 3 and 4 (50 years of age
or older in the Seasons study).
INFERENCES ABOUT POPULATION MEANS: THE ONE-SAMPLE CASE 113
maximum. There is some good news, however. None of the participants have a TC level as
high as 300, a value that would clearly signal a high-risk patient. Nevertheless, the box plot
warns us that cholesterol level may be a problem for many of these patients. One other point
should be noted in the box plot. There are two extreme outliers, TC scores close to 100.
Such scores are very unusual, and it might be wise to consider rechecking these patients'
cholesterol levels. One possibility that should be considered is that the scores represents
clerical errors.
The dot plot at the bottom of the figure shows the distribution of scores in a very
detailed manner. Our impression of symmetry is confirmed and we again see the low
outliers. The curve above the dot plot is a normal distribution having the same mean and
standard deviation as the sample of 117 TC scores. Because the normal distribution is
symmetric, the peak is at its mean, and again we can see that the mean is clearly above the
recommended maximum level indicated by the dashed line. Comparing the dot plot and the
theoretical normal distribution, we find it evident that the distribution in the sample is flatter
and "shorter tailed" than the normal distribution. In the following sections, in which we
draw inferences from our data, we comment on the consequences of this apparent departure
from normality.
To draw inferences about the population mean, based on the normal distribution, we
need three pieces of information: the sample size (N), the sample mean (7), and the sample
standard deviation (s). The mean and standard deviation are readily obtained from any
statistical package; in this example, we have
We are now ready to estimate an interval containing the population mean and to test
whether that mean differs significantly from the theoretical value of 200. The calculations
that follow are based on the normal probability distribution. Strictly speaking, because we
do not know a, but instead have estimated it, the t distribution provides more valid infer-
ences. However, because N is large, there will be little difference between the results based
on the normal and on the t distributions. We have used the normal probability distribu-
tion in this chapter to postpone certain complexities in our discussion of inferences about
5.5.2 A Confidence Interval for u
The sample mean, Y = 224.684, provides a point estimate of u, the population mean TC
score for male participants older than 50 years. However, the sample mean might be close
to the parameter, or it might be considerably in error. To have a sense of the reliability of
such estimates, we calculate a confidence interval (CI), a pair of numbers that provide
bounds for the parameter being estimated. How is this confidence interval calculated? What
assumptions underlie the calculations? How should we interpret the results? We consider
these questions next, first noting that although our example involves the confidence interval
for the population mean, the interpretation of confidence intervals for other parameters is
similar to that for the mean.
Assume that many samples of 117 TC scores are drawn from a population of such
scores. Further assume that the mean of each sample is converted into a standardized (z)
score by subtracting the mean of the sampling distribution and then dividing the remainder
by the standard error of the mean. If the scores are independently sampled from a normal
population, Appendix Table C.2 tells us that .95 of the sampled z values will lie between
-1.96 and 1.96. That is,
To obtain the bounds on u, consider each inequality separately. First, consider
Solving for u, we have the upper bound:
Similarly, we can solve for the lower bound. From the inequality
we arrive at
Putting it all together, we have
Recall that Y = 224.684 and 5 = 31.302. Dividing s by the square root of N, R117, we
have 2.894, an estimate of pY. Because N is large and the statistic s is a consistent estimate
of p, we expect s to be very close in value to p, and we feel justified in using it in our
calculations.3 Substituting into Equation 5.14, we find that the upper and lower bounds of
the .95 confidence interval for u are
Consistent with our earlier look at plots of the data, we see that the lower limit of the CI
is above the recommended maximum TC of 200, but we are reasonably confident that the
population mean is not dangerously high for the population of over-50 males from which
we have sampled.
What exactly do these numerical limits mean? In what sense do we have .95 confidence
that the population mean is contained within them? We cannot say that "the probability is
.95 that u lies between 219 and 230"; ^ is either in this interval or it is not. Equation 5.14
tells us that, if we were to select many samples and find the .95 CI for each sample, in the
long run, .95 of these intervals will contain u. Therefore, our confidence is .95 that the one
interval calculated for this data set contains u.
Ideally, the narrower the interval, the better our estimate of u. Returning to Equation
5.14, we can see that the interval width depends on the SE; the smaller the variability
of the sample mean, the smaller the distance between the two limits. If we recall that
pY = p/RN, it follows that the interval decreases with increased sample size and with
decreased variability. In short, we can increase the precision of our estimate by doing
whatever we can to reduce error variance, and by collecting as many observations as is
practical. A third factor, not immediately obvious in Equation 5.14, also affects the width
of the CI. We refer to the level of confidence. Turning to Appendix Table C.2, note that if
the level of confidence is set at .90, rather than .95, the critical z score is 1.645. Replacing
1.96 by 1.645, we see that the new limits are 219.92 and 229.44; we have less confidence
but a slightly narrower interval. There is a trade-off between confidence and interval width.
5.5.3 A Test of the Null Hypothesis
We originally asked whether the mean of the sampled population of TC scores differed from
a value of 200. The CI limits we just calculated suggest that the answer is that there is a
significant difference. We reason as follows: First, we have .95 confidence that the computed
interval, which has the limits 219 and 230, contains the population mean. Second, that
interval does not contain 200. Thus, we conclude with .95 confidence that the population
mean TC score differs from 200.
Most researchers tend not to calculate the CI and instead directly test whether the pop-
ulation mean equals the theoretical value. We believe this is a mistake because it addresses
the question, Is the mean 200? rather than the question, What is the mean? Nevertheless, the
practice of hypothesis testing is widespread. Furthermore, a presentation of the test permits
us to again address related concepts such as Type 1 and Type 2 errors. For these reasons, we
use the standardized normal distribution to test whether u differs from 200. In Section 5.8.
we present a more detailed discussion of the relation between the CI and the significance
As in our discussion of the binomial test in Chapter 4, we establish a null and alternative
hypothesis; these are again designated H0 and H1, respectively. Letting uTC represent the
mean of the population of TC scores, we can restate our two hypotheses. The null hypothesis
is that u = 200 and is stated as
The alternative hypothesis is
Once these two hypotheses have been formulated, we need a test statistic whose value will
enable us to decide between them. Recall the general form of the z statistic (Equation 5.7):
To test whether the mean TC score is significantly different from 200, we replace V by
YTC, E(V) by the population mean specified by H0 (u hyp ), and pv by the SEM. Conse-
quently, we have
Substituting the values presented in Subsection 5.5.2, we have
This z score informs us that the observed mean is more than 8 standard deviation units
above the hypothesized mean of 200.
Now that we have a numerical value for our test statistic, we must use it to decide
between H0 and H1. We do this by determining those values of z that would lead to rejection
of H0 in favor of H1. Such values constitute the rejection region, the set of possible values
of z that are consistent with H1 and very improbable if H0 is assumed to be true. Indeed,
those values are so improbable if H0 is assumed that their occurrence leads us to reject
H0. An arbitrarily chosen value, a (alpha), defines exactly how unlikely "so unlikely" is.
Traditionally, researchers have set a at .05. Again, we want very strong evidence against
H0 before we reject it.
Once we have decided on a value of a, we can establish a rejection region for our study.
Turning to Appendix Table C.2, we find that 1.96 is exceeded by .025 of the standardized
normal curve; because the curve is symmetric, .025 of the area also lies below — 1.96. Thus,
if the null hypothesis is true, there is only a 5% probability of obtaining a value greater than
1.96 or less than —1.96. Equivalently, we reject H0 if the absolute value of z, |z|, is greater
than 1.96. Obviously, the z we calculated is much larger than 1.96 and therefore we reject
H0.
To summarize the steps in testing the null hypothesis (also see Chapter 4):
1. State the null and alternative hypotheses.
2. Decide on a test statistic; in the present example, this is the z defined by Equation
3. Decide on a value of alpha and establish a rejection region. If a = .05, and the test
is two tailed, reject H0 when z > 1.96 or z < —1.96.
The obtained value of the test statistic falls well into the rejection region, so we can reject
the null hypothesis.
Alternatively, in step 3 we can find the p value, the probability that the value of the
test statistic would be at least as extreme as we actually obtained, if the null hypothesis was
true. We reject the null hypothesis if p < a. Here, the p is p(z > 8.53) + p(z < — 8.53); to
three decimal places, p = .000.
The two-tailed rejection region was selected because we tested whether cholesterol
scores were significantly different from (lower or higher than) 200. However, we might
have decided that low cholesterol scores are good and our interest lies only in detecting
high values. In that case the rejection region would have been one tailed and the null and
alternative hypotheses would have been
In this situation, if the population of scores is normally and independently distributed, and
we know the population variance, we again can use the z test. Accordingly, we turn to
Appendix Table C.2. Again, the rejection region consists of those extreme values of z that
are consistent with the alternative hypothesis. In this case, because our test is one tailed,
the region consists only of the largest 5% of the z distribution. Therefore, again assuming
that a = .05, we will reject H0 if the z calculated from our data is greater than 1.645. Also,
in the one-tailed case, the p value is determined only by the part of the distribution beyond
the value of the test statistic in the direction consistent with the alternative hypothesis.
The choice between one- and two-tailed tests should be made before the data are
collected. To understand why, consider the following scenario. Suppose we originally hy-
pothesized that the average TC score should be above 200; we have a one-tailed hypothesis.
INFERENCES ABOUT POPULATION MEANS: THE CORRELATED-SAMPLES CASE 117
However, upon examining the data, we find that the sample mean is less than 200, and
we now restate our hypotheses, testing at the .05 level for a significant difference in the
direction opposite to that originally hypothesized. Because we have already considered one
tail (mean TC scores above 200) and are now considering the other tail (the mean is below
200), the true alpha level is greater than .05. In essence, we failed to find evidence for one
alternative hypothesis at the .05 level and are now seeking evidence for a different alterna-
tive, again at the .05 level. But 2 x .05 = .10, the true alpha level in this approach to the
Why not always carry out the two-tailed test? Doing so would allow us to test for
departures from the null hypothesis in both directions. The answer lies in a consideration
of power. Note that the two-tailed test requires a cutoff of 1.96 in the right-hand tail of the
normal distribution, whereas the one-tailed test requires a cutoff of 1.645. In other words,
if the alternative hypothesis is that the population mean is greater than 200, the one-tailed
test has a more lenient criterion for rejection. Therefore, as we illustrated with the binomial
distribution of Chapter 4 (see Fig. 4.4), when the null hypothesis is true the one-tailed test
has more power against that alternative. We illustrate the calculation of the power of the
normal probability (z) test and discuss the factors affecting it in Section 5.7. Before doing
this, and also before considering the assumptions underlying the inferences we have drawn,
we first consider a second example.
THE CORRELATED-SAMPLES CASE
A major purpose of the study carried out by researchers at the University of Massachusetts
Medical School was to determine whether changes in the seasons affect various personality
and physical attributes. One measure that might reflect seasonal change is the Beck depres-
sion score. We subtracted the depression score obtained in the spring from that obtained in
the winter for each participant in the Seasons study who had scores in both seasons. Note
that once we have carried out this subtraction and obtained the sample of change (or differ-
ence) scores, our data set resembles the set of TC scores. That is, although we began with
two samples (winter and spring depression scores), we now have a single sample of change
scores. We refer to this as the correlated-scores case because each participant contributes
a winter and spring score to the change score, and the seasonal scores are therefore likely to
be correlated. Correlated scores also are a product of research designs in which individuals
are matched on some measure other than the dependent variable (such as IQ or a pretest
score), or are paired because they are siblings (or, often, twins), and each member of the
pair is randomly assigned to one of two treatments. In these cases, difference scores are
obtained for each pair, and CI and significance test calculations reduce to those previously
illustrated in Section 5.5. In summary, the correlated-scores case is really a one-sample case
in which the sample consists of difference scores. Because one-sample data sets are most
common in studies in which two measures are taken from one individual, or individuals are
matched on the basis of some variable, we focus on the example of seasonal change scores
for the rest of this chapter.
Summary statistics based on the seasonal Beck depression change scores are presented
separately for male and female participants in Table 5.2. These statistics are based on the
TABLE 5.2 DEPRESSION SEASONAL CHANGE (WINTER - SPRING)
SCORES BY GENDER
Males Females
No. of cases 211 215
Minimum -13.054 -16.205
Maximum 22.054 16.500
Median 0.000 0.026
Mean 0.028 0.557
95% CI upper 0.588 1.080
95% CI lower -0.532 0.033
Std. error 0.284 0.266
Standard dev. 4.126 3.897
Variance 17.028 15.185
Skewness(Gl) 1.096 0.235
Kurtosis(G2) 6.527 2.548
Note. Output is from SYSTAT.
difference between winter and spring (Beck_Dl — Beck_D2) scores in the Beck_D file in
the Seasons folder of the CD. We first use these statistics to draw inferences about the mean
difference in depression scores in the female population. In Section 5.10, we investigate the
difference between the mean change scores of the male and female populations.
Substituting the values of the sample statistics for female participants into Equation
5.14 yields the following CI on the mean change score:
These values differ slightly from those in Table 5.2 (.033, 1.080). Because the population
standard deviation is not known, but is estimated, our statistical package used critical values
of the t distribution rather than the normal. For large N, the normal and t distributions are
very similar. If we had substituted the critical value of t, 1.911, rather than 1.96 into Equation
5.14, we would have obtained the confidence limits shown in Table 5.2.
The CI limits we just calculated suggest that there is a significant change in depression
scores between the winter and spring seasons because the computed interval, which has
the limits .04 and 1.08, does not contain zero. Thus, we conclude with .95 confidence that
there is an average difference in the population; the mean depression score is higher in
the sampled population in the winter than in the spring. As in our analysis of cholesterol
scores in Section 5.5, we can carry out a direct test. Letting uchange represent the mean of
the population of change scores, we can state two hypotheses. Assuming that we wish to
determine whether there is a difference in either direction, the hypothesis that the mean
change is zero is the null hypothesis,
THE POWER OF THE Z TEST 119
and the alternative hypothesis is
enable us to decide between them. As in Section 5.5,
Substituting the mean and SE values from Table 5.2 yields
Setting a = .05, we see that the rejection region is z > 1.96 or z < —1.96, and we reject H0
because 2.09 > 1.96. Alternatively, we can find the p value, which is .036. Because .036 is
less than our a, .05, we again conclude that the null hypothesis is false. Note that uhyp need
not be zero. The value of the mean under the null hypothesis might be some other value,
perhaps based on a mathematical model or on norms gathered from a different population,
or in previous studies.
5.7 THE POWER OF THE ZTEST
Following a data analysis in which the null hypothesis was not rejected, the researcher
might wish to know what power the test had, given the effect size that was observed, the
variability of the data, and the number of observations. Another investigator might wish
to know what the power would be if a different sample size were used. In these cases, N,
a (or an estimate), and a specific effect size (such as the mean change score) are known
and power is to be determined. Ideally, researchers should take power into consideration
when planning the experiment. We should ask what sample size we need to have a specified
level of power to reject H0, assuming a specific effect size. In this case, power,CT,and the
effect size are known and TV is to be determined. To reinforce our understanding of what
power means, we provide an example of how power is calculated when N,CT,and a specific
effect size are given. Further examples of the determination of power, and also of N, using
software available on the Internet, are presented in Chapter 6.
5.7.1 Determining the Power of the
Normal Probability (z) Test
Suppose that a research group in another part of the country wants to know if the effects of
seasonal change on female depression scores can be replicated in their area, an area in which
seasonal climates differ from those in Massachusetts. Further suppose that their sample of
female participants is limited to an N of 100. This is a smaller sample than the 215 tested by
the University of Massachusetts researchers. Would this second group of researchers have
reasonable power to reject the null hypothesis if it is false? To answer this question, we have
to follow the steps outlined in Chapter 4, in which power was discussed in the context of
the binomial test. We first have to establish a specific alternative hypothesis. A reasonable
approach is to test H0 against an alternative suggested by the results of the Seasons study;
therefore, we will base the specific hypothesis, HA, on the observed mean change, .557.
Furthermore, we decide on a one-tailed test. Then, the null hypothesis is H0:u Change = 0,
the alternative hypothesis is H1: uchange > 0, and the specific alternative is HA: uchange =
.557. Having specified a one-tailed test, and setting a = .05, we see that the decision rule
is to reject if z > 1.645.
The basic principles in computing power are the same that dictated the power cal-
culations of Subsection 4.7.3. Simply put, we have to calculate the area in the rejection
region assuming HA is true. We do this by finding the distance in SE units between the
mean assuming the alternative hypothesis and the mean assuming the null hypothesis; the
alternative value, uA, is .557 and the value assuming the null hypothesis, uhyp, is zero.
If Y represents the mean change score, its SE is py = pchange/RN. In words, the SEM
is the standard deviation of the change scores divided by the square root of the sample
size. To find the numerical value of the SE, we assume that pchange = 3.897, the value
of Change obtained in the Seasons study. Therefore, SE = 3.897/10, or .390. Now we can
calculate the distance in standardized error units between uA and uhyp. This is a z score;
Figure 5.5 presents two standardized normal distributions. The left one is the distribution
assuming the null hypothesis to be true; its mean is at zero, and the rejection region is the
shaded area to the right of 1.645, the cutoff established when we selected our alpha level.
The right distribution has its mean at uA, which we have just shown is 1.429 standard errors
to the right of uhyp. The power of the hypothesis test is the proportion of this distribution
to the right of z = 1.645 in this alternative distribution. To find the size of this area, do the
1. Find the z score of the cutoff with respect to the alternative distribution. This value
is .216, because if we look at the null distribution, the critical z score is 1.645 —
1.429 = .216 units greater than the z of uA.
2. Turn to Appendix Table C.2 and find the area to the right of a z of .216.
Fig. 5.5 Null and alternative distributions (shaded areas are
rejection regions).
In our example, this area, the power of our test, is approximately .41. In words, if the
mean change score in the population is .557, the probability is .41 of rejecting the null
hypothesis that the mean change is zero. The rather low value of power serves to remind
us of the critical effect variability has on our inferences. Despite what many laboratory
scientists would consider to be a large sample, power is low against what appears to be a
reasonable alternative hypothesis (on the basis of an actual study), and, accordingly, the
Type 2 error rate is very high. The situation is considerably better—though hardly great—
in the actual study in which the N was larger; power against the specific alternative (u =
.557) with an N of 215 instead of 100 is approximately .67. Clearly, the variability in Beck
depression scores makes it difficult to achieve precise parameter estimates or high power
to test hypotheses, even with relatively large samples.
5.7.2 Factors Affecting Power
In general, the power of a test depends on several factors, all of which have effects qualita-
tively similar to those noted in the discussion of power in Chapter 4. First, power increases
as a increases because the increase in a requires an increase in the range of values included
in the rejection region. If alpha is. 10, the critical value in Fig. 5.5 shifts from 1.645 to
1.28, increasing the rejection region, and consequently the area above it. Second, power is
affected by the nature of H0 and H1. If the statement of H\ is two tailed, with alpha still
at .05, the decision is to reject H0 if z < —1.96 or z > 1.96. Then there would be two
critical values in Fig. 5.5: —1.96 and 1.96. Power would correspond to the areas to the right
of 1.96 and to the left of —1.96 under the HA distribution. As we can see in Fig. 5.5, the
probability of z < —1.96 if HA is true is essentially zero and the probability that z > 1.96
is less than the probability that z > 1.645. Therefore, the one-tailed test is more powerful
against the specific alternative, uA =.557. In contrast, this one-tailed test has virtually no
power against specific alternatives of the form u < uhyp, whereas the two-tailed test has
the same probability of rejecting H0 against these alternatives as against those of the form
H- > uhyp.
Two other factors affecting power are the population variance and the sample size;
reduced variance and larger N yield smaller SEMs. As the SEM decreases, the sample mean
is more likely to be close to the true parameter value. As we have noted previously, a smaller
SEM increases the probability of getting values of Y close to the true population mean. Thus
a decreased SEM will result in increased power to reject false null hypotheses. Of course,
as we pointed out in Chapter 4, most null hypotheses are false at least to some extent. We
should always consider whether we have had so large an N that an effect of little practical
or theoretical importance was detected. This is one reason why CIs are an important part of
our analyses. Very large sample sizes may sometimes result in rejection of a null hypothesis
even if the effect is trivial, but the CI, by providing a bounded estimate of the effect, enables
us to assess its importance.
We can influence variability by our choice of measures and experimental design, as
well as by controlling extraneous factors that might contribute to chance variability. How
large an N we need will depend on the other factors noted herein and the power we want,
as well as the smallest size effect we want to be able to reject with that power. A sample
size of as little as 40 would have provided more than the .41 power we calculated if the
variance of the depression scores had been smaller, or if uA had been larger than .557. Many
sources, including books (e.g., Cohen, 1988; Kraemer & Thiemann, 1987), software, and
Web sites, enable researchers to calculate the sample size needed to have a certain level
of power against a specified alternative. We illustrate this important application of power
analyses in later chapters.
Examining power as a function of the variables that affect it has proven useful in decid-
ing between alternative methods of analyses (e.g., Blair & Higgins, 1980, 1985; Levine &
Dunlap, 1982; Ratcliff, 1993; Zimmerman & Zumbo, 1993). For example, power functions
for different statistical tests can be compared. Assuming that the choice among tests is
not dictated by some other factor (such as validity of assumptions, ease of calculations, or
availability of tables), we should choose the test with the higher power function. We can
also consider the effects of violations of assumptions on the power of various statistical
tests. For example, the t test is more powerful than other tests that can be used to compare
two experimental conditions when the population is normally distributed. However, when
power, particularly when a one-tailed hypothesis is tested (Sawilosky & Blair, 1992). One
way to increase power is to increase sample size when practical. Other approaches to the
problem are discussed at various points in this book.
5.8 HYPOTHESIS TESTS AND CIs
The relation between CIs and significance tests may be understood by considering the usual
decision rule for a two-tailed test: Assuming a = .05, reject H0 if
Some algebra will show that this is equivalent to the following rule: Reject if
However, Y ± 1.96 OY are the lower and upper limits of a 95% CI on u. Therefore, the
null hypothesis will be rejected at the .05 level (two tailed) whenever the hypothesized
value of the population mean is less than the lower bound, or more than the upper bound
of a .95 CI. In the example of the seasonal change in depression scores, the value zero was
below the lower limit of the .95 CI, allowing us to reject at the .05 level of significance the
hypothesis of no mean change in the sampled population. Note that the CI permits evaluation
of any null hypothesis; any hypothesized parameter value that falls outside the limits will
lead to rejection by a significance test (assuming a is set at one minus the confidence
level), whereas null hypotheses that assert values of the parameter within the CI will not be
rejected.
We can also use the confidence interval to carry out one-tailed tests of significance. We
might wish to test the null hypothesis of no seasonal change in depression scores against
the one-tailed alternative that the winter-spring difference is positive; that is, that the mean
depression score for women is higher in winter than in spring. Then we test H0: u = 0
against the directional alternative, H1: u > 0. Because the lower bound of our .95 CI is .03,
we have .975 confidence that the true population mean is greater than .03. Therefore, H0:
u = 0 is very unlikely to be true. In fact, we can reject this null hypothesis at the .025 level
of significance. The .90 CI would provide a one-tailed test of the null hypothesis at the .05
VALIDITY OF ASSUMPTIONS 123
A CI provides several advantages over a hypothesis test. First, it provides a bounded
estimate of the population parameter, thus focusing attention on the parameter value rather
than on whether that parameter has one specific value. Second, the CI permits tests of all
possible null hypotheses simultaneously, thus providing considerably more information than
does the hypothesis test. Finally, the interval width provides information about the precision
of the research. A significant result, coupled with a very narrow interval, may suggest that
power was so great as to enable us to reject even a trivial effect. In contrast, a nonsignificant
result, together with a wide interval, suggests that our experiment lacked precision, pointing
to the need for either a less variable measure, more careful application of experimental
procedures, or a larger sample. Note that the width of the interval is influenced by the same
variables that influence power. The narrower the interval, the more powerful a test of any
particular null hypothesis will be. The interval narrows, and power increases, as TV increases
and as s decreases. Furthermore, increasing a and decreasing confidence have parallel
effects. An increase in ct increases power at the cost of increasing the Type 1 error rate.
There is a similar trade-off between confidence and the interval width; decreasing confidence
yields a narrower interval providing a more precise estimate but with less confidence in that
estimate.
5.9 VALIDITY OF ASSUMPTIONS
The validity of the inferences we made based on the calculations of CIs and hypothesis tests
in the preceding sections rests on three assumptions. Scores are assumed to be independently
and normally distributed, and they have a known standard deviation, a. Let's consider each
of these assumptions in turn.
5.9.1 The Independence Assumption
Two scores, Yi and Yj, are independent of each other if the probability of any value of
one is independent of the value of the other. In the notation of Chapter 4, we have two
independent scores if p ( Y j \ Y i ) = p ( Y j ) . In simple English, two scores are independent if
knowing one score provides no information about the value of any other score. If scores
are not independently distributed, the CI for u may be invalid and Type 1 error rates and
power associated with tests of hypotheses about u may be seriously affected. In the Seasons
data, spring and winter scores are likely to be correlated, and therefore not independent;
individuals who are more depressed than others in the winter will also tend to be so in the
spring. For this reason we cannot treat the winter and spring samples as independent of each
other. To draw inferences about the difference in the winter and spring mean depression
scores, we created a single change score for each participant in the study. The statistics of
Table 5.2 are based on these change scores. We have one such score for each participant in the
study, and the assumption of independence is that these change scores are independently
distributed. Assuming that our participants were randomly sampled, we can analyze the
change scores to provide inferences about uchange, the mean of the population of change
scores.
If we treat the spring and winter scores as though they were independent of each other,
using calculations we present later in Section 5.10, we find that the CI will be overly wide
and power will be low relative to that in the correct analysis. The reason for this is that the
standard error of the difference between two independent means is larger than that for two
dependent means.4 By treating the means as independent when they are not, we use too
large an estimate of the variability in this research design. In other research designs, the
result of a failure to take nonindependence into account in the data analysis may result in
an inflation of Type 1 error rate. Chapter 16 presents an example of this.
5.9.2 The Normality Assumption
The skewness and kurtosis values in Table 5.2 indicate that the change scores are not
normally distributed. As we noted in Chapter 2, Gl is zero if the distribution is perfectly
symmetric and G2 is zero for the normal distribution. In Table 5.2, both of these statistics
have values more than twice as large as their SEs, providing evidence against the assumption
that the population of change scores is normally distributed. However, the issue for any
assumption is not whether it is correct but rather whether it is sufficiently close to being
correct that our inferences are valid. In the example of the depression change scores, the
departure from normality is not likely to be a problem. Our inferences are based on the
assumption that the sampling distribution of the mean change score is normal. Even if
the population of scores is not normal, because we have a large number (215) of change
scores, the central limit theorem leads us to believe that the sampling distribution of the
mean is approximately normal. This approach to normality with increasing sample size was
illustrated in Fig. 4.3.
5.9.3 The Assumption of a Known Value
of the Standard Deviation
Although we can be certain that the population standard deviation is not exactly 3.90, the
value of s in Table 5.2 is an unbiased estimate of a, and a consistent one. Consistency
implies that, as the sample grows larger, the probability increases that s lies close to a.
Because our sample size is large, using the sample value of the standard deviation in our
calculations in place of the true (unknown) population value should not present a problem.
We have one indication that this is the case when we compare the CI calculated by using
values from the table of normal probabilities with those in Table 5.2, which were based on
the t distribution. Although the t distribution assumes an estimate of a rather than a known
value, the two sets of results are very similar.
Further evidence that violations of the normality and known-a assumptions are not
critical when N is large derives from a computer study we conducted. We drew 2,000 samples
of 215 scores each from the population distribution in Fig. 5.3(a) and we calculated a CI
for each sample. This population was constructed to have characteristics—mean, standard
deviation, skewness, and kurtosis—similar to those of the sample of 215 scores.5 The
proportion of samples yielding limits containing the mean of the simulated population was
.945, quite close to the theoretical value of .95.6 In terms of a two-tailed test of the null
hypothesis, this implies a rejection rate of .055. The close approximation of confidence
and significance values to the theoretical values indicates that even if the population is not
normally distributed, the normal probability function and an estimate of the population
standard deviation can provide adequate inferences when the sample is large. This raises
the question of how large is large. There is no simple answer to this. Using the population
of Fig. 5.3 and drawing samples of size 30 instead of 215, we found that .940 of the
COMPARING MEANS OF TWO INDEPENDENT POPULATIONS 125
2,000 CIs contained the true value of the population mean, a reasonable approximation to
the theoretical value of .95. However, the results may not be quite as satisfactory with small
samples if the population distribution deviates more markedly from normality.
5.10 COMPARING MEANS OF TWO INDEPENDENT
In the developments thus far, we had winter and spring depression scores for 215 women who
participated in the Seasons study. Because each participant's depression score was obtained
twice, once in each season, the scores are correlated; r= .819. Consequently, although
we estimated and tested the difference between two means, we did this by first obtaining a
single distribution of scores. A change score was obtained for each participant and the mean
of the population of change scores was estimated and submitted to a hypothesis test. In this
section, we consider a comparison of two means that are based on two independently (and
therefore uncorrelated) distributed sets of scores. Specifically, we consider the difference
between the effect of the seasons on changes in male and female depression scores.
The change scores for the women and men are displayed in histograms in Fig. 5.6. The
data are from the Beck_D file. Both distributions are roughly symmetric, although the right
tail is more pronounced than the left, particularly in the sample of male scores. The other
notable difference is that there are more scores in the male than in the female histogram
in the two most frequent categories. These differences are reflected in Table 5.2, presented
earlier in this chapter. The fact that the right tail extends further for the male histograms is
consistent with the higher skewness value for men, and the difference in the peaks of the
distributions is also reflected in the higher kurtosis value. The difference in the right tails
is also reflected in the difference in the maximum values in the table. In summary, there
are proportionately more men than women in the categories close to zero change, and also
slightly more in the most extreme right tail. Turning to measures of location in Table 5.2,
we find the medians either at zero (men) or very close to zero (women). The means present
a somewhat different picture. The mean change score is higher for women, suggesting that,
on the average, they exhibit a greater increase in depression scores from spring to winter.
Also note that the male CI contains zero, whereas the lower limit of the female CI is greater
than zero. Nevertheless, the difference in the mean change scores (.028 vs. .557) seems
small relative to the range of scores. Before we draw any conclusions, it would be wise to
consider this difference further.
Fig. 5.6 Histograms of winter-spring Beck depression scores by gender.
5.10.1 Assumptions
We continue to use the normal distribution, tabled in Appendix Table C.2, as the basis for our
inferences. More specifically, we assume two populations of independently and normally
distributed depression change (winter-spring ) scores, one for men and one for women.
We assume that many pairs of samples of size 211 (male) and 215 (female) are drawn
at random from the two populations. Following each draw, the difference in the sample
means, YF — YM, is computed. Consequently, we generate a sampling distribution of these
differences. The mean and SE of that sampling distribution are uF — uM and pYF-YM,
respectively.
5.10.2 Interval Estimation
Let uM and uF be the means for the male and female populations of change scores,
respectively. We want a .95 CI that bounds uF — uM. It is helpful in deriving that interval
to consider a general form for the CI:
To obtain the .95 CI for uF - uM, V = YF- YM = .551 - .028 = .529 and E(V) =
uF - uM. All that remains is to calculate pv. We begin by noting that pv is pYF-YM, and
Y F— YM is a linear combination of the two means; that is,
In Appendix 5.1 we prove that the variance of the difference between two independently
distributed quantities is the sum of their variances. Therefore,
Because the variance of the sampling distribution of the mean is the population variance
divided by the sample size, the SE of the sampling distribution of the difference between
the means is
and, substituting the values in Table 5.2, we have
We may now rewrite Equation 5.16 for the special case in which V = Y F — YM:
Substituting numerical values, the lower and upper limits for uF — uM are
We have .95 confidence that the difference between male and female seasonal change scores
lies between -.23 and 1.29. It appears that the difference between the male and female
populations is quite small. In fact, the confidence interval is consistent with the hypothesis
that there is no gender difference; uF — uM = 0 lies within the interval. As usual, that
hypothesis can be tested directly and we consider such a test before further discussing the
relation of seasonal depression score changes to gender.
5.10.3 The Hypothesis Test
Testing the null hypothesis that the difference between two population means has some
specified value (usually, but not always, zero) follows directly from the developments of
the preceding section. Equation 5.7 stated that
From our work with the Cl for uF - uM, we know that V = YF — YM — .557 - .028 = .529,
E(V) = uF — uM, andpv=R(P2M/nM)+(P2F/ n F ) = -389. To test H0: uF —uM = 0 against the
alternative uF — uM # 0, assuming the sampling distribution of the difference in means
is normal, we form the decision rule: Reject H0 if \z\ > 1.96. Substituting numbers into
Equation 5.7, we find that z = (.529 - 0) /.389 = 1.36. As the confidence limits indicated,
H0 cannot be rejected.
There is an apparent inconsistency if we consider results obtained when CIs (or signif-
icance tests) are calculated for the mean change scores for men and for women separately.
This can be seen by reviewing the results obtained in each case. Table 5.3 presents the mean
and SE of the sampling distribution, and the upper and lower .95 confidence limits, for each
of three variables: the mean change score for women, the mean change score for men, and
the difference in gender means. Considering the confidence limits, we conclude that there
is a significant seasonal change in mean depression scores for women, but not for men. This
implies that there is a difference between the genders in their mean change scores. However,
the confidence limits for the difference between genders fail to support the inference that
the male and female populations differ in their mean change scores.
This pattern of results raises two issues. The first is why separate tests on the male and
female samples indicate that the null hypothesis of no seasonal change can be rejected for
women but not for men, whereas a test of the mean female change score minus the mean
male change score does not support the hypothesis that the change is greater in the female
population. The explanation lies in consideration of the SEs and the interval widths. Note
that the SE and the interval is larger for the sampling distribution of YF — YM than for
either YF or Y'M. We have less precision in estimating the difference in population means
than in estimating either population mean alone. Consistent with the wider interval for
TABLE 5.3 SUMMARY STATISTICS FOR SEASONAL CHANGE SCORES
Cl Limit
Sex Mean SE Lower Upper
Men .028 .284 -.532 0.588
Women .557 .266 .033 1.080
Gender diff. .529 .389 -.233 1.291
u F - u M , hypothesis tests about the difference between population means have less power
than hypothesis tests about either population mean alone.
The second issue raised by the results in Table 5.3 is, What can we conclude? Is the
mean greater in the population of female seasonal change scores than in the male population,
or isn't it? Problems similar to this frequently arise when the results of data analyses are
reviewed, but there is no simple answer. In the current example, we calculated the power to
reject a specific alternative based on the observed difference in change scores; despite the
large sample sizes, power was low, approximately .27. Consequently, it is difficult to reach a
conclusion about the difference in the mean change scores of the two populations. We have
sufficient evidence to reject the null hypothesis of no seasonal change in the female popula-
tion but lack sufficient evidence to reject the hypothesis of no difference between the mean
changes of the male and female populations. However, given the lack of power in the test
of the difference, there is no support for concluding that the population means are the same.
The medians of the change scores provide a complementary picture of the relation
between the two distributions. Note that the male and female medians in Table 5.2, unlike
the means, are quite similar and are both very close to zero. This suggests that there is no
median change in either population. The significant mean change for women may reflect
a few extreme positive change scores, a suggestion that is supported if we delete all eight
(positive and negative) outliers from the data set; in that analysis, the mean change for
women is no longer significant and both medians and means suggest no difference in
location between the male and female populations. In this particular instance, the medians
seem more representative of the data and we would be inclined to conclude that (a) the
population midpoints (i.e., the medians) are close to zero and differ very little, if at all,
and (b) the female mean change score, although significantly different from zero, probably
reflects a few scores that are considerably more depressed in winter than in spring.
Although the statistics at hand, and the further calculations based on them, are important
in influencing our conclusions, other factors may also play a role. Consider a situation in
which behavioral changes are recorded with two drugs in order to select one to be marketed.
Assume the change is significant with one drug and not with the other but, as in the direct
comparison of men and women, there is no significant difference between the drugs. We
might be inclined to market the drug that produced the significant change. However, the
situation becomes more complicated if we stipulate that the other drug is considerably less
expensive to manufacture. Should we go ahead with the more expensive drug if the direct
comparison does not provide clear evidence of its superiority? This example suggests that
there will be situations in which results will be inconclusive and further research, perhaps
with larger samples or with less variable measures, may be desirable.
5.11 THE NORMAL APPROXIMATION TO THE BINOMIAL
When we worked with the binomial distribution in Chapter 4, probabilities were obtained
either by calculation or by using tables of the binomial distribution. However, for large
samples this is unnecessary, because we can estimate binomial probabilities by using tables
of the normal distribution. In Chapter 4, we showed that as the number of trials increased, the
binomial distribution began to look more symmetric, and more like the normal distribution
(see Fig. 4.3), particularly when the probability of a success on a trial, p, was .5. In fact.
CONCLUDING REMARKS 129
TABLE 5.4 TAIL PROBABILITIES, P(Y > y), FOR THE BINOMIAL
AND NORMAL DISTRIBUTIONS (N = 20)
p=.5 p=.75
y Binomial Normal Binomial Normal
11 .4119 .4115 .9861 .9899
with N as small as 20, the normal distribution provides an excellent approximation to the
tail probabilities of the binomial. To use the normal probability table of Appendix Table
C.2 to get the probability of y or more successes when y is greater than pN, calculate
where pN is the mean value of Y, p( 1 —p)N is the variance of Y, and the .5 is a correction
for continuity. The correction reflects the fact that the binomial is a discrete distribution.
Suppose we want the probability of 15 or more successes in 20 trials. Because 15 may be
viewed as representing a point from 14.5 to 15.5 along a continuum, we "correct" 15 by
subtracting .5; in other words, we find the area above 14.5 under the normal distribution. If
y < pN, we add the .5 instead of subtracting it.
Table 5.4 presents tail probabilities [p(Y > y)] when N = 20, p = .5 or .75, for values
of y from 11 to 20. The normal approximation to the binomial is almost perfect when
p = .5 and it is fair when p = .75. The approximation when p = .75 can be improved by
increasing N. A rough rule of thumb is that in order to use the normal approximation to the
binomial, the smaller of Np and N(1 — p) should be greater than 5.
We have focused on two inferential procedures—CIs and hypothesis tests. Once again, we
emphasize that CIs have inherent advantages. They provide a bounded estimate of the size
of the effect and, at the same time, a sense of the precision of the research through the width
of the interval. Furthermore, the very act of defining a null and alternative hypothesis invites
the researcher to make a dichotomous decision between the existence and nonexistence of
an effect, whereas the establishment of a .95 interval should serve to remind us that the true
effect may lie outside the calculated confidence limits. When hypothesis tests are carried
out, they should be accompanied by power calculations. The result of such calculations can
influence our view of hypothesis test (or CI) results. If power is low (or the interval is wide),
we should be skeptical of nonsignificant results. If we have very high power (or a very
narrow interval) to detect a very small effect, we should consider whether the effect is large
enough to be of interest, or whether statistical significance merely reflects the collection
of a very large amount of data. The minimum effect size of interest can be determined on
the basis of practical considerations in applied research or, in more theoretical work, on the
basis of theory and previous research. Determining this targeted effect size, and therefore
the specific alternative hypothesis, has the healthy consequence of forcing us to think about
how large an effect should be before it is important to detect. We have much more to say
about effect sizes, and we provide some guidelines for what is small, medium, or large, in
The development of CIs and hypothesis tests has been carried out within the context
of the normal distribution because it allowed us to focus on basic concepts and procedures,
postponing certain complexities to subsequent chapters. In addition, although the normal
density function rarely plays a direct role in data analyses, the assumption that the population
is normally distributed underlies procedures based on the T, chi-square, and F distributions.
This assumption is rarely true; data distributions in psychological and educational research
tend to be skewed, or have longer tails than the normal, or have gaps, or lumps. Several of
these characteristics were present in the Seasons data analyzed in this chapter, and those data
are quite typical of other psychometric measures in those respects. Despite this, distributions
of means and of differences among means will tend to be symmetric, and with large samples
will be adequately approximated by the normal density function. With smaller samples, the
consequences of violating assumptions are not readily summarized because they depend
on many factors—for example, the actual shape of the population distribution, the sample
size, and whether the hypothesis test is one or two tailed. We consider these factors further
in subsequent chapters.
A critically important concept that underlies all the developments in this chapter is
that of the sampling distribution. We usually have a single data set from which to draw
inferences about the population of scores that has been sampled. The concept of the sampling
distribution provides a bridge between the sample and the population. Our interpretation of
the confidence level is based on the proposition that if we were to take many random samples,
computing a CI for each, the proportion of intervals containing the parameter of interest
would match the nominal confidence level. Similarly, when we say that the probability of
a Type 1 error is .05, we say in effect that if the null hypothesis is true, .05 of independent
replications of the experiment will result in rejecting that hypothesis. Furthermore, our
criteria for estimates of parameters—unbiasedness, consistency, and relative efficiency—
are based on properties of the sampling distribution.
Another important concept introduced in this chapter was that of linear combinations,
that is, weighted sums or averages. Linear combinations will play an important role in future
developments. Appendix 5.1 notes several important linear combinations such as the mean,
sum, and weighted average. Still another application is provided when the mean of one set
of conditions is pitted against the mean of another set of conditions, as, for example, in a test
of whether depression scores are higher in the fall and winter than they are in the spring and
summer. These more general linear combinations, or contrasts, together with their CIs and
significance tests, are introduced in Chapter 6, and they are discussed further in Chapters 9
Although this chapter focused on the inferential process, exemplified by CIs and hy-
pothesis tests based on the normal density function, it is important to be aware that we
EXERCISES 131
begin to have an understanding of the data by looking at the data. This means a consid-
eration not just of arithmetic means but of the diverse statistics that summarize location,
spread, and shape of the distribution. It means viewing graphs that provide an insight into
the spread among scores, the shape of the distribution, and the presence or absence of
outliers that may influence summary statistics. A close examination of the data will often
influence the interpretation of the results, sometimes suggesting the presence of effects
we did not anticipate, or influencing the interpretation of the results of inferential proce-
dures. In some circumstances, it may indicate that critical assumptions have been violated,
providing a spur to consider other forms of analysis, or to at least qualify conclusions.
Researchers typically invest considerable thought, time, and effort into data collection. Too
often, the data analysis involves considerably less thought, consisting of significance tests
or correlation coefficients commonly calculated with similar data. The data analysis process
should be more extensive, and we should begin it by tabulating and graphing more than
continuous random variable cumulative probability function
probability density function central limit theorem
standardized normal distribution z score
standardized normal deviate standard error
unbiased estimator consistent estimator
relative efficiency dot plot
point estimate confidence interval
correlated scores correction for continuity
In some of the following problems, it may be helpful to consult Appendix 5.1, which
contains equations for the variances of linear combinations.
5.1 A standard IQ test yields scores that are normally distributed with (JL = 100 and a =
15. Y is a randomly selected score on the test.
(a) (i) What is p(Y > 130)? (ii) p(85 < Y < 145)? (iii) p(Y > 70)? (iv) p(70 <
Y < 80)?
(b) What scores define the middle 80% of the distribution?
(c) What is the 75th percentile (score such that it exceeds 75% of the scores)?
(d) What is the probability that a randomly selected student will have a score greater
than 115?
(e) What is the probability that the mean IQ of a group of 10 randomly selected
students will be greater than 115?
5.2 On a new test of logical reasoning, the mean and standard deviations for a population
of men are jx = 170 and cr = 50; for women, |x = 200 andCT= 60.
(a) What is the probability that a randomly selected woman will have a score greater
(b) What is the probability that the mean of a group of 9 randomly sampled women
will be greater than 170?
(c) Assume that many pairs of female (F) and male (M) scores are drawn from their
respective populations and a difference score, d = F — M, is calculated for each
pair. What is the mean and standard deviation of the population of such difference
scores?
(d) What is the probability that a randomly selected woman will have a higher score
than a randomly selected man? Note: p(F > M) = p(F — M > 0).
5.3 Assume that X and Y are independently and normally distributed variables. For
X, |JL = 30 andCT= 20; for Y, & = 20 andCT= 16.
(a) What is the probability of sampling an X score (i) <25? (ii) >60? (iii) between
15 and 40?
(b) What is p(X > jx)?
(c) Let W = X + Y. What is p(W > 35)?
(d) An individual's X score is at the 85th percentile (i.e., it exceeds .85 of the pop-
ulation of X scores); this person's Y score is at the 30th percentile of the Y
distribution. What percentage of the population of W scores does the individ-
ual's W score exceed?
5.4 Assume that a population of scores is uniformly distributed between 0 and 1. This
means that f(y) is a line bounded by 0 and 1 with a slope of zero and that F(y), the
probability of sampling a score less than y, equals y. For example, p(Y < .8) = .8.
The mean and standard deviation of this uniformly distributed population are .5 and
1/12.
(a) (i) What is p(Y < .6)? (ii) What is the probability that a sample of two scores
are both less than .6? Express your answer as a probability raised to a power.
(iii) What is the probability that a sample of 20 scores are all less than .6?
(b) Assume we draw many samples of 20 scores and calculate the mean of each
sample. Describe the shape of the sampling distribution. What is its mean and
variance?
(c) On the basis of your answer to part (b), what is the probability that the mean of
a sample of 20 scores is less than .6?
(d) Briefly state your justification for your approach to part (c). Would the same
approach be appropriate in answering part (a), (iii)? Explain.
5.5 In this problem, we use the normal probability distribution to test a hypothesis about
a proportion.
A population of individuals has a disease, is treated, and symptoms are no longer
present. However, .4 of this population suffers a reoccurrence of the symptoms within
1 year. A new drug developed to prevent recurrence of the disease is tried on a sample
of 48 patients. We wish to determine whether the probability of failure (i.e., recurrence
of symptoms) is less than .4.
(a) Let IT equal the probability of failure in the population sampled. State H0 and
(b) Let p equal the probability of failure in the sample. If H0 is true, the mean of
the sampling distribution of p is IT = .4 and its variance is ir(l — ir)/N. In the
study, only 12 of the 48 participants suffered a recurrence of symptoms after
1 year. Use the normal probability distribution to test the null hypothesis. State
your conclusions.
(c) In part (b), we used the normal probability (z) table to test a hypothesis about a
population probability. (i) What assumption about the sampling probability of p
is implied by this procedure? (ii) What justifies this assumption? (iii) Would the
assumption of normality be as justifiable if the sample had only 10 people in it?
Explain.
5.6 A national survey of a large number of college students in 1983 yielded a mean
"authoritarianism" score of 52.8 and a standard deviation of 10.5. For all practical
purposes, we may view these as population parameters.
(a) Suppose we wish to examine whether authoritarian attitudes have increased in
the years since the survey by examining a random sample of 50 students. State
H0, H1, and the rejection region, assuming a = .05.
(b) Assume that the mean of the sample of 50 scores is 56.0. Carry out the significance
test and state your conclusion.
(c) Suppose the true population mean is now 57.00. What is the power of your
significance test?
(d) On the basis of your sample (and assuming the population variance has stayed the
same), what is the 95% CI for the current population mean of authoritarianism
(e) In part (d) you found the 95% CI. What exactly is a 95% CI? What exactly is
supposed to happen 95% of the time?
5.7 We have a population in which p(X = 0) = .8 and p(X = 1) — .2. Let TT = p(X —
1) and 1 - IT = p(X = 0).
(a) (i) Calculate the population mean, ^x, and variance, crx [E(X) and var(X)]. Note
that var(X) = E(X2)- (E(X)]2 and E(X) = (ir)(l) + (1 - ir)(0). (ii) Assume
we draw samples of size 3 from this population. What would be the variance of
the sampling distribution of the mean [var(X)]?
(b) Assume we draw samples of size 3. If we define the outcome of the experiment
as a value of Y where Y = £]X, there are four possible outcomes. Complete the
following table (S2 is the sum of squares divided by N, whereas s2 is the sum of
squares divided by N — 1):
Y P(Y) X bS bS sS 2
* x x
0 .08 — .512
1 (3)(.82)(.2) = .384
2 (3)(.8)(.22) = .096
3 .23 .008
Using the entries in this table, find (i) E(Y), (ii) E ( X ) , (iii) E ( S 2 ) , and (iv) E(s£).
How do E(Y) and E(X} compare with the value of E(X) obtained in part (a)?
How do E(S^) and E(s^) compare with the value of var(X) obtained in part (a)?
What do your answers to parts (d) and (e) say about which sample statistics are
biased or unbiased estimators?
5.8 Two random samples are available from a population with unknown mean. Sample
1 has n1 scores and has a mean of Y1 sample 2 has n2 scores and a mean of y2.
Consider two possible estimates of the population mean, (xy, that are based on both
samples: One estimate is the unweighted mean of the sample means, UM, where
and the other is the weighted mean of the sample means, WM, where
(a) Is UM an unbiased estimator of the population mean? Show why or why not.
(b) Is WM an unbiased estimator of the population mean? Show why or why not.
(c) Calculate the variance of UM if n1 =20 and n2 = 80, and the population vari-
ance, o~2, equals 4.
(d) Calculate the variance of WM if n 1 = 2 0 and n2 = 80, and the population vari-
ance, a2, equals 4.
(e) Is UM or WM a better estimate of the population mean? Why?
(f) Are either UM or WM better estimators of the population mean than Y1 or Y2?
5.9 Assume that we have a treatment (T) and a control (C) population for which \LJ is
larger than JJLC . Assume that both populations are normally distributed and have the
same variance, cr2.
(a) Let T and C be randomly sampled scores from their respective populations. If ^T
is .5(7 larger than JJLC, express the mean and variance of the sampling distribution
of T — C in terms of (JLJ-, (JLC, and CT.
(b) What is the probability that a randomly chosen score from the treatment popula-
tion will be larger than a randomly chosen score from the control population? It
is not necessary to have numerical values for (xj-, JJLC, and a.
(c) Cohen (1988) has suggested that, for two independent groups, we should consider
a medium-sized effect to correspond to a difference of .5o~ between the population
means. He also suggested that small and large effects be considered to correspond
to differences between the population means of .2a and .8a, respectively. We
have assumed a medium effect in parts (a) and (b). Given this assumption, (i) what
is the probability that the mean of 9 randomly chosen scores from the treatment
population will be larger than the mean of 9 randomly chosen scores from the
control population? (ii) What is the probability if the effect is small?
5.10 A population of voters consists of equal numbers of conservatives and liberals.
Furthermore, .9 of the liberals prefer the Democratic candidate in the upcoming
election, whereas only .3 of the conservatives prefer the Democratic candidate.
(a) What is PD, the probability of sampling an individual from the entire population
who prefers the Democratic candidate?
(b) The variance of a proportion p is p(1 — p)/N (see Appendix B for the proof).
With this in mind, what is the variance of the sampling distribution of pD, the
proportion of Democratic voters in a sample of 50 individuals who are randomly
selected from the population of voters?
(c) Suppose you are a pollster who knows that the population is equally divided
between liberals and conservatives, but you do not know what the propor-
tion of Democratic voters is. You sample 50 individuals with the constraint
that 25 are liberals and 25 are conservatives; this is referred to as stratified
sampling.
(i) From the information presented at the start of this problem, what is the
variance of the sampling distribution of PD\L, the proportion of Democratic
voters in a sample of 25 liberals?
(ii) What is the variance of the sampling distribution of PD\C, the proportion of
Democratic voters in a sample of 25 conservatives?
(iii) The proportion of Democrats in the stratified sample is pD = ( 1 / 2 ) ( p D | L +
PD\C). What is the variance of the sampling distribution of P when stratifi-
cation is employed?
(iv) In view of your answers to (b) and (c), (iii) discuss the effect of stratification.
5.11 Following are summary statistics for the total cholesterol scores for the winter (TC1)
and spring (TC2) seasons for men; the data are in the TC file of the Seasons folder
on the CD.
(a) Find the SE of the difference in the means. Reference to Appendix 5.1 may be
helpful.
(b) Using the result in part (a), find the .95 CI for the difference in the two seasonal
(c) Carry out the z test of the null hypothesis of no seasonal effect; a. = .05.
(d) Using any statistical package you have, check your results for parts (b) and (c),
using the data set in the TC file in the Seasons folder on your CD. Note: there
will be a slight difference because most packages conduct a t test, but with large
N, as in this case, the results should be very close.
5.12 (a) Assuming on the basis of the analysis of the winter-spring TC data (Exercise 5.11)
that (Jd = 20, how large should N be to have a .95 CI of 4 points?
(b) Cohen (1988) has defined a standardized effect as d/s d , and he suggested that
effects of .2, .5, and .8 be viewed as small, medium, and large, respectively.
For the data in Exercise 5.11, the standardized effect = 5.241/21.779 = .24, a
fairly small effect although quite significant. Assume that we wish to replicate
our study of cholesterol differences in a new sample of men. If we have only 100
participants available, what is the power to detect a standardized effect of .24?
Assume a one-tailed test with a = .05.
5.13 In the TC file, we created an educational level (EL) variable. If Schoolyr = 1, 2, or
3, EL = 1; if Schoolyr = 4, 5, or 6, EL = 2; and if Schoolyr = 7 or 8, EL = 3. El = 1
corresponds to individuals with a high school education or less, EL = 2 corresponds
to those with education beyond high school but not including a bachelor's degree,
and EL = 3 corresponds to those with a college or graduate school education.
(a) Calculate a .95 CI for the difference in the TC population means between the
EL = 1 and the EL = 2 groups. What can you conclude about this difference
based on the CI?
(b) Another researcher elsewhere in the country wishes to replicate the study of
effects of education on cholesterol levels (TC). The researcher has two EL groups,
each with 100 participants. Assuming that the each population has a = 40, that
a = .05, that the test is one-tailed, and that the investigator wants to detect a
difference of at least 15 points, what power will the test have?
5.14 We decide to run a study of cholesterol levels in a population of patients. On the basis
of the TC data in the present Seasons study, we assume that the population standard
deviation is 30. We would like power = .80 to detect effects of small size (.2o or
6 points) above a level of 200.
(a) What are the null and alternative hypotheses for the proposed study?
(b) What is the specific alternative hypothesis?
(c) How many participants should we recruit for our study? Assume a = .05.
5.15 In this exercise, we use the Mean D variable in the Beck_D file of the Seasons folder.
This is an average of the four seasonal depression scores for those individuals who
were tested in all four seasons. Note that there are missing values of the Mean _D
measure because not all individuals were tested in all four seasons.
(a) Tabulate descriptive statistics separately for male and female participants, and
compare these. Then graph the two data sets any way you choose, relating
characteristics of the plots to the statistics. Comment on location, spread, and
shape.
(b) Using the statistics you obtained, construct a .95 CI for JJIF — JJLM (female - male
Beck_D population means) and decide whether the male and female means differ
significantly at the .05 level. Do you think the assumption of normality is valid?
(c) Outlying scores frequently influence our conclusions. Considering the male and
female data separately, what values would be outliers?
(d) Redo parts (a) and (b) with the outliers excluded. How does this affect the results
in parts (a) and (b)? Note: If the file is sorted by sex and then by Mean_D, outliers
can more easily be extracted.
Linear Combinations
So far, we have considered the means and variances of individual distributions of scores.
However, we are often interested in combinations of scores. For example, suppose each of
the N students in a class takes three tests. We refer to the score of the ith student on test
1 as Yi1, where the first subscript indicates the student and the second indicates the test.
Similarly, Yi2 indicates the score of the ith student on test 2, and so on. From Chapter 2, we
know that the mean of all the scores on test 1 is
where the subscript ".1" indicates that we are averaging over all N of the scores on test 1.
The variance of the scores on test 1 is given by
MEANS OF LINEAR COMBINATIONS 137
Now suppose we want the total score on the three tests for each student. We can express
this as Ti = (+1)Y,1 + (+1)^2 + (+1)7/3; or we might want the average score of the three
tests, 7/. = (+1/3)7,1 + (+1/3)7,2 + (+l/3)7,3; here the subscript"i." means that we are
averaging over all the scores for the ith student. Perhaps we want the difference between test
land test 2. We can express this difference as Di = (+l)7,1 + (—l)7,- 2 .Or the final grade, G,
might give test 3 twice the weight of test 1 and test 2; G, = (+1)7/1 + (+1)7,2 + (+2)7,3-
All of the above are examples of linear combinations of the 7 variables. They are all of
or simply L = £]. Wj Yj, where, for example, in the equation for G, the weights are w\ —
1, vv2 = 1, and n>3 = 2. L is referred to as a linear combination of the 7s because the 7s
are not raised to a power other than 1 or multiplied by one another. The weights can be any
numbers. As the preceding examples illustrate, they need not be equal to one another, or be
integers, or even be positive numbers.
Many statistics of interest to researchers are linear combinations. For example, the
mean of the three test scores for a student is a linear combination for which the weights
each have the value 1/3. To draw inferences about the population parameters estimated by
these statistics, we need to know something about means and standard deviations of linear
combinations. Therefore, we consider these next.
MEANS OF LINEAR COMBINATIONS
Suppose we want the mean of the Li scores in Equation 5.21. This is
The quantity Yj is the mean score on the jth variable. Equation 5.22 indicates that the
mean of a set of linear combinations is a linear combination of the variable means.
The same general rule holds for a population of values of L. The mean of the population
can be represented as the linear combination of the population means. That is,
We can now show why Equation 5.11, |Xy = E(Y) = (Jty, must be true. We know that
because the expected value of each score drawn in a sample is the population mean. There-
fore,£(F) = (l/AO(A^y) = |Jir.
VARIANCES OF LINEAR COMBINATIONS
The Variance of the Sums and Differences of Two Variables
The variance of a linear combination depends both on the variances of the variables and on
the covariance of each pair of variables. To keep things as simple as possible, first consider
two linear combinations of the variables X and Y — T = X + Y and D = X — Y. The
variance of any linear combination, L, is £]( (L, — L)2/(N — 1).Therefore, for the variance
of X + Y,
where SXY — FXYSX^Y, the covariance of X and Y (see Equation 3.5). Therefore,
The variance of the difference scores has a similar form:
VARIANCES OF LINEAR COMBINATIONS 139
The only difference in the two expressions is in the sign of the covariance term. Note that
if the two variables are uncorrelated, both the variance of X + Y and X — Y is sjf + Sy, the
sum of the variances of X and Y.
Analogous expressions hold when we consider population parameters. The population
variances for X + Y and X — Y are
If X and Y are uncorrelated, the variances of both X + Y and X — Y are <r| + a$. These
expressions are important in the development of many inferential procedures.
The General Case
So far we have considered the variances of only very simple linear combinations: only
two variables have been considered, and they have either been added or subtracted. We
now generalize to linear combinations that deal with any number of variables and weights
other than +1 and — 1. Consider N individuals with a scores each, Y1, Y2,..., Ya. The
general linear combination was defined by Equation 5.21 and its mean by Equation 5.22.
The variance of a linear combination, L, can be proven to be
If the variables are independently distributed, all the covariance terms are all 0, and
The corresponding expression for population parameters is
Of particular interest is the variance of the sampling distribution of the mean. Assuming
that the individual scores are independently distributed, and letting wi = 1/N for all i, we
find that substitution into Equation 5.30 yields
Because the variance of the ith score over many samples is a2, the preceding equation
Estimation, Hypothesis
Tests, and Effect Size:
The t Distribution
When we analyze large data sets, such as those used in Chapter 5, it makes little differ-
ence whether inferences are based on the normal or the t distribution. However, in many
studies, either we have a more limited pool of participants available, or we are interested
in a smaller subset of the entire sample. In those cases, the t distribution of Appendix
Table C.3 is better suited to the data analysis. The primary purposes of this chapter are
to review the applications of the t distribution and to discuss the standardized effect size,
a measure of the importance of the difference between means that we observe in our
sample.
To illustrate applications of the t distribution, we again consider only the winter and
spring Beck depression scores in the Seasons data set. More precisely, we subtracted the
score obtained in the spring from that obtained in the winter to get a single change score
for each participant. Whereas in Chapter 5 we drew inferences from a large sample of
such change scores, in this chapter we limit our attention to change scores of partici-
pants under 36 years of age. Seasonal Beck depression scores for both men (sex = 0) and
women (sex = 1) in this age bracket can be found in the Under 36 file in the Seasons
folder on the CD. Table 6.1 presents the depression change score (the Diffl_2 variable
in the file) statistics for the male and female subgroups that meet this age criterion. A
comparison with the values in Table 5.2 will reveal several differences, but our initial
concern is with the differences in sample size. With over 200 participants, we felt justi-
fied in using the normal probability distribution as the basis for calculations of confidence
intervals (CI) and for testing hypotheses. However, with the smaller number of partici-
pants, the standard error (SE) is likely to be considerably more in error as an estimate
of the standard deviation of the sampling distribution of the mean, and, consequently, the
denominator of the z statistic will be in error. The t statistic provides a remedy for this
INFERENCES ABOUT A POPULATION MEAN 141
TABLE 6.1 STATISTICS FOR MALE AND FEMALE WINTER - SPRING
DEPRESSION CHANGE SCORES
Gender = Male Female
N of cases 30 32
Minimum -8.825 -6.197
Maximum 15.838 8.009
Median -0.178 0.000
Mean -0.745 0.747
95% CI Lower -2.391 -0.599
Standard Dev 4.407 3.733
Kurtosis(G2) 6.260 -0.582
Note. From SYSTAT for subjects under 36 years of age.
6.2 INFERENCES ABOUT A POPULATION MEAN
6.2.1 The t Statistic
In Chapter 5, we considered statistics that had the general form of [V — E(V)]crv. In this
expression, V represents a statistic such as the sample mean, or the difference between two
means, and ov is the standard error (SE) of the sampling distribution of that statistic. If
we assume that ov is a constant, only the numerator of the ratio will vary over samples. If
that numerator has a normally distributed sampling distribution, the ratio will be normally
distributed. If N is large, say more than 40, an estimate of ov based on the data will vary
only slightly across samples. In that case, we can replace ov by the estimate and the normal
distribution of Table C.2 will adequately approximate the sampling distribution of the test
statistic. In many experiments, N is not large; consequently, the estimate of the standard
error of the mean (SEM) will vary considerably over samples. In that case, the distribution
of the ratio is the t distribution tabled in Appendix C.3. The t distribution has a more
prominent peak than the standardized normal, and more of the t distribution's area is in its
tails. This means that ifCTis estimated from a relatively small sample, inferences based on
the normal distribution may be in error.
The t statistic has the general definition
where, as before, V refers to an observed variable that estimates a population parameter,
E(V), and sv is the sample statistic that estimates the SE of the sampling distribution of V.
142 6/f DISTRIBUTION
Note that the t statistic is identical to the z statistic, except that the denominator of Equation
6.1 is an estimate of the SE of V rather than its actual value. As in the case of z, it is assumed
that scores are independently and normally distributed.
With respect to the example of Table 6.1, E (V) is the mean of the population of change
scores, |xchange (i.e., (JLwinter — ^spring), V is Ychange, and Sy is the sample standard deviation,
schange, divided by \/N. There is no single t distribution; rather, there is a family of t
distributions whose members look more like the normal distribution as sample size increases.
This increasing approximation to the normal distribution with increasing N reflects the fact
that sv is a consistent estimator of av; as N increases, the statistic sv becomes less variable
over samples and approaches the parameter, <r v . For large N, values of t closely approximate
the values of the standardized normal deviate, z.
It is a slight oversimplification to tie the shape of the t distribution to N. It really depends
on something called degrees of freedom, frequently referred to as df. The concept of
degrees of freedom is closely related to sample size but is not quite the same thing. Because
degrees of freedom are a parameter of other distributions that play a role in data analysis,
they deserve further discussion.
6.2.2 Degrees of Freedom ( d f ) and the t Distribution
The degrees of freedom associated with any quantity are the number of independent ob-
servations on which that quantity is based. The meaning of "independent observations" is
best illustrated by using an example. Suppose that we are asked to choose 10 numbers that
sum to 50. We can freely choose any 9 values, but the 10th must be 50 minus the sum of
the first 9. In this case, there are 10 scores, but because only 9 can be chosen independently,
there are only 9 df. There is a restriction, because the numbers must sum to 50 and that
costs us a "degree of freedom." The same situation occurs when we calculate a sample
standard deviation. This requires us to subtract each score from the mean. However, as we
noted in Chapter 2, the sum of deviations of all scores about their mean must be zero; that
is, Y^ (Y — Y) — 0. Rewriting this last result, we have £] Y = NY. If the sample mean is
5 and N is 10, we have the original example in which the sum of 10 scores must equal 50.
Therefore, the sample standard deviation is based on 10 — 1, or 9, df.
At this point, it looks as if the degrees of freedom are always just N — 1. That is true
if the statistic of interest involves only one restriction. But suppose we draw two samples
from some population; one sample is of size n1and the other of size n2. We want to estimate
the population variance but we have two estimates. As we will see shortly, these can be
averaged; however, the point now is that there are two restrictions if two sample variances
are computed: the sum of the n1 scores in the first sample must equal n1 Y1 and the sum
of the n2 scores in the second sample must equal n2Y2. There are n\ — 1 df associated with
the variance for the first sample and n2 — 1 df associated with that for the second sample.
The degrees of freedom associated with a statistic involving some combination of these two
variances will be df= (n1 — 1) -f («2 ~ 1) = n\ + n-i — 2. The message is that the degrees,
of freedom are not necessarily the number of scores minus 1. Rather
df = number of independent observations
= total number of observations minus number of restrictions on those
In the two-sample example, there are n\ + n2 observations and two restrictions caused by
taking deviations about each of the sample means.
Turn again to Table C. 3. Each row of the table corresponds to a different value of degrees
of freedom. The columns are proportions of the sampling distribution of t exceeding some
cutoff; note that each column is headed by one-tailed and two-tailed proportions. Find the
column corresponding to a one-tailed proportion of .025 (and a two-tailed proportion of
.05) and the row for df= 9. The critical value in the cell is 2.262. This value is exceeded
by .025 of the sampling distribution of t when there are 9 df; .05 of the distribution is
greater than 2.262 or less than —2.262. Now look down the same column to the row labeled
"infinity." The critical value in that cell is 1.96. This means that the probability is .025 of
exceeding 1.96, and the probability of t > 1.96 or t < —1.96 is .05. This is exactly the
critical value in Table C.2, the normal probability table. In general, the critical value of t
decreases as the degrees of freedom increase, rapidly approaching the critical value in Table
C.2 for the normal distribution. The reason for this is that our estimate of a exhibits less
sampling variability as N increases. In short, as N increases, s more closely approximates cr,
and therefore the distribution of t more closely approximates that of a normally distributed
z score. In general, Table C.3 will provide more accurate inferences than Table C.2, although
there is little difference when sample sizes are large. For example, on the basis of the normal
distribution of Table C.2, we calculated the .95 confidence limits for the mean change score
for the 215 female participants to be .035 and .078. If we use the t distribution on 214 df
instead of the normal distribution, the critical value is 1.971 instead of 1.96, the value in
Table C.2—and the confidence limits are .033 and .080.
6.2.3 Confidence Intervals and t Tests:
We first consider a single sample of scores. Specifically, we draw inferences based on the
sample of seasonal depression change scores for women under the age of 36 years in the
Seasons study. Before considering the confidence interval (CI) for the mean change score,
and the test of the null hypothesis of no change, look at the plots of the data in Fig. 6.1.
Both the box and dot plots suggest that the data distribution is not symmetric; in addition,
the dot plot appears flat, in contrast to the theoretical normal distribution having the same
mean and standard deviation. Looking back at Table 6.1, neither the skewness (Gl) nor the
kurtosis (G2) values are very different from the value of zero that would be appropriate
for normally distributed data; neither statistic exceeds its SE by a ratio of 2 or more. This
suggests that the departure from normality may not be a problem. We discuss the possible
consequences of nonnormality later in this chapter.
Let's turn now to the statistics presented in Fig. 6.1. In this one-sample case, confidence
intervals (CIs) and significance tests follow the procedures developed in Chapter 5. The only
difference is that critical values are obtained from the t, rather than the z, distribution. The
general form of the CI in the one-sample case is
where r/v-i,a/2 is the value of t such that a/2 of the distribution on N — 1 dj'lies to the
right of it. In the example of seasonal changes in Beck depression scores, with 32 female
participants under the age of 36, JJL represents the mean of the population of change scores,
Data for the following results were selected according to:
(AGE < 36) AND (SEX= "FEMALE")
One-sample t test of DIFF with 32 cases; Ho: Mean = 0.000
Mean = 0.747 95.00% Cl = -0.599 to 2.093
SD= 3.733 t= 1.132
c/f= 31 Prob= 0.266
Fig. 6.1 Data plots, CI, and /-test results based
on the female data of Table 6.1 (from SYSTAT).
df — 31, and the critical t value for a .95 CI is approximately 2.04 (by linear interpolation
between 2.042 and 2.021). Substituting the sample mean and SE of the change scores into
Equation 6.2, we have
Completing the calculations, we obtain the values reported in Table 6.1 and Fig. 6.1: —.599
and 2.093.
Note that the CI for the sample of women under 36 years of age is much wider than
that obtained for the entire sample of 215 women (those limits were .033 and 1.080). One
contributing factor to the reduced precision of the estimate is the slightly larger value of s
in the analysis of the smaller data set. A more important factor is the difference in sample
sizes, which affects the CIs in two ways. First, with the smaller sample, the critical / value
is 2.04, rather than 1.97. Second, and more important, s is divided by 14.663 (the square
root of 215) in the earlier analysis, but by only 5.657 (the square root of 32) in the analysis
of the under-36 group. Therefore, even if s were the same in both analyses, the SE would
be roughly 2.6 times larger for the smaller sample and the CI would be that much larger.
To improve our estimates of population parameters, we need to do all we can to reduce
variability and we should collect as much data as is practical.
THE STANDARDIZED EFFECT SIZE 145
Because the CI contains zero, it should be evident that a two-tailed test of //o: ^change = 0
at the .05 level will not yield a significant result. The direct test parallels that developed in
Chapter 5; the only difference is that we replace a by an estimate, s. We calculate
and, if the test is two tailed, reject HQ if \t\ > fy/-i,a/2- In the present example, jLihyp = 0, and
the denominator is .660, the SE in Table 6.1. Therefore, t = .747/.660 =1.132, which is
much smaller than the critical value of 2.040. As reported in Fig. 6.1, the actual probability
of obtaining a value oft greater than 1.132 or less than —1.132, assuming HQ to be true, is
.266 . If we were to reject HQ based on these results, there would be more than a one in four
chance of making a Type 1 error. Generally, researchers consider these rather poor odds.
6.3 THE STANDARDIZED EFFECT SIZE
The mean difference between the winter and spring depression scores for women under
the age of 36 years (.747) is the raw effect size. The advantage of the raw effect size is
that the units are on the original scale and therefore differences on that scale should be
meaningful to the researcher. The disadvantage of the raw effect size is that it is on the
original scale, and therefore it is difficult to make comparisons with results of research
conducted with other measures of depression. When comparing effects across groups, even
if the same measurement scale is used, we often find it useful to consider differences
between group means relative to their standard deviations. Because differences in variability
and in the measurement scale make direct comparisons of raw effect sizes difficult to
interpret, the standardized effect size,1 Es, is an important tool for understanding our
data. It provides a scale-free index of the importance of the effect, something that neither
CIs on the raw effect nor the p values associated with t tests can do. Furthermore, Es
provides information required for estimates of the power of hypotheses tests, and it is used
in meta-analyses, analyses that combine results from several studies. The CI for the raw
effect, although informative, is dependent on the original measurement scale. The p value
is often misleading; very small effects, perhaps unimportant in any practical sense, may be
very significant because the sample sizes are large and, conversely, large effects may not be
significant because the sample was too small, or variability too great, to have much power.
It is not unusual to find in a comparison of two variables that the one associated with the
lower p value actually has the smaller effect size.
There is increasing agreement that measures of effect size are important and should
be reported along with the results of statistical tests. A growing number of journals now
explicitly require that measures of effect size be reported, and the fifth edition of the Publi-
cation Manual of the American Psychological Association (2001) states, "For the reader to
fully understand the importance of your findings, it is almost always necessary to include
some index of effect size or strength of relationship in your Results section" (p. 25).
6.3.1 Estimating ES
In the example of the seasonal change scores, we conceived of a population of such scores.
The mean of that population is ^change (i.e., M-winter — ^spring), and its standard deviation is
^change- In general in the one-sample case, we define the standardized effect score as
where (jLhyp is the mean assuming the null hypothesis to be true and \IA is the mean under
an alternative hypothesis. In the example of the change scores, |Xhyp = 0, and JJL^ is the true
mean of the population of change scores, |xchange- In the example summarized by Table 6.1
and Fig. 6.1, Es = ((^change — 0)/achange and is estimated as
In the example of the seasonal change scores for the sample of 32 women under the age
of 36, we substitute the mean and standard deviation of the seasonal change scores (from
Table 6.1 or Fig. 6.1) into Equation 6.5; then,
In the one-sample case, there is a simple relation between£5 and t:
A rough rule of thumb suggested by Cohen (1988), and widely adopted, is that effect sizes of
.2, .5, and .8 should be considered small, medium, and large, respectively. According to that
guideline, the seasonal change in the mean Beck scores is small relative to the variability in
the change scores. Note that "small" does not necessarily mean "unimportant." If we were
testing a new cancer treatment, even a small improvement in remission rate, or increase in
life expectancy, might be worthwhile. In theoretical work, small effects—particularly when
unexpected, or not predicted by a competing theory—might prove important. Cohen's
guidelines are just that—guides, not mandates, and each effect size should be evaluated in
terms of the researcher's goals and knowledge of the relevant research and theory.
Confidence limits for E$ may aid our evaluation of the effect size. For example, if E'$ is
.75, Cohen's guidelines would suggest it is large. However, it is just an estimate; we would
feel surer that the population effect size was large if confidence limits were close to this
value than if they were widely separated. Steiger and Fouladi (1997; also see Gumming &
Finch, 2001) describe one method for calculating limits on Es. We have included a brief
illustration of this method in the Supplementary Materials folder of the accompanying CD.
Hedges and Olkin (1985) describe a second, approximate method, and the entire August,
2001 issue of Educational and Psychological Measurement is devoted to the topic of CIs
for measures of effect size. Interested readers may consult these sources.
6.3.2 p Values and Effect Sizes
As we suggested earlier, the significance level, or p value, obtained in a study can be
misleading about the importance of an effect, or about the relative size of effects in two
studies. Table 6.2 presents the results of two experiments that differed only in the number
of participants and in the dependent variable; in both cases, two scores were obtained
from each participant, allowing a comparison of two treatments. The higher t and lower
p values in Experiment 2 might lead us to believe that the measure in Experiment 2 was
more sensitive; the larger raw effect in Experiment 2 would seem to support that inference.
However, the standardized effect size is almost twice as large in Experiment 1, indicating
POWER OF THE ONE-SAMPLE / TEST 147
TABLE 6.2 COMPARISON OF (TWO-TAILED) P VALUES
AND EFFECT SIZES
Experiment 1 Experiment 2
N 16 64
F,-F2 15 40
s 33 152
t 1.818 2.105
P .089 .039
Es .455 .263
that Experiment 1 may have lacked power to reject the null hypothesis because of the
relatively small sample. Note that, according to Cohen's (1988) guidelines, the effect in
Experiment 1 is medium whereas that in Experiment 2 is small.
6.4 POWER OF THE ONE-SAMPLE f TEST
Here we consider two uses of power functions: The post hoc calculation of power refers
to the calculation of the power of the hypothesis test to detect an effect estimated from
previously collected data; the a priori calculation of power refers to the estimation of the
sample sizes needed to achieve a specified level of power to detect a specific value of Es. In
terms of the example of the seasonal depression change scores of the 32 women under age
36, we might wish to know what power our experiment had to reject HQ if the population
effect size was .2, the value of Es we estimated from our data. Once we find that value of
power, assuming it is lower than we wish, we might determine what sample size would be
needed in subsequent research to achieve a specific value of power that was higher than
that in the study already run. Of course, a priori calculations do not require that we have
previously collected data. Any basis for specifying an effect size can be used. If effect sizes
in an area of the literature are all small, we might specify the effect size for a new study as
.2, and calculate the sample size needed to achieve some desired level of power.
Power can be calculated by several standard software programs, including SAS's
CDF module, SYSTAT's Design of Experiments (under "Statistics") module, and SPSS's
NCDF.F program (under "Compute"). Those lacking access to these packages will find
programs for calculating power freely available from several Internet sources.2 We describe
the use of two such sources—a Web site developed by the UCLA statistics department,
and a downloadable program, GPOWER. However, before illustrating the calculations, we
consider the concept of the power of the t test.
6.4.1 The Noncentral t Distribution
As with the binomial (Chapter 4) and normal probability (Chapter 5) tests, we can conceive
of two distributions: one when the null hypothesis is true and one when it is false by a
specifiable amount. For example, we have one distribution corresponding to H0: Es = 0
and an alternative distribution corresponding to HA: Es = .2. When the null hypothesis is
true, the t distribution has a mean of zero. The area exceeding any particular value of t
depends on the degree of freedom, and this "tail probability" may be found in Appendix
Table C.3. We refer to the distribution of t under the null hypothesis as the central t
distribution. To calculate the power to reject a specific alternative hypothesis, we also
must consider the alternative distribution; for example, we must consider the distribution
of t when Es = .2. In this case, the mean of the t distribution is no longer at zero, and
the distribution of t is referred to as a noncentral t distribution. As with other statistical
tests, we first determine the critical region—the values of t that lead to rejection of the
null hypothesis. The critical value can be obtained from Table C.3, or from most statistical
software packages. Once we know the rejection region, we can calculate power by finding the
probabilities of the values in that region, assuming the alternative distribution. This second
step is more difficult than it was with the normal probability test because the noncentral t
distribution is not merely displaced from the null distribution; it has a different variance and
The location, variance, and shape of the noncentral t distribution is determined by
the degree of freedom and a noncentrality parameter, 8 (the Greek letter delta). This
parameter incorporates information about variability (a), sample size (N), and the effect
size under the alternative hypothesis. Assuming a sample of N difference scores, as in the
example of the change between winter and spring depression scores, we see that the formula
for 8 is
where (JLChange is the mean of the sampled population of difference, or change, scores and
o~change is the standard deviation of that population. The noncentrality parameter is closely
related to the standardized effect size; when considering a population of change scores, we
see that ES = M-change/o"; therefore, we can rewrite Equation 6.7 as
Some software programs require a value of 8 in order to calculate power, whereas others
require ES (or, equivalently, Cohen's d) and the value of N (or the degree of freedom). In
either case, we need to estimate the effect size. We can estimate 8 by first estimating ES
and then using Equation 6.8. However, from Equations 6.6 and 6.8, we can see that 8 is also
estimated by the value of the test statistic, t.
6.4.2 Post Hoc Power Calculations
Previously, we found that the standardized effect of the change in seasons (from winter to
spring) on the depression scores of 32 women under 36 years of age was .2. We determine
the power of the experiment to detect this effect size by using two different methods based
on software freely available on the Internet.
The UCLA Calculator. One useful Internet site provided by the UCLA department of
statistics at this time is http://www.stat.ucla.edu/calculators/cdf/. When you enter this site,
a table appears. The rows provide a choice of distributions, including the normal and the
central t (labeled "Student"), and the columns allow calculations or plots based on either
POWER OF THE ONE-SAMPLE f TEST 149
the cumulative or probability density functions. Assuming that we have already obtained
the critical value from Appendix Table C.3 (if not, we can obtain it by clicking on the cell
in the "Student" row and "CDF" column), click on the CDF (cumulative distribution func-
tion) calculator in the "Noncentral Student" (noncentral t distribution) row of the table.
Then, follow these steps:
1. For the "X value," provide the critical t required for significance. For 31 df, a — .05,
and a one-tailed test of H0: (x = 0 versus HI: jx > 0, the critical t is 1.696.
2. Type the degrees of freedom (31) in the "Degrees of Freedom" row.
3. Type the estimate of 8 in the "Noncentrality Parameter" row. In the current example,
ES = .20 and N = 32; then from Equation 6.8 our estimate of 8 is 1.1314. Power
is the probability of exceeding the "X value" in a noncentral t distribution whose
location is determined by the value of 8.
4. Type ? in the "Probability" row and click on "Complete Me!"
The probability returned is always the probability of a value less than the critical t. If we were
to look at a plot of the alternative (noncentral) t distribution we specified, the probability
returned would correspond to the area to the left of the critical value of t. If the critical t is
positive, this probability is (b,the probability of accepting a false null hypothesis. Therefore,
power is obtained by subtracting that value from 1. If the critical t is negative, the values to
the left of the critical t form the rejection region and, in this case, the probability returned
will correspond directly to power. In the current example, completing the table returns the
value .705 in the Probability row, so the power is 1 — .705 = .295. Verify that if we estimate
the power for Es = .2 with a new N of 120, we obtain a value of .703 (remember that
the critical t should be based on the new degrees of freedom, and 8 will also change because
of the change in N).
GPOWER. An alternative that is versatile, simple to use, and can be down-
loaded to your own computer is the free software, GPOWER, available at the
Web site http://www.psychologie.uni-trier.de:8000/projects/gpower.html. Documentation
(Erdfelder, Faul, & Buchner, 1996) is available at this site but the software is quite simple
to use. It can be downloaded from the World Wide Web, and it requires less than half of
the space of a standard diskette. Both Macintosh and PC versions are available. We illus-
trate GPOWER's use, continuing with the example of the hypothesis test based on seasonal
change scores. Figure 6.2 presents a reproduction of the screen for a post hoc calculation
in the one-sample (or correlated-scores) case.
Moving to the main screen, we click on "Tests," which brings up a menu that includes
the t for two means, the t test of the correlation coefficient, "Other t Tests," chi-square tests,
and F tests. In the present instance, although we have two seasonal means, they are not
independent because they are based on the same set of participants. We really have a single
sample of change scores. Therefore, click on "Other t Tests." Then:
1. Click on "Post hoc" under "Analysis" to determine what power the t test had to
reject H0: |xchange = 0 against the specific alternative Es = .2 (the effect size we
estimated from our data).
2. Select "One tailed" from the choice of one and two tails.
3. In the choice between "Speed" and "Accuracy," choose "Accuracy" unless you
have a very slow computer.
150 6/t DISTRIBUTION
Fig. 6.2 GPOWER screen for a post hoc analysis.
4. Double click on "Effect size"3 and type ".2."
5. Enter the values of alpha (.05), N (32), and df(31).
6. Click on "Calculate" and the question marks on the screen are replaced by
Delta = 1.1314
Critical t(31) = 1.6955
Power = .2952
The delta value is the noncentrality parameter that we obtained by using Equation 6.8 and
entered into the UCLA calculator. The power is the same low value obtained with the UCLA
software. To increase power, we must either use a less variable measure of depression than
the Beck scale or increase the sample size.
6.4.3 A Priori Power Calculations
At this point, we have an estimated standardized effect size that was not large enough to
allow rejection of H0. We might wish to ask the following question: If the actual effect was
.2, what sample size should we use in future research in this area to achieve a power of,
say, .8? With N = 32, power was about .3; it is evident that to increase power to .8, we will
need many more participants. Let's see just how many "many more" is.
GPOWER has an a priori option under "Analysis" but this is not available for "Other
t Tests." However, by trying various values of N and degrees of freedom (= N — 1) in the
"Post hoc" analysis, we quickly find that N = 155 provides .80 (actually .7979) power.
Although the UCLA calculator does not calculate the N needed directly, as with GPOWER
POWER OF THE ONE-SAMPLE t TEST 151
we can try various values to obtain the desired power. Note that for each value of N we
try with the UCLA calculator, we would have to insert values of the critical t, degrees of
freedom, and estimated 8 based on that value.
Our calculations indicate that if we are serious about studying seasonal depression
score changes, either we are going to have to collect a large data sample, or we are going to
have to improve our measuring instrument so as to reduce variability. Another possibility
is to attempt to standardize data collection to reduce variability. For example, some of the
winter scores were collected early in the winter; others were collected later. The same is
true, of course, for depression scores placed in the spring category.
6.4.4 Consequences of Violations of Assumptions
The inferences we have drawn thus far in this chapter rest on two assumptions: first, the
scores are independently distributed, and second, the population distribution is normal. We
discussed these assumptions in Chapter 5 with respect to the use of the normal probability
tables. As we noted there, it is reasonable to view the seasonal change (winter- spring) scores
as independently distributed. We also noted in Chapter 5 that, although the population was
unlikely to be normally distributed, the sampling distribution of the mean would tend to be
normal because each mean was based on 215 scores. However, the sampling distribution of
the mean will be less well approximated by the normal distribution in the current example, in
which we have only 32 scores. In general, with samples of approximately this size, problems
may arise if the population distribution is very skewed. We have drawn 10,000 samples of
sizes 10 and 40 from the right-skewed exponential distribution depicted in Fig. 6.3. This
distribution has a high peak at its minimum point and then the density quickly decreases
as the scores increase. The population from which we sampled had a mean of one and we
tested H0: |x = 1 at the .05 level. The results of these computer experiments can be seen in
Table 6.3. In each case, the theoretical error rate is .05. We can see that when the alternative
was H\: JJL > 1, there were far too few rejections. When the alternative hypothesis was H1:
Fig. 6.3 An exponential distribution.
TABLE 6.3 TYPE 1 ERROR RATES FOR THE DISTRIBUTION OF FIG. 6.3 (a = .05)
Sample Size (N) Alternative Hypothesis Type 1 Error Rate
10 |JL > 1 .014
10 IJL < 1 .131
10 u=1 .097
40 IJL > 1 .025
40 JX<1 .096
40 M=l .072
(JL < 0, there were many more than .05 rejections. Two-tailed tests also resulted in inflated
Type 1 error rates.
Although our data are frequently skewed, they are rarely as skewed as in the exponential
distribution plotted in Fig. 6.3. The resulting Type 1 error rates of Table 6.3 represent a worst-
case scenario. Furthermore, the situation is clearly better when N = 40 than when N = 10.
If we used still larger samples, the empirical error rates would eventually stabilize at .05
even in the exponential condition; this is a consequence of the central limit theorem. Larger
data sets not only yield increased power and narrower CIs, but also tend to offset clear
violations of the normality assumption.
If there is evidence of extreme skew, efforts should be made to collect larger samples.
If small samples are drawn from very skewed populations, there are few remedies. One
possibility is to transform the data by performing some operation on all the scores and then
applying the t test. For example, the logarithm of exponentially distributed scores will tend
to be more nearly normally distributed than will the original scores. We have more to say
about transformations in Chapter 8.
Departures from normality may also affect the power of the t test and the precision of
interval estimates. For example, if the distribution of scores is symmetric but has a longer
tail than the normal distribution, the SEM may be large and, consequently, power may be
low and CIs may be wide. Precision of estimation and the power of hypothesis tests are
often improved by "trimming" the data, which consists of deleting the most extreme scores
prior to calculating a modified t statistic (Wilcox, 1997). An alternative to the t test based
on the ranks of the scores, the Wilcoxon Signed Rank Test, may also improve power. This
alternative to the standard one-sample t test is discussed in Chapter 13.
6.5 THE t DISTRIBUTION; TWO INDEPENDENT GROUPS
Looking back at Table 6.1, we see that the average change scores for men and women seem
quite different. The mean winter-spring change in the Beck depression score is —.745 for
men but .747 for women. In other words, the means of the seasonal change scores suggest
that men in this age group are more depressed in spring than in winter, but women are more
depressed in winter than in spring. The medians, however, suggest a different picture; the
male median is negative (—.18) but close to zero and the female median is zero. Figure 6.4
presents box plots of the seasonal change scores for the two groups that should help clarify
the reasons for the discrepancy between comparisons of medians and means. Recall that the
horizontal line dividing the box into two segments represents the median and that the ends of
THE t DISTRIBUTION: TWO INDEPENDENT GROUPS 153
Fig. 6.4 Box plots of male and female seasonal
change scores.
the box are the hinges, approximately the 25th and 75th percentiles. In the female data, the
25% of the scores between the median and the lower hinge are mainly negative but close to
zero, whereas the 25% of the scores between the median and the upper hinge include values
further from the median, thus moving the mean in the positive direction. The situation is
reversed in the male data, where the scores between the median and the upper hinge tend to
be closer to zero (with the exception of one extreme outlier represented by a small circle)
than the mostly negative values between the median and the lower hinge. The other aspect
of the box plot that merits our attention is that there is a considerable range of scores, and
there are outliers in both directions in the male data. In view of the apparent variability, and
the somewhat conflicting picture presented by means and medians, we will further analyze
the data.
Before we can calculate confidence limits or conduct significance tests, we need to
consider the relevant sampling distribution, its SE, and how that SE should be estimated.
We turn next to these matters.
6.5.1 The SE of the Difference Between Two
Independent Means
Suppose we drew many independent random samples of size n1 and n2 from two inde-
pendently and normally distributed populations of scores. If we carried out this sampling
procedure, we could compute a difference between the means for each pair of samples
drawn. The standard deviation of the sampling distribution of this difference, the SE, is the
quantity to be estimated by the denominator of the t in the two-group study. The issue is
how to calculate that estimate.
In Chapter 5, we noted that if the n1 + n2 scores are independently distributed, the SE
of the sampling distribution of the difference between the two group means is (Equation
The t distributions of Appendix Table C.3 require one additional assumption, that the two
population variances are equal; that is,
This is usually referred to as the assumption of homogeneity of variance, or homoscedas-
ticity. Given this assumption, we can rewrite the SE of the sampling distribution of the
difference between the means as
We have a single population variance (a2) and two possible estimates of it—the variances
of the two groups sampled in an experiment. To obtain the best single estimate ofCT,we need
to average the two group variances and take the square root of the result. Because variance
estimates are consistent statistics, the estimate based on the larger group is more likely to
be close to the true variance. Therefore, the best estimate of cr2 is a weighted average of the
two group variances. This is often referred to as the pooled-variance estimate, or spooled,
and it is calculated as
Note that the weight on each group variance in Equation 6.9 is obtained by dividing the
degrees of freedom for that group by the sum of the degrees of freedom for the two groups
[(n1 — 1) + (n2 — 1)]• The degrees of freedom rather than the ns are used in these weights
because this yields an unbiased estimate of o-y{ _y, • The pooled-variance estimate can also
be written as
where the SS for the j th group is a sum of squared deviations of the scores about the mean
of the group; that is,
This quantity is usually referred to as the sum of squares, and it plays an important role in
the remaining chapters.
We can now state the expression for the estimate of the SE of the sampling distribution
of the difference of two independent means:
When n1 = n2 = n, as is frequently the case in experimental research, Equations 6.10 and
6.12 simplify:
To illustrate the calculation of Sdiff, let's substitute values from Table 6.1. In other
words, let's get a numerical estimate of the SE of the sampling distribution of the difference
between the means of the male and female change scores. From Equation 6.12,
6.5.2 CIs and t Tests
With a formula for the SE of the difference between the means, we can estimate confidence
limits on the difference in mean change scores for the under-36 groups of men and women.
We begin by stating the general form of the CI; if the confidence level is 1 — a , then
and the confidence limits are
where, as usual, V is the statistic that estimates the relevant population parameter, sv is the
estimate of the SE of the sampling distribution of V, and r^/ a/2 is the value of t exceeded
by a/2 of the t distribution on n1 + n2 — 2 df. Replacing the V in Equation 6.14 by the
difference between the male and female change means, we have
We can now substitute values from Table 6.1 to obtain the .95 limits on |xf — JJLM, the
difference between the means of the two populations of change scores. The difference in
the group means is YF - YM = .747 - (-.745) = 1.492, and the estimate of the SE of
this difference, Sdiff, was computed previously as 1.035.
Turning to Appendix Table C.3, we see that the two-tailed t required for significance
at the .05 level when df = 60 is 2.000. Substituting these values into Equation 6.15, we find
that the confidence limits are
The interval bounds give a sense of the likely range of differences between the male and
female change scores. The interval is wide, more than 4 points, indicating that our estimate
of the difference in mean change scores is not very reliable. We also note that the interval
contains zero. Therefore, despite the fact that the mean seasonal change score was positive
for women and negative for men, we cannot reject the null hypothesis of no difference.
However, the wide interval suggests that power may be less than desirable. We consider the
power of the significance test in Section 6.7.
A direct test of H0: [i\ — (Jt2 = 0 against the alternative H1: |xi — 1^2 =£ 0 follows from
the preceding developments. The test statistic is
Substituting numerical values, we have
Because this value is less than 2.00, the result is clearly not significant. However, the
CI is quite wide, wider than any encountered previously in either Chapter 5 or 6, indicating
that the point estimate of the difference between the sexes may be in considerable error. The
wide CI also suggests that power to reject H0 will be low, if we assume the current group
sizes and the estimated values of the population parameters. In the following sections, we
proceed as we did in Sections 6.3 and 6.4, calculating first the standardized effect size for
the two sample case and then the power of the significance test.
6.6 STANDARDIZED EFFECT SIZE FOR TWO
Following the developments of Section 6.3, we define the standardized effect size Es as
where |j>1 and 1x2 are the actual population means, and A (Greek uppercase delta) is the
population difference assuming H0 is true; A is usually, but not always, zero. In the two-
sample case, ES is estimated as
In the example of the contrast of male and female seasonal change scores, we previously
determined that YF — YM = 1-492 and spooled = 4.073. Therefore,
Following Cohen's (1988) guidelines, we find that the standardized effect size of gender
on the seasonal change in Beck depression scores for individuals less than 36 years of age
falls somewhere between small and medium.
Although the measure of effect size we have presented is the most common one, others
are possible and, particularly when homogeneity of variance is suspect, such alternatives
should be considered. Grissom and Kim (2001) provide an excellent review of the as-
sumptions involved in calculating effect size, and they discuss the pros and cons of several
alternatives to Es.
POWER OF THE TEST OF TWO INDEPENDENT MEANS 157
6.7 POWER OF THE TEST OF TWO INDEPENDENT MEANS
If the actual effect size in the population equals the estimate, .37, what is the probability
of rejecting the null hypothesis? In other words, if the experiment was replicated many
times and the true Es was .37, in what proportion of those replications would the null
hypothesis be rejected? The underlying process of obtaining a numerical value of power
involves the same two stages involved in finding power for the normal probability and
one-sample t tests. Assuming H0 to be true, and given the nature of the alternative (one
or two tailed) and the value of a, we use Appendix Table C.3 to decide on the criti-
cal region, those values of t that will lead to rejection of H0. Then, we find the proba-
bility of obtaining a value of t that falls within that region under a specified alternative
Power calculations can be performed by several standard statistical software packages,
as well as by programs at the UCLA Web site or by using GPOWER. With GPOWER,
use the default "t test for means." As in the one-sample case, we again need to specify
an effect size. One possibility is to use the value of Es determined from the data (.37 in
our example). This is the practice followed by many researchers. Alternatively, because
.37 falls about midway between Cohen's (1988) guidelines for small and medium effects,
we might compute power by assuming ES = .2 or .5. Whatever value is used, the process
of calculating power is straightforward with GPOWER. Indicate that the calculation is
post hoc and enter the effect size value. The default value of alpha is .05, but this can
be changed if we want to test at a different significance level. Complete the input by
inserting the values of n1 and n2, and by indicating whether a one- or two-tailed test is
desired. Then click on "Calculate." For a two-tailed test against the specific alternative,
HA- Es = .37, power = .29. It appears that even if the null hypothesis was false, we had
little power to reject it. This is consistent with the wide interval we noted when the CI was
calculated.
Using the noncentral Student CDF calculator at the UCLA Web site to calculate power
yields the same result. For the two-tailed test with 60 df, the critical values of the t distribution
are ±2.00. Entering 2.00 for "X Value," ? for "Probability," 60 for "Degrees of Freedom,"
and the observed t, 1.44, for "Noncentrality Parameter" returns a probability value of .71.
Because this is (3, the area below the cutoff, the power associated with the upper region
of rejection is 1 — .71 = .29, the same value obtained by using GPOWER. No additional
power is obtained by using the lower region of rejection. If we insert —2.00 instead of
+2.00 for "X Value," the probability returned is .000. In this instance, because a negative
value of t was inserted, the probability returned corresponds directly to power. No power
is gained by considering the area below the lower critical value.
Ideally, we should determine the n required to obtain a specific level of power prior
to collecting our data. From previous work with the Beck depression scores, we might be
aware that small effects were generally obtained. In that case, using GPOWER, we set the
effect size parameter at .2, select "A priori" under "Analysis," indicate the desired level of
power, and click on "Calculate." For a two-tailed test with a = .05, if Es = .2, GPOWER
estimates that we will need n = 394 in order for the power to equal .80. No a priori power
calculations are directly available with UCLA's noncentral Student calculator, although
we could use it to find the power for different sample sizes. If we wish to use the UCLA
noncentral t calculator (or the noncentral t functions in statistical packages such as SPSS)
to calculate a priori power for two-group t tests, we should note that the noncentrality
parameter can be estimated from
6.8 ASSUMPTIONS UNDERLYING THE TWO-GROUP f TEST
When using the t distribution as a basis for inferences about two independent population
means, we assume that the two populations of scores are independently and normally dis-
tributed and have the same variance. As we discussed earlier, violation of the independence
assumption will often result in distorted Type 1 error rates. Nonnormality is less of a prob-
lem, at least in large samples. The impact of heterogeneity of variance also depends on the
absolute sample sizes, as well as on the relative sizes of the two samples. Let's consider
these issues more closely.
6.8.1 The Assumption of Normality
From our discussion of the central limit theorem, we know that the sampling distribution
of the difference of means approaches normality as the combined sample size (n1 + n2)
increases. As a consequence, the actual Type 1 error rates associated with the test statistic
will closely approximate the values for the t distribution in Table C.3 if the combined
sample size is moderately large. "Moderately large" may be as small as 20 if n\ — n2 and
if the two populations have symmetric distributions, or even if they are skewed but have the
same direction and degree of skewness. Our rather liberal attitude with respect to skewness
may seem surprising in view of the fact that the one-sample t required quite large ns to
achieve honest Type 1 error rates when the parent population was skewed. However, here
we are concerned with the sampling distribution of the difference between independent
means. If two populations are skewed in the same direction, and if the samples are equal in
size, then the differences in sample means are as likely to be positive as negative, and the
sampling distribution of those differences will tend to be symmetric. For most situations the
researcher will encounter, combined sample sizes of 40 should be sufficient to guarantee
an honest Type 1 error rate.
When populations are skewed or have outliers, the sampling distribution of Y \ —Y 2
will tend to be long tailed. In such cases, estimates of the difference between population
means will be less precise, and the t test will be less powerful than when the normality
assumption holds. As in the one-sample case, trimming extreme scores (Wilcoxon, 1997)
and tests based on ranks often will be more powerful than the t test. We consider the latter
approach in Chapter 13.
6.8.2 The Assumption of Homogeneity of Variance
The denominator of the equation for the two-sample t test is based on the pool of two vari-
ance estimates (Equation 6.9); the underlying assumption is that the two group variances
estimate the same population variance. If this is not true—if the population variances are
heterogeneous—then the sampling distribution of the t statistic of Equation 6.16 may not
ASSUMPTIONS UNDERLYING THE TWO-GROUP f TEST 159
TABLE 6.4 TYPE 1 ERROR RATES FOR THE f TEST AS A FUNCTION
OF POPULATION VARIANCES AND SAMPLE SIZES
HI r>2 cr,2/a22 a =.05 a = .01
5 5 4 .060 .014
5 5 16 .061 .019
5 5 100 .066 .024
15 15 4 .056 .010
15 15 16 .054 .015
15 15 100 .059 .017
5 10 4 .095 .031
5 10 .25 .021 .003
10 15 .25 .040 .006
have a true / distribution. Table 6.4 gives some sense of what may happen in this case.
We drew 2,000 pairs of samples of various sizes from two normal populations with iden-
tical means but different variances. Proportions of rejections for a = .01 and a = .05 are
presented.
Several points about the results should be noted. First, if the two sample sizes are
equal, the difference between the empirical and theoretical Type 1 alpha rates tends to
be at most 1%, except when n is very small (i.e., n = 5) and the variance ratio is very
large (i.e., of/cr^ = 100). Second, when ns are unequal, whether the Type 1 error rate
is inflated or deflated depends on the direction of the relation between sample size and
population variance. The reason for this can be understood by considering the fact that the
denominator of the t is based on a weighted average of two variance estimates; the weights
are proportions of degrees of freedom. Therefore, when the larger group is drawn from
the population with the larger variance, the larger variance estimate receives more weight
than the smaller estimate. The denominator of the t test tends to be large and the t small;
the rejection rate is less than it should be. Conversely, when sample size and population
variance are negatively correlated, the smaller variance estimate gets the larger weight;
the denominator of the t statistic tends to be small and the t large; the rejection rate is
inflated.
Unequal sample sizes should be avoided when possible. However, we recognize that
there will be many cases in which sample sizes will differ, often markedly. For example,
the response rate to questionnaires may be quite different for two populations such as male
and female, or college and noncollege educated. We do not advocate discarding data from
the larger sample; this would increase sampling variability for statistics computed from that
sample. Instead, we recommend an alternative to the standard t test. One such alternative
is a t that does not use the pooled estimate of the population variance. The denominator of
this statistic would be that of the z test for two independent groups with variance estimates
instead of known population variances. We define
This statistic is sometimes referred to as Welch's t (Welch, 1938). If the scores have been
drawn from normally distributed populations, t' is distributed approximately as t but not
with the usual degrees of freedom. The degrees of freedom are
where s4 is the square of the variance, s2. The degrees of freedom are rounded to the nearest
integer when the value of t' is evaluated or when CIs are obtained.
Most statistical packages compute values of both t' and t when an independent-groups
t test is performed. For example, SYSTAT outputs both a separate-variance t and a pooled-
variance t, and SPSS outputs a result for "Equal variances not assumed" and "Equal vari-
ances assumed." The former is the t' of Equation 6.19 and the pooled-variance t is the
standard t statistic of Equation 6.16. Tables 6.5a and 6.5b present SPSS's output of the t
statistics and CIs for the comparison of mean seasonal depression change scores of men and
TABLE 6.5a THE t TEST COMPARING MEAN SEASONAL CHANGE SCORES FOR MEN AND
WOMEN: GROUP STATISTICS
Std. Error
SEX N Mean Std. Deviation Mean
DIFF MALE 30 -.7450 4.4074 .8047
FEMALE 32 .7469 3.7328 .6599
Note. Output is from SPSS.
TABLE 6.5b THE t TEST COMPARING MEAN SEASONAL CHANGE SCORES FOR MEN AND WOMEN: INDEPENDENT
SAMPLES TEST
Levene's Test for
Equality of Variances t-test for Equality of Means
95% Confidence
Interval of the
Mean Std. Error Difference
F Sig. t df Sig. (2-tailed) Difference Difference Lower Upper
DIFF Equal variances .038 .847 -1.441 60 .155 -1.4919 1.0350 -3.5623 .5785
assumed
Equal variances - 1.434 57.003 .157 -1.4919 1.0406 -3.5758 .5920
not assumed
Note Output is from SPSS.
CONTRASTS INVOLVING MORE THAN TWO MEANS 161
women under 36 years of age. The values of the t based on the pooled-variance estimate
("Equal variances assumed") and that of t' ("Equal variances not assumed") are almost
identical. This is because the group sizes are very similar, as are the standard deviations.
Further evidence of homogeneity of variance is provided by Levene's (1960) test, which
has a clearly nonsignificant result. This test evaluates the difference between groups with
respect to the average absolute deviation of scores about their respective means. We have
more to say about this test in subsequent chapters, in which we also discuss the Brown-
Forsythe test of homogeneity of variance (Brown & Forsythe, 1974a). That test is similar to
Levene's but is based on the average absolute deviation of scores about the median instead
of the mean.
6.9 CONTRASTS INVOLVING MORE THAN TWO MEANS
Contrasts are linear combinations in which the weights sum to zero; for example, the
contrast between the winter and spring mean depression scores may be represented by
(1)(uWinter) + (—1) U-spring)- Occasionally, we are interested in contrasts involving more
than two means. We designate contrasts by U (the Greek letter, psi); all contrasts have the
general form
with the added condition that the weights sum to zero; that is,
where J is the number of conditions in the contrast (two in our example), and the Wj are
the weights on the means (1 and —1 in our example). Note that when the total number of
groups is more than two, some weights may be zero.
Suppose we wanted to estimate the contrast between the average winter score and the
average score in the other three seasons. This would be a repeated-measures contrast because
several seasonal measures are obtained from each participant in the study. An example of
a contrast involving several independent group means might be the difference between
the mean depression score for college graduates and the combined mean for high school
and vocational school graduates. Because the repeated-measures and independent-group
contrasts involve slightly different calculations, we consider them separately.
6.9.1 Repeated-Measures Contrasts
The contrast between the population mean of Beck depression scores obtained in the winter
season and the mean for the other three seasons is
which, following the form of Equation 6.21, we can rewrite as a linear combination of the
population means:
To obtain a CI for U, we first must estimate the contrast:
To carry out the calculations for the CI, we obtain a contrast score for each participant,
subtracting the average of the depression means for spring, summer, and fall from the
individual's winter score. These scores can be found in the column labeled PSI_D in the
Under 36 file. Most statistical packages perform transformations such as this (and more
complicated ones) quite easily. Once we have the set of contrast scores, the mean and
standard deviation are obtained. Then CIs are constructed, and a hypothesis test is conducted,
using Equations 6.2 and 6.3 for a single sample. We calculated contrast scores from the
depression data for 24 female participants under 36 years of age; 8 under-36 participants
did not have a score for all four seasons and their data were therefore omitted from the
analysis. The mean contrast score, U, was 1.289 and its SE was 3.030A/24, or .619. The
critical t value, t23,.025, is 2.069, and, after substitution into Equation 6.2, the .95 confidence
limits on u are
The CI provides limits for our estimate of the contrast, indicating that the mean depression
score for the under-36 population of women is higher in winter than in the remainder of the
year because zero is not contained in the interval.
If only the null hypothesis H0: u = 0 is of interest, we can calculate the t statistic
directly. The t for these data is u divided by its SE. Therefore,
Because this exceeds the critical t value on 23 df (2.07), we again have evidence that
the mean winter depression score is higher than the mean of scores in the other three
seasons.
Converting u to an estimate of the standardized contrast, Us, requires only that
we divide U by its standard deviation. In our example, Us = 1.289/3.030 = .425. The
effect is slightly less than half the standard deviation greater than zero, a medium sized
We may wish to assess the power of the test of H0: U =0 against the alternative, HA:
U = .425, with a = .05. Many software programs for calculating power require a value of
the noncentrality parameter, 8. In the repeated-measures design, this was defined in Equation
6.8 as
Substituting our estimate of Us (.425), and the N (24), we find that 8 = 2.082, the value
we calculated for t. Using the UCLA calculator (or any other available software that has
the noncentral t function such as SAS's CDF module), we can find the power of the test.
Assuming a one-tailed test of H0, enter 1.714, the critical value that cuts off the upper .05
of the t distribution with 23 df in the "X value" row. Also enter 23 in the "Degrees of
Freedom" row and 2.082 in the "Noncentrality Parameter" row. The value of "Probability"
returned is .354. This is (3, the probability of a Type 2 error; therefore, the power of the t
test is 1 — .354 = .646. GPOWER returns the same result. The only procedural difference
is that the values of Es and a, rather than 8 and the critical t, are input to the program.
Using either of these programs, or any other software, verify that if we obtained the same
effect size with 60 participants, the noncentrality parameter would be (.425) S60 = 3.292,
and the test of the null hypothesis would have a power of about .95.
We sometimes lose track of just what we are doing when we use these programs.
With any statistical test, power is the probability of exceeding some critical value (e.g.,
1.714) when we have a specific alternative to the distribution hypothesized under H0. Es
or 8, specifies the alternative distribution. No matter how we obtain a value of power, we
basically go through the same steps: first, decide on a critical region based on the null
hypothesis, the direction of the alternative (that is, is it one or two tailed?), the value of a,
and degrees of freedom (or sample size in the case of some tests); second, find the probability
that the test statistic will fall within that critical region when the true distribution is a specific
alternative to the null distribution.
6.9.2 Independent-Groups Contrasts
Myers, Hansen, Robson, and McCann (1983) investigated the relative effectiveness of three
methods of teaching elementary probability. They wrote three texts, which they referred
to as the Standard (S), the Low Explanatory (LE), and the High Explanatory (HE) texts.
Each text was studied by a different group of 16 undergraduates, none of whom had any
previous formal exposure to probability . The participants were then tested on two series of
problems—formula problems and story problems. The data are in the PL_data file in the PL
folder. The means and variances of the proportion of story problems correct are presented
The mean proportion of correct responses is higher in the HE group than in the other
two groups, which differ only slightly from one another. Myers et al. carried out a test of
the HE mean against the average of the other two means. In terms of the three population
means, the implied contrast is
Replacing population means by sample means, we estimate the contrast:
The CI for U follows the general form of CIs developed in this chapter; specifically,
TABLE 6.6 MEANS AND VARIANCES FROM THE
MYERS ET AL. STUDY
S LE HE
Mean .396 .406 .531
Variance .031 .038 .023
Equation 6.22 provides a basis for calculating U, and a formula for the SE of U was
derived in Appendix 6.1 (Equation 6.32):
Using these equations, we can calculate the .95 confidence limits on U. The calculations
are presented in Table 6.7. The CI does not contain zero and therefore a two-tailed test of
HO can reject the null hypothesis. The t statistic has also been calculated in Table 6.7 and
leads to the same conclusion.
A sense of the magnitude of the effect can be obtained by estimating the standardized
contrast, Us. This also provides a basis for comparison with the effects of other methods,
or with the effects of these methods on other types of problems. The standardized contrast
TABLE 6.7 CALCULATIONS OF CONFIDENCE LIMITS, t TEST, AND STANDARDIZED CONTRAST
1. First calculate an estimate of the population contrast:
U = [.531 - (1/2)(.396 + .406)] = .130
2. Because the ns are equal, the formula for the pooled standard deviation simplifies to
Spooled =S E S 2 / J ,where J is the number of groups. Therefore
3. Then,
4. The critical values of t on 45 df for a = .05, two tailed, can be obtained by linear
interpolation in Table C.3. The critical value that cuts off the upper .025 of the distribution is
2.015. Substituting this and the results of the previous steps into Equation 6.23, we find that
the .95 confidence limits are CI = .130 ± (2.015)(.054) = .02, .24.
5. To conduct the t test,
t = U/sU = .130/.054 = 2.41
The result is clearly larger than the critical value of 2.015, and so we reject the null
hypothesis of no difference between performance with the HE text and the average
performance with the other two texts.
6. Finally, we calculate the standardized contrast: U s = U/s pool = .130/. 175 = .74.
Note. Calculations are based on the statistics of Table 6.6.
CORRELATED SCORES OR INDEPENDENT GROUPS? 165
is defined as
Assuming homogeneity of variance, we estimate Us by
Numerical values have been substituted into Equation 6.25 as indicated in Table 6.7. Accord-
ing to Cohen's (1977,1988) guidelines, the standardized contrast, .767, would be considered
We can use a program such as SAS's CDF function or the UCLA calculator to find
the power of the test of the null hypothesis. Suppose we wish to assess the power of a two-
tailed t test of H0: Us = 0 against the specific alternative H A : U S = .743. Using the UCLA
program, we provide the "X value," the "Degrees of Freedom," and the "Noncentrality
Parameter." Because the test is two tailed, we reject H0 when t > 2.015 or t < —2.015. For
the "X value," enter 2.015, and also the df, 45. The noncentrality parameter is, as before, a
function of the standardized effect, and it is defined as
Substituting values from Table 6.7, we find 8 = 2.407. When the three numbers (2.015,
45, and 2.407) are entered, a "Probability" of .347 is returned. This is the area below
2.015 in the noncentral t distribution determined by 8 = 2.307. The area above 2.015
plus the area below —2.015 is power, the probability of rejecting HO when the test is two
tailed. The area above 2.015 is 1 — .347, or .653. The area below —2.015 may be found
by replacing the "X value" with —2.015. That area is essentially zero, so the power is
approximately .65.
6.10 CORRELATED SCORES OR INDEPENDENT GROUPS?
In Chapters 5 and 6, we encountered analyses of data sets in which each participant con-
tributed more than one score, such as a score for each of two or more seasons. The scores
for the different seasons will be correlated. Such designs are often referred to as repeated-
measures or within-subjects designs. Correlated scores will also result from matched-pair
designs. For example, we might wish to compare the effects of two instructional methods on
a participant pool of 40 students. The scores of the students might be ordered on the basis of
a pretest of ability. The students would then be divided into 20 pairs, with the members of
each pair having approximately equal scores on the pretest. In an independent-groups
or between-subjects design, scores in the different conditions are independent of each
other. This was the case in our analyses of gender and educational effects. The study of
instructional methods could also be carried out in an independent-groups design. In that
situation, assignment to instructional methods would be completely random except for the
restriction that there would be 20 children in each condition. What are the advantages and
disadvantages of the two designs?
When both types of design are feasible, the independent-groups design has two advan-
tages. First, it involves more degrees of freedom in the t test. Because each group provides
an independent estimate of the population variance, each based on n scores, the t is dis-
tributed on 2(n — 1 ) d f . The matched-pairs design involves only n — 1 df because there is a
single set of n difference scores. Looking at Appendix Table C.3, we find it evident that the
critical value of t becomes smaller as degrees of freedom increase. In fact, the power of the
t test increases and the CI is narrower for larger degrees of freedom. A second advantage of
the independent-groups design is that it does not require an additional measure for matching
participants. Sometimes, such a measure can be difficult to obtain.
Why, then, should we use a repeated-measures or matched-pairs design? The answer
lies in a comparison of the denominators of the t statistics (the SE of the difference between
the means) for the two designs. In the independent-groups design, the SE is based on the
pooled variability of the individual scores, whereas in the repeated-measures design, the
SE is based on the variability of the n difference scores. The latter SE will generally be
smaller. To understand why, consider a data set in which the scores are perfectly correlated.
For example, the scores for 10 subjects tested under both an experimental (E) and control
(C) condition might be
Group 1 2 3 4 5 6 7 8 9 10
E 5 10 3 7 12 13 9 4 8 15
C 7 12 5 9 14 15 11 6 10 17
Note that when two sets of scores are perfectly correlated, the difference is constant across
subjects. In the repeated-measures design, the variance of the difference is the denominator
of the t. In the preceding example, this variance would be zero, and any difference between
the E and C means would be significant. This would be true no matter how variable the
scores within a condition are. Although we never have correlations this high, the correlations
usually achieved by testing each subject under both conditions, or by matching subjects, will
usually reduce the SE of the mean difference considerably compared with the variability of
the individual scores.
A more formal argument is as follows. From Appendix 5.1, we may write the variance
of the sampling distribution of the difference of two means as
where p is the correlation between the means. Because a2 = a2/n, and assuming homoge-
neous variances, we can rewrite the equation as
That correlation should be zero for the independent-groups design but greater than zero
in both matched-pairs and repeated-measures designs. As a result, a2 will be smaller in
KEY CONCEPTS 167
correlated-scores designs (acor) than in independent-groups designs (a ind ). In fact, acor/aind
will equal 1 — p. Therefore, correlated-scores designs will have smaller denominators and
consequently larger t ratios than independent-group designs. There will be some trade-off
because the t statistic for the correlated-scores design is distributed on fewer degrees of
freedom than that for the independent-groups design. Nevertheless, if the same participants
are tested under both conditions, or if participants are matched on the basis of a measure
that is related to the dependent variable in the experiment, p usually will be large enough to
more than offset the loss in degrees of freedom. Ordinarily, the t test will be more powerful
and the CI narrower when scores are correlated than when scores are independent. The
repeated-measures design is used extensively in behavioral research, largely for the reasons
just presented. However, not all variables can be manipulated in this way. It would make no
sense to use both methods of arithmetic instruction on the same participants. Nor is gender
readily manipulated within participants. Furthermore, the researcher should be aware of
the possibility of "contrast effects." Some experimental treatments (e.g., one amount of
reward) have a very different effect when the same participant has been exposed to other
treatments (e.g., other amounts of reward) than when the participant has experienced only
that treatment.
This chapter focused on various applications of the t distribution and on concepts related
to the t statistic. We discussed CIs and t tests for differences between means, together
with formulas for effect size, with examples for both repeated-measures and independent-
groups designs. Examples of the use of computer software to perform post hoc and a
priori power calculations were also presented. Attention to all of these tools should help
researchers to understand their data. To focus only on significance tests is to risk ignoring
considerable additional information that is readily available. CIs focus on estimates of
population effect sizes and provide a sense of the precision of those estimates. Standardized
effect sizes, Es, permit comparisons of results obtained under different conditions, or from
different measurement scales, or from different experiments or laboratories. As a result of
Cohen's (1988) work, we also have guidelines to judge whether the effect is small, large,
or somewhere in between. Finally, Es provides an input needed for calculating power, and
the n required to achieve a specified level of power. Such power calculations can now be
easily carried out with one of several statistical packages or with programs freely available
on the Internet. In this chapter, we illustrated the use of two of these, GPOWER (which can
be downloaded to your computer) and the UCLA calculator. We have occasion to refer to
these again in future chapters.
t statistic degrees of freedom
raw effect size standardized effect size
post hoc calculation of power a priori calculation of power
central t distribution noncentral t distribution
noncentrality parameter, 8 homogeneity of variance (homoscedasticity)
pooled-variance estimate sum of squares
separate-variance t pooled-variance t
Welch's t statistic contrast
standardized contrast repeated-measures (within-subjects) designs
matched-pairs design independent-groups (between-subjects) design
6.1 An investigator wants to determine whether the difficulty of material to be learned
influences the anxiety of college students. A random sample of 10 students is given
both hard and easy material to learn (order of presentation is counterbalanced). After
part of each task is completed, anxiety level is measured by using a questionnaire.
The anxiety scores are as follows:
Task 1 2 3 4 5 6 7 8 9 10 11 12
Hard 48 71 65 47 53 55 68 71 59 31 80 77
Easy 40 59 58 51 49 55 70 61 57 32 70 69
(a) Find the 95% CI for the difference in the population means corresponding to the
two conditions.
(b) Test whether anxiety is significantly different in the two difficulty conditions by
using a matched-group t test. State H0 and HI and indicate the rejection region
for a = .05.
(c) Redo parts (a) and (b), assuming that the experiment had been done with two in-
dependent groups of 12 participants each. What are the strengths and weaknesses
of each design? Note any differences in results of the analyses and the reasons
for them in your answer, as well as any other considerations that you believe are
6.2 A sample of nine 30-day-old protein-deficient infants are given a motor skills test.
The mean for a normal population is 60. The data are
(a) Find a .90 CI for the mean of the protein-deficient population.
(b) Is the mean score of the protein-deficient children significantly below that of a
normal population?
(c) Our estimate of the mean of the protein-deficient population is very imprecise.
Using the value of 5 calculated in part (a), estimate the sample size needed to
have a CI width of only 12 points.
(d) After 3 months on a normal diet, the nine children have scores of
Estimate the mean of the population after 3 months of a normal diet. Calculate
the .90 CI and test whether this population mean is below the normal value of
60. Assume a = .05.
(e) Calculate difference scores and test whether there has been an improvement from
the first test to the second. Assume a = .05.
(f) Calculate the standardized effect of the normal diet.
6.3 For a matched-group design, we wish to test H0: UD = 0 against H 1 : U D > 0 at
a = .05. Using a sample of 16 participants, we find D — 2.0 and SD = 5.6.
(a) Carry out the t test.
(b) Calculate the standardized effect size.
(c) What power did the t test have to reject the null hypothesis, given the value of
Es calculated in part (b)?
(d) What power would the t test have to reject the null hypothesis, given the value
of Es calculated in part (b) and N = 36 instead of 16?
(e) What N would be required to have power equal to or greater than .80?
(f) In parts (c) and (d), you should have found the power of the t test. Redo the power
calculations in part (c), using the standardized normal distribution (see Chapter 5
for a review of the method) for N = 16 and for the N in your answer to part (e).
How good an approximation are these results to the results you obtained with the
t distribution? Is the approximation better or worse as n increases? Why might
this be?
6.4 In an independent-groups design, we find
Group 1 Group 2
n12 = 18 n2 = 14
s = 16 s2 = 20
Y1 =30.1 Y2 = 27.7
(a) Find the 95% CI for u1 — u2. Assuming we wish to test H0: u1 = u2 against a
two-tailed alternative, what can we conclude?
(b) Calculate the standardized effect size. With this effect size, what power did the
experiment have to reject the null hypothesis?
(c) Suppose we wished to redo the study with equal n and want .8 power to reject
H0, assuming the effect size calculated in part (b). What size n would we need?
(d) Using the n from part (c), and assuming the variances given in part (a), what
would the width of the new CI be?
6.5 In an independent-groups design we have
n 1 =21 n2 = 11
s2 = 8 s2 = 30
(a) Test the null hypothesis at a = .05 against a two-tailed alternative by using the
pooled-variance t test.
(b) Test the null hypothesis at a = .05 against a two-tailed alternative by using the
separate-variance (Welch) t test.
(c) Explain any differences in your conclusions in parts (a) and (b).
6.6 An arithmetic skills test is given to 8- and 10-year-old boys and girls. There are
10 children in each of the four cells of this research design. The means and standard
deviations are given as follows:
Children 8 Years 10 Years
Y 58 , 72
s 2.7 2.1
Y 53 60
(a) (i) Calculate a .90 CI for the difference in population means for 8- and 10-year-
old girls (u10.G — u 8,G )- (ii) Assume you wish to test the null hypothesis against
H1: u10,G > u8,G. What can you conclude on the basis of the CI?
(b) There is considerable data showing that boys do better than girls on tests such as
this arithmetic test. An interesting question is whether this advantage increases
with age. In other words, is the difference between boys and girls greater at age 10
than at age 8? (i) State H0 and H1 in terms of a linear combination of the four
population means, (ii) Carry out a t test of your null hypothesis, briefly reporting
the conclusion.
6.7 Three groups of participants are required to solve problems under varying levels of
environmental stress (noise: low, medium, and high). The experimenter has hypothe-
sized an inverted U-shaped function, with the belief that performance should be best
under medium stress and about equal for the high- and low-stress conditions. The
necessary information for the three groups is presented as follows:
Y 67.333 70.611 66.048
s 6.102 6.137 6.128
To test his or her theory, the experimenter carries out two statistical tests. In each
case, state H0, H1, and the rejection region, and carry out the test, reporting your
(a) According to the theory, the average performance of low and high populations
should not differ from each other.
(b) According to the theory, the average of a medium population should be higher
than the average of the combined low and high populations.
(c) Calculate the standardized effect associated with each of the two contrasts.
6.8 Several researchers have compared laboratory reading (participants knew they would
be tested for recall) with natural reading (participants read material without knowing
they would be tested). In one such study, two groups of 9 participants each (lab, natural
groups) were tested twice on the same materials, once on each of two different days.
Free-recall percentages (correct responses) were as follows:
Group Percentage
Day 1 45 60 42 57 63 38 36 51
(a) For each group, find the .95 CI for the population mean of the change in recall
over the 2 days.
(b) We wish to compare the two groups on Day 2. Assuming a two-tailed test, can
we reject H0 at the .05 level?
(c) From part (a), we have a change score for each subject. We wish to test whether
the amount of change is the same for the two populations of readers. State the
null and alternative hypotheses. Do the test at the .05 level.
6.9 The data for this problem are in the TC file in the Seasons folder of the CD.
(a) Calculate the standardized effect size (Es) for the winter - spring difference in
TC scores (TC1 - TC2) for the sayhlth = 2 (very good) and for the sayhlth = 4
(fair) group. How would you characterize the effects in terms of Cohen's guide-
lines? (See Chapter 6 to review the guidelines.)
(b) Calculate the winter - spring CIs for the two sayhlth groups of part (a). In which
is the CI narrower? Also calculate the t statistic for each. Which has the larger t?
The lower p value?
(c) Considering the various statistics, discuss the effects of seasons (winter versus
spring) on TC level.
The next problems are open ended but represent the task faced by the investigator
with a large data set.
6.10 The Royer_acc file on the CD contains subtraction, addition, multiplication, and mean
percentage correct for male and female third to eighth graders who had accuracy
scores for all three arithmetic operations. Considerable attention has been given to
the relative quantitative skills of male and female students. What differences, if any,
are there between the sexes in performance? Support your conclusions with graphs,
and any statistics—including significance test results, CIs, and effect sizes—that you
find relevant.
6.11 Using the Royer_rt file, which provides response times paralleling the accuracy scores
in this exercise, discuss differences, if any, between the sexes, again supporting your
answer with graphs and statistics.
The SE of a Contrast
To obtain a CI, or to test the null hypothesis about u, we require an estimate of the SE
of the sampling distribution of U. To derive an expression for that estimate, we note that
u is a linear combination of the sample means. In Appendix 5.1, we proved that a linear
combination (L) of independently distributed variables (V) of the form
has variance
where the Wk are weights and a2 is the variance of Vk. Replacing L by U and substituting
parameter estimators into Equation 6.27, we have
Assuming homogeneity of variance, we may rewrite Equation 6.29 as
where spooled is the weighted (by degrees of freedom) average of the group variances; that
is,
and / is the number of groups. Substituting the pooled-variance estimate into Equation 6.30
and taking the square root yields an expression for the estimate of the SE of U :
The Chi-Square and
F Distributions
The chi-square (x 2 ) distribution is of interest both because of its role in data analyses and
its relation to the normal, t, and F distributions. With respect to data analyses, the most
common application is to frequency data. In such applications, the x2 distribution is used
to determine how well a theoretical distribution fits an observed distribution, and in testing
whether two or more categorical variables are independent. An example of the goodness-
of-fit application would be in testing whether the distribution of letter grades in a class
conformed to some theoretical distribution such as 15% As and Fs, 20% Bs and Ds, and
30% Cs. An example of the test of independence would be one in which we ask whether the
distribution of letter grades is the same in the populations of male and female students. That
is, is the distribution of grades independent of gender? An introduction to these applications
of the chi-square statistic may be found in almost every introductory statistics textbook, and
entire textbooks have presented detailed treatments, particularly of tests of independence
in multifactor designs (e.g., Bishop, Fienberg, & Holland, 1975; Fienberg, 1977; Fliess,
In this chapter, we limit our presentation of x2 to defining the distribution and illustrat-
ing how it can be used to draw inferences about population variances. If the scores in the
population are distributed independently and normally, confidence intervals (CI) may be
calculated for the variance, and hypothesis tests about the variance may also be carried out.
We also consider the consequences of violating the assumptions upon which the chi-square
test rests. After this discussion of the application of the chi-square statistic to inferences
about variances, we develop the relation of x2 to the F distribution. Whereas the x2 distri-
bution allows us to draw inferences about a single population variance, the F distribution
provides a basis for inferences about the ratio of two population variances.
Perhaps the most common application of the F distribution is in analysis of variance
(ANOVA), a focus of much of this book. In an ANOVA, the numerator of the F ratio is a
174 7 / CHI-SQUARE AND F DISTRIBUTIONS
variance of a set of sample means, and the denominator is a measure of chance, or error,
variance. The F test in an ANOVA addresses the question of whether the variability of the
group means is greater than what could reasonably be attributed to chance. If the F ratio is
large enough, the null hypothesis that the population means are all equal can be rejected.
In the present chapter, we define the F statistic, and we apply it to situations in which we
compare the variances of two independent samples in order to test the null hypothesis that
the two sampled population variances are equal. We also develop relations that exist among
the chi-square, t, and F statistics.
Assume that N scores are randomly sampled from a population of scores that are indepen-
dently and normally distributed with mean u and standard deviation a. Each of the sampled
scores is transformed into a z score by first subtracting U and then dividing the result by
a. These z scores are then squared and summed. Translating our verbal description into an
equation for the chi-square statistic, we have
If many such random samples of size N are drawn from a normal population, and x2 is
calculated for each sample, the sampling distribution will have a characteristic density that
will depend on degrees of freedom. The quantity in Equation 7.1 has N df(as indicated
by the subscript on x2) because N independent values of Y have entered into the statistic.
Because the values of U and a are values of population parameters, there are no constraints
on the N values of Y and therefore no degrees of freedom are lost.
As Figure 7.1 illustrates, the x2 distribution is skewed; the skew is less pronounced
as the degrees of freedom increase. Several other properties of the distribution should be
noted. First, the mean of the distribution is equal to its degrees of freedom and its variance
is twice the degrees of freedom:
Second, the sum of independently distributed x 2 variables also has a x 2 distribution. The
degrees of freedom associated with the sum is the sum of the degrees of freedom for the
component values of x 2 . For example, a quantity obtained by summing a chi-square statistic
with 2 df and one with 3 df will have 5 df. This is the additive property for independent
chi-square statistics.
Appendix Table C.4 presents critical values of x 2 for various numbers of degrees of
freedom. Because ax2 with 1 df is just a squared z score, the values in the first row of
Table C.4 are related to those in Table C.2 for the normal distribution. For example, .95 of a
normally distributed population of scores lies between z = —1.96 and z — 1.96. Therefore,
.95 of x2 values on 1 df should lie between 0 and 1.962, that is, between 0 and 3.8416. This
in turn implies that .05 of scores should exceed 3.8416. This is, in fact, the .05 critical value
of x2 when df= 1 in Table C.4. Note that the mean of any x2 distribution is equal to its
INFERENCES ABOUT THE POPULATION VARIANCE 175
Fig. 7.1 Two x2 distributions. For a = .05, the critical values are 3.84 (df= 1) and 25.00 (df= 10).
degrees of freedom. Therefore, as the degrees of freedom increase, the distribution shifts to
the right and the critical values of x2 for any a level become larger.
In Equation 7.1, we subtracted the population mean from each of the N scores in
the sample. Suppose we repeated our sampling experiment but, this time, we subtract the
sample mean from each score instead of the population mean. In other words, we draw a
sample of N scores from a normally distributed population. For each score, we subtract
the sample mean, square the result, and then divide by the population variance. Then we
sum these transformed scores. Equation 7.2 summarizes these operations:
we often find Equation 7.2 written as
This statistic is also distributed as x 2, but on N — 1 df in contrast to the quantity in Equation
7.1, which is distributed on N df. The loss of a degree of freedom is caused by taking
deviations of N scores about their mean; because the sum of these deviations must equal
zero, 1 df is lost. By capitalizing on the additive property of x2, Appendix 7.1 demonstrates
more formally why the statistic in Equation 7.2, or 7.3, is distributed on N — 1 df.
7.3 INFERENCES ABOUT THE POPULATION VARIANCE
Equation 7.3 provides the basis for drawing inferences about variances. We consider that
application of the chi-square statistic next. In the example we present, we have a hypothesis
that predicts both the mean and variance of a set of scores. The t test of Chapter 6 provides
a test of the prediction about the mean. Here we illustrate the application of the chi-square
statistic to a test of the prediction about the variance.
7.3.1 An Example
Consider a population of chronically ill patients whose symptoms fluctuate unpredictably
over time, even when the patients are maintained on standard drug therapy. When given
monthly tests, the patients are equally likely to have "good" and "bad" months; that is,
they are equally likely to exhibit symptom levels that are better or worse than their baseline
levels. Thirty of these patients are selected for a study in which they are given a new drug
and tested monthly for 1 year. If the new drug is no more effective than the old one, and
if month-to-month fluctuations can be considered to be independent of one another, the
number of "good" months for each patient tested should follow a binomial distribution
with P = .5. From Chapter 4, we know that the mean number of good months for patients
should be Pm, where m is the number of monthly tests. In this example, that would be
(.5)(12), or 6. We know something else as well about the binomial distribution; the variance
is P(l — P)m. Therefore, the theoretical variance of the number of good months out of the
12 months tested is (.5)(.5)(12), or 3. In summary, the hypothesis that the new drug produces
the same chance variation over months that the old one did implies two null hypotheses: H01 :
U = 6, and H02: a2 = 3, where U is the mean number of "good" months in the population
of patients, and a2 is the variance of the number of "good" months.
The observed mean number of good months (Y), averaging over the 30 patients, is 6.4,
and the observed variance (s2) is 5.1. A t statistic on 29 df provides a test of H01 : U = 6;
Thus, the observed number of good months is not large enough to cause us to reject the
hypothesis that the new and old drugs are equally effective.
We can also use the sample variance to further investigate the effectiveness of the new
drug. We first construct a CI for a2. Assume that the population of scores (in our example,
a score is the number of "good" months in the 12 months tested) is normally distributed.
Then it follows from Equation 7.3 that (N — 1)s 2 /a 2 is a chi-square statistic and therefore
where X N - 1 a / 2 is exceeded by a/2 of the x2 distribution on N — 1 df. For example, if we
want a .95 CI, a = .05, and xN-1, a/2 is the value of X2 exceeded by .025 of the distribution
for 29 (because we have 30 patients) df. The value of x2-1,1-a/2 is that value exceeded
by .975 of the x 2 distribution. Finding the .975 and .025 critical x 2 values from Appendix
Table C.4, and replacing N by 30 and s2 by the observed value, 5.1, we can rewrite Equation
What we want, however, are bounds on a2 . Algebraic manipulation of Equation 7.4 yields
the general form of the CI:
Substituting numerical values into this equation, we have
and, doing the arithmetic,
Two points are worth noting about the preceding result. First, in Chapters 5 and 6,
CIs based on the normal and t distributions were symmetric about the statistic estimating
the population parameter; for example, in the estimation of U, the bounds were equidistant
from Y. However, because the x2 distribution is not symmetric, the CI just calculated is
not symmetric about s2, our estimate of a2. The second point is that because the theoretical
variance, 3, does not fall within the .95 CI, we may reject at the .05 level the null hypothesis,
H02: a2 = 3, in favor of the two-tailed alternative, H1 : a2 = 3. We can also test H02 directly.
Table 7.1 presents the test statistic and summarizes the test procedure against three possible
forms of the alternative hypothesis.
The test of the null hypothesis against a two-tailed alternative indicates that the variance
is larger than we would expect it to be if the test and standard drugs were equally effective
and the use of the binomial distribution was appropriate. Because the sample mean is
consistent with the hypothesis that the drugs are equally effective, one possibility is that
the new drug is more effective than the standard drug for some patients, but less effective
for other patients—leading to a situation in which, on the average, the drugs are equally
effective, but with too much variance for the drugs to be equally effective for each patient. If
this was the case, a next step in the research would be to look for differences in two subsets
of patients—those who improved with the new drug therapy and those who did not. Another
possibility is that the assumption of random short-term fluctuations of symptoms over time
is not valid, but rather that during the course of the disease, the patients have periods during
which either good months or bad months occur more frequently. If this is the case, the use
of the binomial distribution to calculate the theoretical variance is not appropriate, because
the assumption of independence does not hold. To obtain a more definitive answer about
drug effectiveness, we would really need to also have a control group of patients treated
with the standard drug, and we would have to look at the variances and the sequences of
good and bad months in both groups. Although our example is an artificial one, medical
treatments may have positive effects for some individuals and negative effects for others.
In such cases, the mean may be little affected but the treatment may increase variability.
The model testing procedure we have just illustrated has certain risks. Failing to reject
the null hypothesis that the population variance (or any other parameter whose value is
predicted by the model) has the predicted value provides support for the model. Sloppy
researchers who have small Ns may find support for their position because of a lack of
power. Confidence intervals are helpful in assessing the validity of a conclusion based
on a statistical test. If the interval containing the population variance is wide, we should
178 7/ CHI-SQUARE AND F DISTRIBUTIONS
TABLE 7.1 USING THE CHI-SQUARE STATI:STIC TO TEST HYPOTHESES ABOUT VARIANCES
1. If H1 is Reject HO if
a > ahyp x2 > xa
< ahyp X2 < X 1-a
= ahyp X 2 < X 1 - a / 2 or X 2 > Xa/2
, (N - 1 ) ( s 2 )
x2 =
ahyp
If a = .05, and df= 29 (as in our example),
If H1, is Reject H0 if
a2 > ahyp X 2 > 42.557
< ahyp X 2 < 17.708
a2= ahyp X2 < 16.047 or x 2 > 45.722
2. Now calculate the value of x 2 based on the data. In the example presented in the text,
X 2 = [(29)(5.1)]/3
= 49.30
Note that the value of a2 is the value assumed under H0.
3. Compare the result with the appropriate critical value. If the alternative hypothesis is
a2 > a hyp , or if the test is two tailed, the null hypothesis can be rejected. If the null
hypothesis is true, the probability of exceeding 49.3, the observed value of x 2 in our
example, is .011. This exact probability can be obtained from many sources, including the
transformation menus in SYSTAT or SPSS, or from the UCLA Web site described in
view a failure to reject the null hypothesis with some skepticism; a wide range of predicted
parameter values would also have been consistent with the data. If the interval is very narrow,
and the observed variance is close to the theoretical value, rejection of H0 may imply high
statistical power; we may be rejecting a model that—although not perfect—does a very
good job of accounting for the data.
7.3.2 Violation of the Normality Assumption
We calculated the CI and conducted the hypothesis test of Table 7.1 under the assumption
that the population distribution was normal. However, in the example of this section, we
were dealing with a binomial distribution. Does this invalidate the inferences drawn?
The answer is that in this application we have no problem. The reason lies in the central
limit theorem, which states that sums, and more generally linear combinations, tend to be
normally distributed as N increases. Each of the scores in our study is actually a sum of ones
(correct responses) and zeros (errors); because there are 12 responses contributing to each
score, they will tend to be normally distributed. Consequently, the true Type 1 error rate
will tend to be approximately .05 when the significance test is carried out at the .05 level.
THE F DISTRIBUTION 179
What about the validity of the test in other situations? There are many mathematical
models that yield numerical predictions for means and variances of various statistics, and
a test of whether the observed variance is consistent with that predicted by the model
would contribute to an assessment of the model. Unfortunately, in many of these cases,
the distribution whose variance is of interest may well be very skewed. Frequently, the
theoretical distribution of interest looks much like the exponential distribution of Fig. 6.3
(see, e.g., Bower, 1961). In such cases, the true Type 1 error rate may be much greater
than the nominal significance level. This returns us to one of our favorite themes: always
plot the data or, when testing a theoretical model, plot the theoretical distribution of the
variable of interest. If either the theory, or the data, indicate a clearly nonnormal population
distribution, inferences about variances based on the chi-square statistic are likely to be
invalid. Alternative procedures exist that do not depend on the normality assumption. The
jackknife and bootstrap methods are two possible alternatives that are described in several
sources (Efron, 1982; Efron & Gong, 1983; Miller, 1974; Mosteller & Tukey, 1977).
7.4 THE F DISTRIBUTION
To understand the F distribution, we begin with a sampling experiment. Assume the exis-
tence of two independently and normally distributed populations with variances of and a2,
respectively. Suppose we draw a random sample of size n 1 from the first population and a
sample of size n2 from the second population. The F statistic is the ratio
If the two population variances are the same, we may write the preceding ratio as
This provides the basis for tests of whether the population variances estimated by the sample
variances are the same. If the ratio in Equation 7.7 is much smaller or much larger than one,
it suggests that the sampled populations have different variances. We will shortly consider
how we can determine whether the ratio is very large or small. The distribution of the F ratio
is determined by both the degrees of freedom of the numerator, df 1 , and of the denominator,
df2.
Equation 7.6 provides a way of relating F to x2. Recall that
Dividing the left and right side by the degrees of freedom, n — 1, we have
Comparing this with the components that form the F ratio of Equation 7.6, we find that
it follows that the F statistic is essentially a ratio of two independent chi-square statistics,
each divided by its degrees of freedom. Therefore, we can also write the F ratio as
where the subscripts refer to the populations being sampled.
If we repeatedly draw random samples of sizes n1 and n2 from their respective pop-
ulations, the sampling distribution of the ratio of sample variances will be distributed as
F. Turning to Appendix Table C.5, we find critical values of the F distribution for various
values of a, df1 and df2. The tabled values are those exceeded by a of the samples, as-
suming that the sampled populations are independently and normally distributed and have
equal variances. The df1 are the degrees of freedom associated with the numerator of the
F distribution, and df2 are those associated with the denominator. When we form a ratio of
sample variances from two populations, as in Equation 7.6, the numerator and denominator
degrees of freedom are n1 — 1 and n2 — 1, respectively. In future chapters in which we
consider other applications of the F ratio, the degrees of freedom take on other values.
From the fact that the critical values of F depend on the degrees of freedom, it should
be evident that the F, like the t and x2 distributions, is not one distribution but a family
of distributions, one for each possible combination of numerator and denominator degrees
of freedom. If we were to repeatedly draw samples, computing the F ratio each time,
the average of the Fs would be a function of the denominator degrees of freedom; more
precisely,
As the denominator degrees of freedom increase, the expected value of F approaches one.
Therefore, ratios of sample variances much less than, or much greater than, one indicate that
the two sample variances are not estimating the same population variance. How different they
must be in order to draw this conclusion depends on the degrees of freedom. For example,
if df1 = 8 and df2 — 10, from Table C.5 we find that an F of 3.07 will be exceeded with
probability .05 when the two sample variances estimate the same population variance. In
terms of the hypothesis testing logic applied in previous chapters, an F larger than 3.07
suggests that either the population variances are equal and by chance a large value of F has
been obtained, or the null hypothesis of equal variances is false.
Although Table C.5 provides critical values, at best it provides only a rough sense of the
F distribution. We can improve our sense of the distribution by viewing the two examples
in Fig. 7.2. Several points should be noted. First, as is typical for the F distribution, the
distributions are skewed to the right. Second, the values of F are positive because F is a ratio
of variances, and variances are sums of squared quantities. Third, the value of F exceeded
by .05 of the distribution is smaller when there are 20 denominator degrees of freedom
than when there are 5. In general, a smaller critical value is required as either numerator
or denominator degrees of freedom increase. If we think about this for a moment, it makes
good sense. If our samples were very large, the ratio of sample variances would closely
approximate the ratio of population variances, and therefore even an F only slightly different
from one would suggest that the population variances differed.
One other point, not evident from the figure, should be noted. Because we are usually
interested in whether the F is significantly large, tables such as Table C.5 present only
critical values in the right tail. We may occasionally be interested in probabilities associated
Fig. 7.2 Two F distributions. For a = .05, the critical values are 6.39 (df 1 ,
df2 = 5) and 2.90 ( d f 1 = 5, df2 = 20).
with very small F ratios. For example, in subsequent chapters on the ANOVA, the logic is
such that very small Fs would suggest a failure of certain assumptions. More immediately,
a CI on the ratio of two population variances requires a lower bound and depends on finding
a critical value in the left (lower) tail. One way to find critical values from the lower tail of
the distribution is to note that such critical values are related to those in the upper tail. Let
the critical value that is exceeded with probability a be designated as Fadf1 ,df2 and the value
that is exceeded with probability 1 — a be designated as F\-a,df 1 ,df 2 The relation between
these is
Suppose that a = .05, df1 = 8, and df2 = 12. Then the critical value in the right tail is
F.05,8,12 = 2.85. From Equation 7.9 we have
F.95,8,12 = 1/F.05,12,8 = 1/3.28 = .305
With access to various software packages such as SYSTAT, SPSS, and SAS, or to
the Internet, we find that the solution to the "lower-tail problem" is simpler. If you do not
have software capable of providing exact p values, access to the Internet can solve the
problem. One solution is to use the F CDF calculator at the UCLA Web site referenced in
Chapter 6. Supply a question mark for "X Value," and .05, 8, and 12 for the probability,
and numerator and denominator degrees of freedom slots, and the answer, .3045, is quickly
provided. The UCLA calculator is also useful for data sets for which one or both degrees of
freedom values are not in Table C.5. GPOWER, the downloadable program we described
in Chapter 6, provides a second solution. Select "Other F Tests" from the "Tests" menu,
and then (ignoring other inputs) input the a, and the numerator and denominator degrees
of freedom. For the .05 lower-tail value to be obtained, the a should be .95. Note that the
UCLA calculator requires that p be the area below the F you want, whereas GPOWER
requires that you enter the area above the required F.
7.5 INFERENCES ABOUT POPULATION VARIANCE RATIOS
We are usually interested in comparing measures of the location of distributions, in par-
ticular, means. Nevertheless, there are times when a comparison of measures of variability
may be of interest. The reason usually given for such comparisons is to determine whether
the population variances meet the assumption of homogeneity of variance underlying the
t test (and the ANOVA). However, that rationale is not very compelling. As we saw in
Chapter 6, the variances must be very unequal, or the ns must differ, before the error rate of
the t test is distorted. In those cases, the separate-variance t test is available. Of greater inter-
est is the possibility that two treatments differ in their effects on variability. For example, we
may find little difference in the mean of a behavioral measure for two groups given different
medication for depression, but greater variability in the scores of one of the groups. This
would raise the possibility that the more variable data set represents some individuals who
improved and others who became more depressed, something certainly worth investigating
further. In some areas of research, we may be able to formulate precise enough theories to
predict the effects of certain factors on variability as well as on averages, thus providing a
more sensitive test of the theory. In such situations, comparisons of measures of variability
in different conditions should be made. In what follows, we will use a subset of the Seasons
data to illustrate some approaches to drawing inferences about population variances.
7.5.1 A Confidence Interval
Table 7.2 presents summary statistics based on the anxiety scores of men and women over
60 years of age. These data were extracted from the Seasons data file. Figure 7.3 presents
histograms of the means over seasons. From Table 7.2, we see that the variance of the female
mean scores is more than twice as great as that of the male mean scores. The histogram
in Fig. 7.3 reinforces our sense that variability is greater in the sample of female anxiety
scores. Confidence limits on the ratio of variances will provide a better sense of the possible
range of values of this ratio.
TABLE 7.2 ANXIETY STATISTICS FOR MEN AND
WOMEN OVER 60 YEARS OF AGE
Sex = Male Female
Minimum 2.000 1.250
Maximum 6.500 7.750
Variance 0.860 1.840
Skewness(G1) -0.836 -0.280
INFERENCES ABOUT POPULATION VARIANCE RATIOS 183
TABLE 7.3 CI FOR THE RATIO OF THE VARIANCES IN TABLE 7.2
In our example, let a2 be the variance of the female population and a2 be the variance of the male
population. The F ratio, s2F/s2M, = 2.14 and is distributed on 40 numerator df and 54 df. From the
UCLA Web site (see Chapter 6), we find the F values required by Equation 7.1 1:
F.925.54.40 = .564 and F.025,54.40 = 1-819
Substituting these values and the values of the variances from Table 7.2 into Equation 7.10, we have
Completing the arithmetic, we find that the upper and lower confidence limits are 1.207 and 3.892,
respectively. Because the lower bound is greater than one, we can reject the null hypothesis of
homogeneous variances at the .05 level.
Fig. 7.3 Histograms of anxiety data for men and women over 60 years of age.
We begin by establishing limits on the variance ratio of Equation 7.6. From that defi-
nition of the F statistic, we have
Algebraic manipulation and application of Equation 7.9 yields the general form of the CI:
Table 7.3 illustrates the application of Equation 7.10 by using the variances and sample
sizes in Table 7.2. The confidence bounds indicate that the variance of the population of
female anxiety scores is at least somewhat larger than that for the males.
Note that for the degrees of freedom in this data set, critical F values are not available
in Appendix Table C.5. In the past, researchers had to resort to linear interpolation in tables,
or to obtaining an approximate p value. With the availability of various statistical packages,
such approximations are no longer necessary. As we pointed out in Section 7.4, The UCLA
CDF calculator or GPOWER, as well as many other programs, can provide the necessary
F values. Commonly used packages such as SYSTAT, SPSS, and SAS will also provide
the exact probability associated with many statistics such as F, t, z, and X2.
7/ CHI-SQUARE AND F DISTRIBUTIONS
TABLE 7.4 APPLYING THE F RATIO TO TEST HOMOGENEITY OF VARIANCE
1. Let F = s 2 /s 2 . Then
IfH1, is Reject H0 if
a2 > a2 F > Fa
a2 < a2
a2 < a2 F < F1-a
= a2 F > F a / 2 orF < F 1 _ a/2
In our example, let a be the variance of the female population and a2 be the variance of
the male population. The F ratio, s2F/s2M, is distributed on 40 numerator df and
54 denominator df. From the UCLA Web site, we find that
F.05,40.54 = 1.617, F.95.40,54 = 0.606, F.025.40.54 = 1.773, and F.975.40.54 = 0.550.
Therefore, the decision rule for this example is
If H1 is Reject H0 if
a2 > a2
a2 > a2 F > 1.617
a2 < a2 F < .606
a2 = a22
a2 = a F > 1.773 or F < .550
2. From the statistics in Table 7.2, the F ratio in our example is
F = 1.840/.86 = 2.14
Because we have no prior reason to expect greater variability in the scores of either gender,
the test in our example is two tailed. Because 2.14 > 1.773, the null hypothesis of equal
population variances is rejected. With df1 = 40 and df2 = 54, the two-tailed p = .01.
7.5.2 The F Test
We can directly test the null hypothesis of equal variances. To illustrate the application
of the F test of variances, we again apply it to the Seasons anxiety scores of men and
women over 60 years of age. Table 7.4 presents decision rules for the test, and calculations
based on the statistics of Table 7.2. We again conclude that the population variances are
not equal. It may not be immediately obvious, but, as is the case with other statistical tests,
the test of the null hypothesis derived from the confidence limits in Table 7.3 and the direct
test in Table 7.4 are equivalent. We illustrate the relation between the two procedures in
Appendix 7.2.
The validity of the F test of the variance ratio illustrated in Table 7.4 rests on the assump-
tion that the two populations are normally distributed. When the population distributions
are skewed or have longer tails than the normal distribution, Type 1 error rates associated
with this test are often inflated. This distortion increases as sample sizes increase because
the shapes of the sample distributions more closely approximate those of the populations.
Deciding whether a sample may reasonably be viewed as drawn from a normally distributed
RELATIONS AMONG DISTRIBUTIONS 185
population is difficult. A useful graphic device is the normal probability plot that we de-
scribed and illustrated in Chapter 2. As previously noted, such plots are available in many
statistical packages. The expected values under the normality assumption are plotted as
a function of the observed scores, and marked departures from linearity indicate nonnor-
mality. In addition, it can be helpful to plot box plots and histograms, looking for obvious
discrepancies between sample means and medians, and seeing if kurtosis and skewness
values are large relative to their standard errors (SE). For confirmation, researchers can test
for nonnormality by using the Shapiro-Wilk W test, available in SPSS and SAS (Shapiro &
Wilk, 1965). This test has good power for samples of 50 smaller. For larger samples,
D'Agostino, Belanger, and D'Agostino (1990) describe alternative tests that take kurtosis
and skewness statistics as inputs.
When the investigator is interested in assessing differences between conditions in vari-
ability and when the validity of the normality assumption is in doubt, the Levene (1960) or
Brown-Forsythe (1974a) tests described in Chapter 6 provide an alternative to the F test
of variances. Although neither directly tests whether the population variances are equal,
they do permit a comparison of the average spread of scores. As described in Chapter 6,
the Levene test is based on the average absolute deviation of scores about the group mean,
and the Brown-Forsythe test is based on the average absolute deviation of scores about
the median. If the normality assumption is valid, these tests will be less powerful than
the F test of the variances. However, the Levene and Brown-Forsythe tests are less af-
fected by violations of the normality assumption than the F test based on the sample
variances. The Brown-Forsythe test has been found to be more powerful than the Levene
test under some conditions and therefore provides a reasonable alternative when normality
is suspect, though users should be aware that when the population distributions differ in
shape from one another, the actual Type 1 error rate may differ from the nominal error
7.6 RELATIONS AMONG DISTRIBUTIONS
In this chapter, we have noted the relation of x2 and F to each other, and to the normal
distribution. Both are also related to the t distribution. We consider these relations next. We
may write the one-sample t statistic of Chapter 6 as
In order to show the relation of t to x 2, we rearrange the quantities in Equation 7.11, divide
numerator and denominator by a2, and then square the result:
If Y is normally distributed, the numerator on the right-hand side of Equation 7.12 is
distributed as x2 on 1 df; the denominator is distributed as x2 on N — 1 df, divided by
186 7/CHI-SQUARE AND F DISTRIBUTIONS
N — 1. Therefore,
The t statistic for the two-sample case has a similar form. Beginning with the square
of the usual formula for t, we have
Dividing the numerator and denominator of Equation 7.14by a 2 (l/n 1 + 1/n2), we have
If the two population variances are the same, then a 2 (1/n 1 + l/n2) is the variance of the
sampling distribution of Y1 — ¥2- If, in addition, Y is normally distributed, the numerator
of Equation 7.15 is distributed as x2 on 1 df.
Under the assumptions of normality and homogeneity of variance, the denominator of
Equation 7.15 is also related to x2. We can rewrite the denominator as
If the sums of squares (the SS quantities) are numerators of estimates of the same population
variance, this quantity is distributed as x 2 divided by its degrees of freedom. In that case,
the two-sample t statistic is of the same form as Equation 7.13.
If we square the right-hand side of Equation 7.13 and slightly rewrite the numerator,
In other words, t2 is the ratio of two chi-squared variables, each divided by its degrees of
freedom. But this is exactly the definition of F given by Equation 7.8. Therefore, a squared t
statistic is an F with one numerator degree of freedom.
This chapter had two major goals: first, to introduce the x 2 and F distributions, two promi-
nent players in statistical inference; second, to consider applications of these distributions
to inferences about variances. With respect to the latter goal, it is worth repeating that when
evaluating the effects of an independent variable, researchers focus on means or, less often,
other measures of location, almost exclusively. However, knowledge of how other aspects
of the data distributions are affected can contribute to both practical applications and the
development of theory. This chapter provided one set of tools for investigating one aspect
of a distribution's shape—its variance.
Inferential procedures based on the distributions considered thus far in the book play
a major role in data analysis. The t and F distributions, in particular, have starring roles
throughout the remainder of this book. This reflects the prominence of these distributions
in data analyses reported in the research literature, a prominence resulting from ease of
calculation, applicability to data from many research designs, and a reliance on statistics
that are good estimators of population parameters under many conditions. Perhaps most
importantly, when their underlying assumptions are met, the tests based on these distri-
butions are uniformly most powerful tests; no alternative test will have greater power
to reject the null hypothesis. Bear in mind, however, that when assumptions are violated,
other procedures may provide more valid inferences. These may be tests based on other
distributions, such as alternatives to the t test based on ranked data, or modifications of
the usual test statistic, such as t' when variances are heterogeneous. With this in mind,
we emphasize that there are several considerations prior to choosing a method of analysis:
First, do the data indicate departures from assumptions? Second, if so, are these departures
severe enough to distort CIs, increase Type 1 error rate, or decrease power? Third, if so,
are there alternative procedures that are likely to yield more valid inferences? Answering
these questions requires that we begin our data analysis by looking at the data. Summary
statistics and data plots available in most statistical packages will aid this process.
goodness-of-fit test of independence
analysis of variance (ANOVA) chi-square statistic
additive property of x2 F statistic
uniformly most powerful tests
7.1 Equation 7.5 states that (N — I)s2/a2 is distributed as a chi-square variable if each
score in the sample is randomly drawn from an independently and normally dis-
tributed population of scores. Let's see what this implies.
(a) Suppose we draw many samples of size N from a normally distributed population.
We calculate the ratio, (N — 1)s 2 /a 2 , for each sample. If N = 6, (i) what is the
probability that this ratio is less than 9.236? (ii) What is the probability that the
ratio lies between 1.145 and 6.626?
(b) The population sampled in part a has a variance of 10. If we still assume N = 6,
in what proportion of samples will s2 be less than 8.703?
7.2 We are interested in a new method of teaching arithmetic. We use the new method with
a sample of 31 students, and at the end of a trial period we give them a standardized
test. In the past, the population of scores on the test has been approximately normally
distributed with a mean of 64 and a variance of 10. Following the training period,
the mean test score is 66 and the variance is 14.
(a) Has the new method of teaching led to a significant improvement in test scores?
Let a = .05.
(b) One interpretation of the increased variability on the test is that the new method
helps some students but hurts the learning of other students. Is the increase in
variability significant? Let a = .05.
(c) We might better consider the variability under the new method by obtaining a CI.
Find the .90 CI for a2.
7.3 On the basis of a review of large amounts of data, it is well established that the
variance of a population of ratings of the quality of a particular wine is 12.64. A
new method for training raters is established in hopes of reducing the variance. In a
sample of 10 judges trained under the new method, s2 is 3.51.
(a) Would you conclude that the new method has effectively reduced the variance of
ratings? Explain your reasoning.
(b) Suppose the population of scores was not normally distributed. Why is this a
problem for the approach you took in part (a)? Would it be less of a problem if
your sample size were larger? Explain.
7.4 A sample of 7 scores is selected randomly from a normally distributed population.
The scores are 22, 2, 0, 30, 28, 26, and 32.
(a) Find the 90% CI for the population variance.
(b) Assuming a = .05, test the null hypothesis that the population standard deviation
is 10 against the alternative hypothesis that it is greater than 10. Relate your
conclusion to the confidence limits you calculated in part (a).
7.5 We have samples of reading scores from 5 boys and 11 girls. We form a ratio of the
variances of the two samples, s 2 / s 2 ; call this F in accord with Equation 7.7.
(a) If many samples of sizes 5 and 11 are drawn, (i) what is the proportion of F values
greater than 2.61 that we should expect? (ii) What is the proportion less than 4.47?
(b) What assumptions are implied in your approach to answering part (a)?
7.6 Samples of scores are obtained from 9 male and 13 female subjects. Assuming
a = .05, answer the follwing.
(a) What is the rejection region if the researcher wishes to detect a difference in the
variances?
(b) How large must s 2 / s 2 be for you to conclude that the variance is greater in the
population of boys' scores?
7.7 In Exercise 6.5, we carried out a test of the means against a two-tailed alternative,
using the pooled- and separate-variance t test. It is of interest to decide whether
the separate-variance test is justified. Assuming that the scores were sampled from
a normal population, test whether the variances are equal against the alternative
hypothesis that they are not. The ns were 21 and 11, and the variances were 8 and
30, respectively.
7.8 An experimenter drew four independent random samples each of size 5 from one
normally distributed population. The variance of the four sample means about their
grand mean is 84.
(a) (i) Estimate the variance of the population from which the samples were drawn,
(ii) How many degrees of freedom are associated with your estimate?
The experimenter also has a sample of 15 independent observations that are
believed to be from the same population from which the original four samples
were taken. The variance of the 15 scores is 384.
(b) Perform a statistical test to determine whether the 15 scores do come from the
same population as the four samples of 5.
7.9 In viewing summary statistics for multiplication accuracy in the Royer data
(Royer_acc file in the Royer folder), we noted that in the fourth grade the male
multacc scores seemed to be considerably more variable than the female scores.
APPENDIX 7.1 189
(a) Find the .95 CI for the ratio of population variances (s2M/s2F) for fourth-grade
multiplication accuracy.
(b) Is there a significant difference in the variances? State the null and alternative
hypotheses.
(c) Plot the two density distributions in any way you find helpful. This can include
box plots, density plots, histograms, or stem-and-leaf plots. Does the plot provide
any insight into the reason for the difference in variances? If so, can you suggest
a further analysis of the data?
(d) Do you see any problem in the calculation of confidence limits and significance
tests for this data set?
7.10 Further analyzing the Royer_acc data of Exercise 7.10, calculate the Brown-Forsythe
t statistic to test whether there is a significant effect of gender on the variability of
the fourth-grade multiplication accuracy scores.
7.11 The Royer_rt file contains response times for addition, subtraction, and multiplica-
tion, a well as a variable labeled rt; this is the mean of the three measures.
(a) Plot the mean and standard deviations of the rt variable as a function of grade,
from Grade 3 to Grade 8. Describe the two functions.
(b) Test whether the variances of the sixth- and eighth-grade response times (RTs)
differ at the .05 level. State H0 and H1.
APPENDIX7.1
Chi-Square and Degrees of Freedom
We begin with the identity
Squaring both sides of the equation, summing the N quantities, and applying the summation
rules of Appendix A, we have
Because E (Y — Y) = 0, the preceding equation reduces to
Rearranging terms, and dividing both sides by the population variance, a2, yields
However, the leftmost quantity is a chi-square statistic distributed on N df. The quantity
subtracted from it is a chi-square statistic on 1 df because a single value (Y) is subtracted
from its expected value (U), and divided by the variance of its sampling distribution. From
the additive property of chi square, it follows that the rightmost term is distributed on N — 1
df.That is, x 2 - X 2 =xN-1
Relation Between the F test and the Confidence
Interval for a2/a2
Equation 7.10 established confidence limits on the ratio of population variances:
Let Fobs stand for the observed F ratio, s2/s2. Let FL and FU stand for the lower and upper
critical F values. Then we can rewrite the preceding equation:
With respect to the CI, we reject H0 if the value 1.00 does not fall within the interval.
That is, we reject the null hypothesis if 1 < FLFobs or 1 > FUFobS. Note that we can
rewrite 1 < FLF0bS as 1 / F L < F0bs. However, 1/F L = FU is the upper critical value of the
F distribution; therefore, asking if one is less than the lower limit of the CI is the same as
asking if the upper critical value of the F distribution is less than the observed F computed
from the data. Similarly, asking whether the upper bound of the CI is less than one is
equivalent to asking whether the observed F is less than the lower critical value of the
F distribution. In summary, we can reject H0 (that the population variances are equal) if
the observed F is greater than the upper critical value of the F distribution or less than the
lower critical value.
Between-Subjects Designs:
One Factor
This chapter deals with the analysis of data from a research design that serves as a building
block for the more complex designs we consider in subsequent chapters. In the one-factor,
between-subjects design, the scores of several groups of participants are analyzed in order
to decide whether the means of the treatment populations, the populations represented
by the groups, are equal. In such studies, participants may either be selected from existing
populations or be randomly assigned to one of several experimental conditions, or treatment
levels. An example of the former is the Seasons study in which individuals were sampled
from populations differing with respect to several factors, including gender, educational
level, and occupation. Strictly speaking, that study would be classified as an observational
study. True experiments involve random assignment of participants to levels of an indepen-
dent variable; the independent variable is said to be manipulated and the design is often
referred to as completely randomized. An example we mentioned in Chapter 6 was the
experiment by Myers et al. (1983). In that experiment, participants were randomly assigned
to study one of three texts presenting elementary rules of probability.
Whether the levels of the independent variable are observed or manipulated, the data
analysis has much the same form and the underlying assumptions are the same. What
characterizes the designs of this and the following chapter is that each participant yields a
single score. These designs are between-subjects designs; all the variability in the data is
due to differences between participants. In within-subjects designs the same participants
are tested under several conditions, and there is variability within each participant's set of
scores. Within-subject designs are also called repeated-measures designs. In later chapters
we also consider mixed designs, in which there are both between- and within-subjects
factors.
Between-subjects designs have the advantage of simplicity. Inferences require fewer
assumptions than are required by designs in which each participant responds on several
192 8/ BETWEEN-SUBJECTS DESIGNS: ONE FACTOR
trials or under several conditions. Each additional assumption underlying the derivation of
the test statistic is one more assumption that may be violated, possibly undermining the
validity of the statistical inference. The between-subjects design also has the advantage
of computational simplicity relative to other designs. This is less important in the present
era of electronic calculators and fast computers than it was in the past. Nevertheless, it
is useful to be able to obtain and check results quickly. The chief disadvantage of the
between-subjects design is its relative inefficiency. Because individuals differ on so many
dimensions, chance variability will often tend to be great, sometimes obscuring real effects
and reducing the power of the statistical test. As we discussed in Chapter 6, matching
participants on the basis of some measure other than the dependent variable, or testing each
participant in several conditions, will often lead to a reduction in chance variability. We
consider matching, or blocking, designs in Chapter 12 and repeated-measures designs in
Chapter 13.
In this chapter, we begin by looking at part of the data on which the article by Myers
et al. was based. As usual, this means generating and looking at descriptive statistics and
plotting the data in several ways. We then use the data to illustrate a conceptual framework
in which each score is viewed as a sum of components, and the total variability of the
scores is viewed as a sum of the variabilities of those components. This leads us into the
analysis of variance, or ANOVA, a partitioning of the total variability into parts that
provide the basis for an F test of the hypothesis that the treatment population means are
the same. Having illustrated the ANOVA with our data set, we consider the underlying
theory. We present a structural model, a model of the relation between each score and
the population parameters, and we state other assumptions necessary to justify the F test.
Following this, we present several measures of effect size, and we consider the power of the
F test. Finally, we consider the consequences of violations of assumptions, and we examine
possible remedies when those violations are thought to be severe enough to threaten the
validity of our inferences.
This is a rather extensive menu of theory and calculations. To avoid presenting too
many new ideas at once, we have placed the following restrictions on the presentation of
material in this chapter:
1. We consider only the subset of between-subjects designs that involve a single
independent variable; these are one-factor designs. We extend the development
to multifactor between-subjects designs in Chapter 11.
2. We consider only fixed-effect variables. This means that we view the population
of levels of the independent variable as consisting only of those that have been
selected for the experiment. We have more to say about this in Chapter 13. At that
point, we also introduce the concept of random-effect variables. The levels of
these variables are assumed to be randomly sampled from a population of levels.
3. We consider only tests of the omnibus null hypothesis: u1 = U2 = ... Uj = ...
...uj=... where ua, is the mean of a population of scores of individuals tested under
Aj, the jth level of the independent variable, A, and there are a levels of A in the
study. In Chapters 9 and 10, we extend the analysis of data from between-subjects
designs to contrasts among means of subsets of conditions, and to the analysis of
mathematical functions of quantitative independent variables, such as number of
trials, time in therapy, or hours of sleep deprivation.
EXPLORING THE DATA 193
8.2 EXPLORING THE DATA
In the probability-learning experiment conducted by Myers et al. (1983), each of 48 par-
ticipants studied one of three texts that presented elementary probability concepts such as
the addition and multiplication rules, and conditional and joint probability. The participants
were tested on 6 story and 6 formula problems immediately following study, and they were
tested on a second set of 12 problems 2 days later. Half of each group of participants re-
ceived the story problems first, and the other half received the formula problems first. The
researchers were primarily interested in whether the relative performances of the groups
differed on the two types of problems.1 However, we will analyze the proportion correct
of the 12 story problems, ignoring other aspects of the design and proceeding as if we had
a one-factor design with three levels, and 16 participants at each level. The story problem
scores are presented in Table 8.1, together with various summary statistics for each group.
The standard text is abbreviated by S, the Low Explanatory text by LE, and the High Ex-
planatory text by HE. The texts differed with respect to the kinds of examples used and the
presence or absence of illustrations such as tree diagrams.
Figure 8.1 presents box plots of the three groups of data. Several aspects of the plot
stand out. First, there is little difference between the S and LE data sets, whereas the HE set
is noticeably different in several respects. It appears that the HE distribution is somewhat
skewed because the median lies above the midpoint of the box. Also, with the exception of
one outlying score, the HE data seem less variable. We note the apparent skew of the HE
group and its smaller interhinge distance because, as with the t test of Chapter 6, normality
and homogeneity of variance are assumptions underlying the F test of means. However,
the skew is not so pronounced nor, as the variances in Table 8.1 indicate, are the spreads
TABLE 8.1 PROPORTION OF CORRECT SCORES IN THE MYERS
ET AL. (1983) STUDY
.083 .083 .333
Y.j = .396 .406 .531 Y.. = .444
s2 = .031 .038 .023
194 84 8/ BETWEEN-SUBJECTS DESIGNS: ONE FACTOR
Fig. 8.2 Dot-density plot of the data in
different enough to cause concern about the F test we present shortly. We can also see that
there is an outlier in the HE plot. This suggests the possibility that the higher HE mean in
Table 8.1 is merely a reflection of this one score. However, other aspects of the plot suggest
this is not the case; the median, which is not influenced by the outlier, is also higher in the
HE condition and the entire distribution is displaced upward relative to the other two. This
displacement is also apparent in the dot-density plot of Fig. 8.2. Such plots provide an
alternative to histograms, with each small circle or "dot" representing a score. In Fig. 8.2,
the advantage of the HE condition is reflected in the fact that almost one third (5 of 16)
of the scores in the S and LE conditions lie below the lowest score in the HE condition.
Whether this advantage of the HE group is large enough relative to the variability in the
data to enable us to conclude that it holds for the populations represented by these three
groups of scores is a question we consider shortly.
THE ANALYSIS OF VARIANCE 195
8.3 THE ANALYSIS OF VARIANCE
Before participants were assigned to the text conditions, they could be viewed as a sample
from a single, infinitely large population. Assume that each individual in this parent popu-
lation is randomly assigned to one of the three treatments (texts) applied in the experiment.
There are then three treatment populations, very large populations of scores that potentially
differ systematically from each other. Within this framework, we may view each of the sets
of 16 scores in Table 8.1 as a random sample from the corresponding treatment popula-
tion. Usually, the first question of interest is whether there are any differences among the
means of the three populations of scores. More precisely, we wish to test the omnibus null
In general, we assume that there are a levels of the treatment, A, and the omnibus null
hypothesis is
The alternative hypothesis is that there is at least one inequality among the means of the a
treatment populations.
Suppose the null hypothesis in the probability-learning experiment is true; the three texts
do not differ in their effects on the proportion correct on the story-problem test. Even if this
were so, the three group means would differ from each other. By chance, the 16 individuals
in one group may be more motivated, more alert, or more knowledgeable about probability
than those in another group. In addition, there may be chance variations caused by other
factors that affect test performance, such as the times at which the participants are tested.
The test of the null hypothesis is really a test of whether the group means differ more than
would be expected on the basis of these chance factors. If they do, then something more
than chance variation is involved. That "something more" is presumably the effect of the
independent variable (the text studied), and the null hypothesis should be rejected.
In the ANOVA, we attempt to determine whether more than chance variability is
involved by comparing two independent estimates of the population error variance, estimates
that do not differ significantly if the null hypothesis is true. One of these, the between-groups
mean square, is based on the variance of the group means and is influenced by both chance
variability and—if H0 is false—the effects of the independent variable. We label this mean
square MS A. The second estimate of variance, the error mean square, is the average of
the variances of scores in each group; it reflects only chance variability. We refer to this as
MSs/A (the mean square for subjects within levels of A). If MSA is much larger than MSS/A,
we may decide that the spread among the group means is too large to have resulted only
from chance variability. If so, we will conclude that the treatment population means are not
all equal; that is, H0 is false.
In the following sections, we develop formulas for the mean squares. We begin by
partitioning the total variability in the data set into two components that provide the numer-
ators of the mean squares. We discuss why the two mean squares are estimates of the same
chance, or error, variability when HQ is true and why, therefore, their ratio is distributed as
F. Following this, we present a more formal discussion, including a model of the structure
of the scores in terms of population parameters, and an explicit statement of assumptions.
196 8 / BETWEEN-SUBJECTS DESIGNS: ONE FACTOR
8.3.1 Partitioning the Total Variability
Mean squares are ratios of sums of squared deviations to degrees of freedom. As we noted
in Chapter 6, these sums of squared deviations are referred to as sums of squares (SS).
In this section we show that SSA and SSs/A together account for the total variability in the
data of the one-factor design. We begin by partitioning the deviation of the ith score in the
jth group (Yij) from the grand mean (Y..), the mean of all an scores, into two components:
first, the deviation of the score from the mean of its own treatment group ( Y . j ) , and second,
the deviation of the group mean from the grand mean:
Table 8.2 illustrates this partitioning, using the data of Table 8.1. The table is divided
into three sets of three columns. The first three columns contain the values of Yij — Y..
for each of the three texts. The next three columns contain the effects of the three texts,
Y.j — Y.j These treatment effects actually reflect both the effect of the treatment and
chance variability. If there is a treatment effect in the population—that is, if the treatment
population means differ—that population effect should be reflected in differences among
the group means, and therefore in a deviation of any group mean from the grand mean. But
even if H0 is true, the group means will still differ, because there are different individuals
in each group and they will perform differently because of many factors. The residual
terms, Yij— Y.j, are contained in the last three columns, and they reflect the variation in
the performance of individuals who have received the same treatment. By squaring both
sides of Equation 8.1, summing, and applying the rules of summation of Appendix A, we
TABLE 8.2 BREAKDOWN OF THE SCORES FROM THE MYERS ET AL. (1 995) EXPERIMENT
Scores - Grand Mean = Text Effect + Residual
(Yij-Y..) = (Y.j-Y..) + (Yij-Y.i]
-.361 -.361 -.111 -.049 -.038 .087 -.313 -.323 -.198
-.111 -.111 .056 -.049 -.038 .087 -.063 -.073 -.031
-.111 -.027 .056 -.049 -.038 .087 -.063 .011 -.031
-.027 -.027 .139 -.049 -.038 .087 .021 .011 .052
.056 -.027 .139 -.049 -.038 .087 .104 .011 .052
.056 .056 .139 -.049 -.038 .087 .104 .094 .052
SStot = Ej E i (Y ij - Y..)2 = (-.361)2 + (-.277)2 + • • • + (.473)2 = 1.561
SSText = n Ei (Y.j - Y..)2 = (16)[(-.049)2 + (-.038)2 + (.087)2] = .182
SS Res = £j £i (Y ij - Y.j)2 = (-.313)2 + (-.229)2 + • • • + (.386)2 = 1.379
find that the total sum of squares (SStot) is partitioned into two component sources of
variability, the between-groups sum of squares (SSA, the sum of squares for A), and the
within-groups sum of squares (SSs/A, the sum of squares for subjects within levels of A).
We have carried out this squaring and summing operation with the numbers in Table 8.2,
and, as indicated at the bottom of the table, the SSA and SSs/A do indeed sum to the SStot. A
general proof that SStot = SSA + SSs/A is presented in Appendix 8.1. The end result is
The SStot of Equation 8.2 is the numerator of the variance of all an scores about the
grand mean. Accordingly, it is distributed on an — 1 df. The within-groups sum of squares,
SSs/A, is the sum, or "pool" of the numerators of each of the group variances. Because
each group variance is distributed on n — 1 df, SSs/A is distributed on the sum of the group
degrees of freedom, or a(n — 1) df. The between-groups sum of squares, SSA, is n times
the numerator of the variance of the a group means about the grand mean and is therefore
distributed on a — 1 df. Note that
Equation 8.3 demonstrates that the degrees of freedom are partitioned into two nonover-
lapping parts corresponding to the sums of squares. This partitioning of the degrees of
freedom provides a partial check on the partitioning of the total variability in more complex
designs in which some term in the analysis may be overlooked, or the total variability may
be misanalyzed in some other way. When designs have many factors, and therefore many
components of the total variability, it is wise to find the degrees of freedom associated with
each term and to check to see if these add up to the total number of scores minus one.
Equation 8.2 defines the sums of squares for the one-factor between-subjects design,
and, accordingly, we refer to the component terms as definitional formulas. Using such
formulas with a calculator can result in rounding errors, and therefore textbooks have
generally provided so-called computational or raw-score formulas. Such formulas are of
less practical use today because most analyses are carried out by computer packages, or by
calculators that have a high degree of accuracy. Therefore, we ordinarily will not include
raw-score formulas. However, Appendix 8.2 presents the raw-score equivalent of the terms
in Equation 8.2, and it provides some rules that will generally enable the calculation of
sums of squares in more complex designs provided the user knows the correct degrees of
freedom for the terms. The use of the raw-score formulas is illustrated in Appendix 8.2 with
the data of Table 8.1.
Dividing the sums of squares by degrees of freedom results in the mean squares. These
quantities provide the components of the F statistic that tests the omnibus null hypothesis.
We consider these mean squares next.
8.3.2 Mean Squares, F, and the ANOVA Table
Suppose we draw a samples of n scores from the same population; we represent the variance
of the population (the error variance) by cr 2 . The variance of the a sample means is an
estimate of the variance of the sampling distribution of the mean; that is,
The caret above the o stands for "estimated." Recall that the variance of the sampling
distribution of the mean is the population variance divided by n. Therefore, we can rewrite
the preceding equation as
and, multiplying both sides by n,
The left side of Equation 8.4 is the SSA divided by dfA; or MS A - The entire equation states
that MSA, the between-groups mean square, is an estimate of chance, or error, variance
when the a groups of scores are sampled from the same population.
Now assume that each group of scores is sampled from one of a treatment populations,
and that the population parameters are identical; that is,
Under this assumption, the situation is the same as if we sampled the a groups of scores
from one population. Therefore, if the null hypothesis is true and there is homogeneity of
variance, the between-groups mean square is an estimate of the error variance common
to the a treatment populations.
The within-groups mean square, MSs/A, provides a second estimate of error variance.
We may rewrite Equation 8.5 as
The expression in the square brackets on the right side is the variance of the jth group of
scores, and the entire right side is an average of the a group variances. Therefore, MSs/A is
an average of a estimates of the population variance, a2.
The point of the preceding development is that, if the null hypothesis is true, MSA
and MS S/A both estimate the same population error variance. Therefore, their ratio should
be about one. Usually the ratio will be a little more or a little less than one; it would be
surprising if two independent estimates of the same population variance were identical. If
H0 is true, the ratio, MS A /MS S / A , is distributed as F on a— 1 and a(n— 1) df. As we noted in
Chapter 7, critical values are tabled in Appendix C.5 and can also be obtained from various
software packages and Web sites.
Suppose the null hypothesis is false. For example, suppose that the text studied does
affect performance on story problems in the Myers et al. experiment. Then the means of
the groups of scores in Table 8.1 will differ not only because the scores in the different
groups differ by chance but also because the participants studied different texts. In other
words, if H0 is false, MSA, which is n times the variance of the group means, reflects not
only chance variability but also variability that is due to the independent variable. However,
the within-group variance should not be affected by the independent variable because all
participants in a group receive the same treatment. Therefore, when HO is false, the ratio
MSA/MSs/A should be greater than one.
In summary, under the assumptions of the null hypothesis, homogeneity of variance,
and independently distributed scores, MSA and MSs/A are two independent estimates of the
population error variance, o2. From Chapter 7, we know that if we add the assumption that
the population of scores is normally distributed, the ratio of two independent estimates of
the same population variance has an F distribution. Therefore, under these assumptions, the
ratio MS A /MS S / A is distributed as F. Because the numerator is an estimate of the variance of
a population means, it has a — 1 df. The denominator has a(n— 1 ) d f because the variance
estimate for each group is based on n— 1 df and the a group variances are averaged.
Table 8.3 summarizes the developments so far, presenting the formulas for sums of
squares, degrees of freedom, mean squares, and the F ratio for the one-factor between-
subjects design; Table 8.4 presents SYSTAT ANOVA output for the probability-learning
data of Table 8.1. Despite the apparent advantage of the HE text in Figs. 8.1 and 8.2, the
p value, .062, is slightly larger than the usual standard for statistical significance, p =
.05. We cannot reject H0 : us = ULE = U HE - However, neither can we accept the null
hypothesis. The usual criterion for p, .05, is not a magic number, and it is good to keep in
mind that the p value reflects not only the variance of the treatment population means but
also error variance and sample size. As we have noted in previous chapters, effects large
enough to be of practical or theoretical importance may not be statistically significant, and
trivial effects may be if enough data are collected. With this in mind, we should look at one
or more indices of importance. We discuss several in Section 8.5, but we now briefly note
one index that accompanies the ANOVA output in Table 8.4, and that is also provided by
other statistical packages, such as SPSS.
TABLE 8.3 ANOVA FOR THE ONE-FACTOR BETWEEN-SUBJECTS DESIGN
SV df SS MS F
Total an — 1 EE(yij-Y..)2
A a- 1 nE(Y.j-Y..)2 SSA/dfA MSA/MSS/A
S/A a(n - 1) EE(Yij-Y.j)2 SSS/A /dfs/A
Note. S V = source of variance.
200 88/BETWEEN-SUBJECTSDESIGNS: ONE FACTOR
TABLE 8.4 ANOVA OF THE DATA IN TABLE 8. 1
Categorical values encountered during processing are:
TEXT (3 levels)
HE, LE, S
Dep Var: Y N: 48 Multiple R: .341 Squared multiple R: .1 1 6
Source Sum-of-Squares df Mean-Square F-ratio P
TEXT .182 2 .091 2.965 .062
Error 1.379 45 .031
Note. ANOVA output is from SYSTAT.
8.3.3 The Coefficient of Multiple Determination
The sums of squares provide one possible index of the importance of the independent
variable. The ratio, SS A /SS t o t , is the proportion of the total variability attributable to the
independent variable. This ratio is often called m,2 (eta squared) and is similar to r2, the
coefficient of determination introduced in Chapter 3. Recall that we defined r2 as SSregression/
SS Y , where SSregression(the sum of squares for regression) was the sum of the squared
deviations of the predicted scores about the grand mean and SSy was the sum of squared
deviations of the actual scores about the grand mean. The best prediction we have for each
score is the mean of the group to which it belongs. Replacing Regression by SSA and SSY by
SStot, we have the coefficient of multiple determination:
This ratio of the between-groups to the total sum of squares may be viewed as a measure of
how well the group means predict the individual scores. The better this prediction—that is,
the larger the portion of the total variability accounted for by the independent variable—the
higher the value of R2.
The R2 for the probability-learning data is reported in Table 8.4 as a squared multiple
R of .116 (SPSS reports it as eta squared). Following Equation 8.6, the SSA was divided by
SStot (SSA + SS S/A ):
The multiple R of .341 is the square root of .116.
R2 overestimates the actual proportion of variability in the population that is due to
the independent variable. To understand why, assume that the treatment population means
are identical. In that case, none of the variability in the population would be due to the
independent variable; therefore, R2 should be zero. However, R2 will be greater than zero
because—in contrast to the population means—the sample means will vary because of
chance. To correct for this, Wherry (1931) proposed a formula for shrunken R2 (also called
THE MODEL FOR THE ONE-FACTOR DESIGN 201
the adjusted, corrected, or attenuated R2, depending on the source):
where N= na. For the current example, the adjusted R2 is
and the corresponding Radj is .277.
Given that the participants in the study probably varied with respect to motivation,
mathematical background, and aptitude, as well as other factors that can affect learning, it
is not surprising that the independent variable appears to account for only a small portion of
the total variability, roughly 8%. Of course, this does not mean that the effects of the text are
unimportant. Nor are they as small as it might appear. According to guidelines suggested by
Cohen (1988), an adjusted R2 of .077 corresponds to a medium-sized effect. In any event,
judgment of the importance of the independent variable should be guided by knowledge of
the research situation and the potential application. If the difference of approximately 13%
correct between the mean in the HE condition and the other two conditions held for the
population, most instructors would consider it an important gain in performance.
R2 is only one possible index of the importance of the independent variable. It has the
advantage of being easy to compute and understand, but other measures deserve consider-
ation and are discussed in Section 8.5. These measures are best interpreted in terms of the
parameters of the ANOVA model. Therefore, we first consider the model in a more formal
way than we have so far.
8.4 THE MODEL FOR THE ONE-FACTOR DESIGN
In Subsection 8.3.2, we presented a somewhat informal statement of assumptions and
justification of the ratio of mean squares as an F ratio. In this section, we take a closer look
at these assumptions and see how they lead to an important concept, that of expected mean
squares.
8.4.1 The Structural Model
We can view an observed score as consisting of two components—the treatment population
mean and an error component. We represent this as
where uj is the mean of the jth treatment population and is a component of all the scores
in that population, and eij (the Greek letter epsilon) is the unique contribution to the score
of the ith individual in the jth treatment population. Any differences among the treatment
population means reflect differences in the effects of the treatments, whereas variation in
the error components reflects differences caused by differences in characteristics of the
individuals (such as ability or motivation) and differences in conditions of measurement
(such as the time of testing).
Equation 8.8 is unchanged if on the right side we both add and subtract the constant JJL
resulting in
where u = Euj/a, the average of the a treatment population means. We may rewrite the
preceding equation as
where aj = uj — jx is the effect of treatment Aj. Equation 8.9 provides a structure within
which we view a score as a sum of the following three components:
1. The parent population mean, JJL. This quantity may be viewed as the average of
the treatment population means and is a constant component of all scores in the
data set.
2. The effect of treatment Aj, aj. This is a constant component of all scores obtained
under Ay but may vary over treatments (levels of j). The null hypothesis asserts
that the a values of ay are all zero.
3. The error, eij. This is the deviation of the i th score in group j from uj and reflects
uncontrolled, or chance, variability. It is the only source of variation within the
y'th group, and if the null hypothesis is true, the only source of variation within
the data set.
Two other points about treatment effects should be noted. First, researchers often refer
to the effects of an independent variable as in "A had a significant effect upon perfor-
mance." A somewhat wordy, but precise, translation of such a statement is that one or more
of the levels of A had an effect. In terms of the structural model, at least one of the ay
was not zero. Second, a distinction that will be important is that between fixed-effects
variables, independent variables whose levels have been arbitrarily selected, and random-
effects variables, variables whose levels have been randomly sampled. When levels
have been arbitrarily selected—as when we select three texts for study—it is as if we have
exhausted the population of levels. There is no statistical basis for generalizing from the
results of this experiment to draw conclusions about the effects of texts not included in
the experiment. In this situation, Ej(uj-U)=0 (or equivalently, Ej aj = 0) because
the sum of deviations of all values about their mean is zero. We deal exclusively with
fixed-effect treatment variables in this chapter.
Equation 8.9 is not sufficient for deriving parameter estimates and significance tests.
In addition, the following assumptions about the distribution of eij are required:
1. The eij are independently distributed. This means that the probability of sampling
some value of eij does not depend on other values of eij in the sample. An important
consequence of this is that the £ij are uncorrelated.
2. The eij are normally distributed with mean zero in each of the a treatment popu-
lations.
3. The distribution of the eij has variance a2 in each of the a treatment populations;
that is, a2 = a2 = • • • = a2 = • • • = o2 = a2. This is usually referred
to as the assumption of homogeneity of variance.
Although we never have access to the populations of scores, plots of the data will provide
information about the validity of our assumptions. Chapter 2 described some procedures
that will prove helpful, and we illustrated one other (the dot-density plot) in this chapter. In
later sections of this chapter, we discuss the problems that may arise when our assumptions
are violated, and how alternative data analyses may provide a solution to these problems.
For now, we assume that Equation 8.9 is valid and that the populations of error components
are independently and normally distributed with equal variances. This provides the basis
for a more formal justification of the F test presented in this chapter.
8.4.2 Expected Mean Squares
In Section 8.3 we argued that the ratio of mean squares, MS A /MS s / A , was a reasonable test
statistic for the null hypothesis of equality of the treatment population means. The idea
is that if Ho is true, MSA and MSS/A are both estimates of oe2,the treatment population
error variance. Because they both estimate the same quantity, their average values over
many random replications of the experiment should be about the same size. If H0 is false,
however, MSA reflects the differences among the treatment population means in addition
to chance variability. In that case, on the average, MSA will tend to be larger than MSs/A
In this section, we provide a somewhat different version of this argument, one based on the
average values of MSA and MSs/A over many replications of the experiment.
Suppose we draw a samples of n scores from their respective treatment populations, and
calculate the two mean squares. Now suppose that we draw another a samples of n scores,
and again calculate MSA and MSs/A . We could repeat this sampling experiment many times,
and arrive at two sampling distributions, one for MSA and another for MSs/A . Given the argu-
ments of Section 8.3, the average value of MSA will reflect both error variance and treatment
effects, whereas the average value of MSs/A will reflect only error variance. These averages
of the sampling distributions of the two mean squares are the expected values of the mean
squares, or the expected mean squares (EMS). They play an important role both in under-
standing the analysis of variance (ANOVA) and in deciding a number of practical issues. To
cite just one application, in more complex designs there will be many possible sources of
variance; the EMS dictates the appropriate error term for any particular source. Given the
structural model of Equation 8.9, and assuming that the eij are independently distributed
with variance ae2, we can derive the EMS of Table 8.5 (Kirk, 1995; Myers & Well, 1995).
Look again at Table 8.5. Note that if the null hypothesis is true (i.e., if the uj are all
equal), both expectations equal oe2; in any one experiment, the two mean squares we have
calculated will not be identical, but they rarely should be very different if H0 is true. In
contrast, if the uj differ, MSA has a larger expected value than MSs/A and therefore their ratio
should be greater than one. When we look at Table 8.5, it appears that this ratio increases
TABLE 8.5 EMS FOIR THE ONE-FACTOR
BETWEE N-SUBJECTS DESIGN
SV EMS
A a2 + n02
S/A a2
Note.0 2 = [E/OJ-./ - fA) 2 ]/(a - 1). We use the
6 notation rather than a2 to remind us that the treatment
component of the EMS involves division by degrees of
freedom rather than by the number of levels, as it would
in the formula for a population variance.
with increases in n, with increases in the spread among the jxy, and with decreases in the
error variance. Therefore, we can expect greater power when we run more subjects, when
the effects of the independent variable are larger, and when error variance is reduced. The
error variance will depend on the subject population sampled, the measure selected, and
the experimental design. An example of the last factor—the experimental design—was the
comparison of the correlated-scores and independent-groups designs in Chapters 5 and 6.
In the one-factor design, there are only two mean squares and therefore it requires no
great insight to decide that MSs/A is the error term—the denominator of the F test, against
which MSA is to be tested. The choice of an error term is more complicated in designs in
which there are several possible error terms. However, that choice is guided by a simple
Choose an error term such that its EMS and the EMS of the term to be tested are identical when the
null hypothesis is true.
In summary, if the eij are independently distributed with variance 02, the mean squares
have the expectations presented in Table 8.5. If, in addition, the null hypothesis is true,
02 — o, and the two expectations are the same. Finally, if the eij are normally distributed,
the ratio of mean squares, MSA/MSs/A, is distributed as F and the probabilities of exceeding
various values are those tabled in Appendix C.5.
8.4.3 ANOVA With Unequal Group Sizes
The ns in conditions in a study may vary for one of several reasons. The populations may
be equal in size but data may be lost from some conditions, perhaps because of a mal-
function of equipment, or a participant's failure to complete the data-collection session.
Usually, individuals can be replaced, but sometimes this is impossible. In other instances,
the treatments may affect the availability of scores; for example, animals in one drug con-
dition may be less likely to survive the experiment than animals in another condition. In
still other instances, usually when we collect data from existing populations, some condi-
tions may naturally have more individuals available for participation than others will. For
example, in the Seasons data, we find different numbers of participants at various levels of
educational experience, and in different occupations. In educational and clinical settings,
there may naturally be different numbers of individuals in different conditions, such as
grade levels, or diagnostic categories. Discarding participants to equalize the group ns will
reduce error degrees of freedom and, consequently, power, and this may also misrepresent
the relative size of the populations sampled. In the latter case, the effects of some con-
ditions may be weighted too heavily or too lightly in the data analysis. In all of these
circumstances involving unequal n, the ANOVA is a straightforward modification of the
equal-n case, at least in the one-factor bet ween-subjects design. (Complications arise when
more than one factor is involved; these are treated in Chapters 11 and 21.) Table 8.6 presents
the ANOVA formulas and EMS for the unequal-n case. Note that if the nj are equal,
these formulas reduce to the definitional formulas in Table 8.3 and the EMS formulas of
Table 8.7 presents statistics based on the average (over seasons) Beck depression
scores (Beck_D) for four groups of male subjects who participated in the University of
Massachusetts Medical School research on seasonal effects (the Seasons data). For the pur-
poses of this example, we excluded some participants (those having only no or some high
TABLE 8.6 ANOVA FOR THE ONE-FACTOR BETWEEN-SUBJECTS DESIGN WITH UNEQUAL GROUP SIZES
SV df SS MS F EMS
a nj 2
Total N -1 E E (Yij -Y..)2
j =] i = 1
A 0-1 En ( Y . - Y . . )
SSA/dfA MSA/MSS/A oe2 +1/a-1
Enja2j
j j
a nj
S/A N -a E E(Y
j=1 i=1 ij-Y.j)
SSs/A /df s / A a2e
Note, nj is the number of scores in the jth group and N = Eaj=1 nj.
school education, and those with vocational training or an associate's degree). The remain-
ing groups are HS (high school diploma only), C (some college), B (bachelor's degree),
and GS (graduate school). The data may be found in the Male_educ file in the Seasons
folder.
The group of males participants with high school diplomas have higher average depres-
sion scores, both means and medians, than those of the other three groups; whereas the three
groups with more education have means ranging from 3.331 to 4.847, the HS group has a
mean depression score of almost 7 (6.903). The groups with the highest depression scores
(HS and GS) also have the highest variances, a finding that suggests that heterogeneity of
variance may affect the validity of conclusions based on the ANOVA of these data. Skew
and kurtosis are most pronounced in the B group. This is of interest because nonnormality
is most problematic when distributions vary in shape (Lindquist, 1953, pp. 78-90). Box
plots obtained from SPSS are presented in Fig. 8.3. They provide a more direct view of the
distributions, and the differences in location, spread, and skew are quite evident, as is the
presence of outliers in the B and GS groups. These characteristics of the plot suggest that
the assumptions of the ANOVA are violated. In Section 8.7 we discuss those assumptions,
together with alternative analyses developed to respond to violations of them.
TABLE 8.7 SUMMARY STATISTICS FOR BECK DEPRESSION
SCORES IN FOUR EDUCATIONAL LEVELS
HS C B GS
N of cases 19 33 37 39
Median 6.272 2.875 2.265 3.031
Mean 6.903 3.674 3.331 4.847
Variance 34.541 5.970 9.861 26.218
Skewness(Gl) .824 0.368 2.047 1.270
Kurtosis(G2) .168 -0.919 5.837 .745
Note. HS = high school diploma only; C = some college; B =
bachelor's degree; GS = graduate school. Data may be found in the
Male_educ file in the Seasons folder.
Fig. 8.3 SPSS Box plots of Beck depression scores as a function of educational level.
Note that the numbers next to the outliers are case numbers and the asterisk represents an
extreme outlier.
The output is based on an analysis of Beck depression means for those participants
for whom scores in all four seasons were available. It contains considerable information
in addition to the ANOVA table. First note the squared multiple R of .079. Under Cohen's
(1988) guidelines, this would be a medium-sized association between the dependent and
independent variable. The ANOVA table follows and it is clear that the omnibus null hy-
pothesis can be rejected. The last thing to note is the warning that two scores are outliers,
using the studentized residual as a criterion.2 Upon examining these two cases, we found
that one belonged to the HS group and the other to the GS group. We redid the ANOVA with
these cases excluded, and the result was no longer significant; the new p value was .065.
Deleting just 2 of 128 scores changed the p value from .016 to .065, and a finding of statisti-
cal significance to nonsignificance. Despite the fact that statistical significance depended on
the presence of the two outliers, there are reasons to suspect that the population distributions
of depression scores do differ. First, even after the two outliers were excluded, the medians
of the three groups with some education beyond high school were about 3 or less, whereas
that for the HS group was more than twice as high. Second, examination of the four distri-
butions revealed that 26% of the HS scores were above 10, the cutoff for dysphoria, roughly
defined as a state of dissatisfaction (16 is the cutoff for depression) and 20% of the GS
group exceeded that cutoff; in contrast, only one individual in the other two groups had an
average Beck depression score exceeding 10. The point we wish to make is that we need to
look beyond the significance test and examine the sample distributions to better understand
ASSESSING THE IMPORTANCE OF THE INDEPENDENT VARIABLE 207
TABLE 8.8 ANOVA OF THE DATA SUMMARIZED IN TABLE 8.7
Dep Var: D_SCORE N: 128 Multiple R: .282 Squared multiple R: .079
Source Sum-of -Squares df Mean-Square F-ratio P
EDUC$ 186.501 3 62.167 3.562 .016
Error 2164.061 124 17.452
*** WARNING ***
Case 14 is an outlier (Studentized Residual = 3.665)
Case 114 is an outlier (Studentized Residual = 3.622)
Durbin-Watson D Statistic 1.995
First Order Autocorrelation -.006
our data. In addition, we should supplement the omnibus F test results with measures of
importance, such as those we consider in Section 8.5. Finally, CIs and significance tests on
specific contrasts (for example, the HS group against the combined mean of the other three
groups) will be informative. We introduced such contrasts in Chapter 6 and have more to
say about them in Chapter 9. For example, there we pursue the question of whether the HS
mean differs significantly from the mean of the other three groups combined, as the means
and medians suggest.
Significance tests and confidence intervals (CIs), examination of contrasts, measures
of importance, examination of group distributions—these combine to provide a sense of
whether the sampled populations differ and in what ways. The cause of the differences
can be elusive. For example, it is not clear why high school graduates (and possibly also
those with graduate school education) are more likely to be depressed than individuals in
the other two categories, if this difference proves to be reliable. Two possible causes of
such differences are occupational and age differences among the groups. Individuals with
only a high school education may be more dissatisfied with their jobs, and individuals with
graduate school education may feel overqualified for theirs. If the four groups differ in
age, there may be differences in health and, therefore, in depression scores. In summary,
even if we conclude that the populations differ in some respects, we must ask why they
differ.
8.5 ASSESSING THE IMPORTANCE OF THE
INDEPENDENT VARIABLE
Too often, researchers view .05 as a boundary dividing real effects from null effects. How-
ever, there is nothing magical about p = .05; it is merely one possible index of the effect
of the independent variable. Usually, when p is less than or equal to .05, researchers reject
the null hypothesis. But should we really have so much less confidence that there is an
effect if p = .06? And if p = .01, does this statistically significant result reflect an effect
of practical or theoretical importance, or does it reflect a researcher with access to a large
pool of subjects? There is no statistic that carries a magic dividing line; however, there are
several measures that complement p, and that are more directly related to the variance of
the treatment means, and therefore are more easily interpreted. Using the example of the
probability-learning experiment, we considered one such measure, R2, the proportion of
variability accounted for by the independent variable. R2 may also be viewed as a measure
of the strength of association between the dependent and independent variable. In what
follows, we consider another measure of the strength of association, as well as a standard-
ized effect statistic that parallels the Es presented in Chapter 6. The development of both
statistics follows from our knowledge of the expected mean squares.
8.5.1 Measuring Strength of Association:
w2 (Omega Squared)
When we perform an ANOVA, R2 is the ratio of the between-groups sum of squares to
the total sum of squares. An alternative measure of the strength of association between the
dependent and independent variable is the ratio of the population variance of the means to
the total population variance. Following Hays' (1994) notation, we label this ratio as w2.
The numerator of the ratio is the variance of the treatment population means (the u7j) or,
equivalently, the variance of the treatment effects (the a,):
The denominator of o2 is the total population variance; that is, it is the treatment population
error variance, v2, plus the variance of the treatment population means, crj. Therefore,
We cannot know this ratio but we can use Equations 8.10 and 8.11 to derive an estimate
of it. More precisely, we derive estimates of a2 and a2.. We begin by restating the EMS
equations of Table 8.5:
To obtain an estimate of o^ we first subtract Equation 8.13 from Equation 8.12, and divide
by n; then we have
where the symbol = means "is an estimate of." However, the numerator of o2 as defined
by Equation 8.11 involves a2, not 02. Because a2 = [(a — l)/a]62, our estimate is
where the "caret" above the a indicates "the estimate of." We now have estimates of the
numerator and denominator of to2, and, therefore, substituting into Equation 8.11, we have
an estimate of o2: for the one-factor between-subjects design, we have
We may write Equation 8.15 in a different form, one that allows us to calculate or from
knowledge of the F ratio, a, and n. The advantages are that the expression is somewhat
simpler, and, perhaps more important, because most research reports contain this informa-
tion we can estimate the strength of association for data collected by other investigators. We
begin by defining FA = MS A /MSs/A- Then, multiplying the numerator and denominator of
Equation 8.15 by an, and dividing by MS$/A, we have
Let's review what Equation 8.15 (or 8.16) represents. If we replicate the experiment
many times, the average value of the right-hand term will approximately equal w2, the
proportion of the total variance in the a treatment populations that is attributable to the
variance of their means. We say "approximately equal" because the expected value of a
ratio is not the same as the ratio of expected values. The approximation is reasonably
accurate and the expression is much simpler than that for the correct expression.
One other aspect of Equation 8.16 should be noted. If the null hypothesis is true, it is
quite possible that the F will have a value less than one, because in that case we have two
independent estimates of the error variance and either one could be the larger of the two.
Then, u2 would be less than zero and we should conclude that w2 = 0; that is, none of the
total population variance is attributable to the independent variable.
We can apply Equation 8.16 to the probability-learning experiment. In that experiment,
a = 3, n = 16, and (from Table 8.3) F = 2.965. Then, inserting these values into Equation
8.16, we have
This is very close to the value we calculated earlier for R2. That the values of R2- and co2
are so close is not unusual; Maxwell, Camp, and Arvey (1981) found that the two rarely
differ by more than .02. With respect to assessing the importance of either measure, Cohen
(1988) suggested that values of ,01, .06, and .14 may be viewed as small, medium, and
large, respectively. According to those guidelines, the proportion of variability accounted
for may be judged to be medium. Again, however, we caution that the importance attached
to any value must be assessed in the context of the research problem and the investigator's
knowledge of the research literature.
8.5.2 Measuring Effect Size
R2 and w2 have intuitive appeal because their values are on a scale of zero to one and
they can be directly interpreted in terms of the variability accounted for by the independent
variable. However, standardized effect sizes such as those presented in Chapter 6 have other
advantages. They play an important role in meta-analysis, a procedure for combining the
results of several experiments (Hedges & Olkin, 1985), and, as we saw in Chapters 5 and 6,
standardized effect sizes, together with a and n, determine power. In Chapter 6, we defined
the standardized effect size as
Cohen (1988) suggested another standardized effect-size measure, /, which is a useful
adjunct to ANOVA:
The quantity under the square root sign in Equation 8.17 is the ratio of C2, the variance
of the treatment population means, to c 2 ,the population error variance. An estimate of c2
is provided by Equation 8.14 and the error variance is estimated by MSs/A- Therefore, our
estimate of / is
Substituting values from the output in Table 8.4 into the last equation, we have
Cohen (1988) has suggested that / = .10, .25, and .40 corresponds to small, medium,
and large effect sizes, respectively. Using these guidelines, we judge the effect to be of
medium size. This corresponds to the conclusion reached when we estimated co2. This is
not surprising because the two measures are directly related:
Estimates of either of two other parameters are often used in obtaining the power of
the F test. When the null hypothesis is false, the ratio of mean squares has a noncentral
F distribution, with noncentrality parameter, A. (lambda). This parameter serves as
one of the inputs to software programs for finding power, such as SAS's CDF module
TABLE 8.9 SOME PARAMETERS AND STATISTICS FOR THE ONE-FACTOR
BETWEEN-SUBJECTS DESIGN
Parameters Estimators
and the UCLA calculator. Another parameter, used in conjunction with power charts that
are found in many textbooks, is O) (phi). These two parameters, as well as / and w2,
together with formulas for estimating them, and relations among them, are presented in
8.5.3 Measures of Importance With Unequal Group Sizes
When there are different numbers of scores in each condition, we need a different way of
estimating a2A than that presented in Table 8.9; we cannot just divide by n because there is
no single value of n. However, if the ns vary by chance and are not too different, the average
n might replace the n in the denominator of the estimator of Table 8.6. That is, we might
The first line in Equation 8.19 is identical to that for the equal-n case (Equation 8.14), except
that n has been replaced by the average of the ns.
We can arrive at Equation 8.19 in a somewhat different way. Suppose the populations
do differ in size. Then the definition of c2 is
where the PJ sum to one and are weights reflecting the relative population sizes. From the
EMS in Table 8.6,
which is a reasonable estimate of the parameter defined by Equation 8.20 if nj/N is an
adequate estimate of P . Although the rationales differ, the left-hand side of this equation
is identical to the last line of Equation 8.19.
To obtain estimates of the other parameters in Table 8.9 in the unequal-n case, we need
only substitute the expression for d^ in Equation 8.19 into the other expressions in the
8.5.4 Measures of Importance: Limitations
In an introductory chapter to an edited collection aptly titled What if There Were No Sig-
nificance Tests?, Harlow (1997, pp. 5-6) reported that 11 of the book's other 13 chapters
"were very much in favor" of reporting measures such as R2, w2, and Es, and the remaining
two contributors "at least mildly endorsed such use." Similar support for measures such as
these can be found in The American Psychological Associations's guidelines for statistical
usage (Wilkinson, 1999), which urge researchers to report effect size statistics. Neverthe-
less, there are potential pitfalls. Values of these statistics may depend on the experimental
design (for example, between or within subjects), the choice and number of levels of the
independent variable, the dependent variable, and the population sampled. Another concern
is that squared coefficients tend to be small and it is sometimes easy to dismiss an effect
as trivial because of a small value of co2. These arguments suggest that we must be careful
in generalizing the results of any one study, or of making comparisons across studies that
differ with respect to the factors just cited. In addition, we should treat guidelines such
as those set forth by Cohen (1988) as suggestions, not as definitive boundaries between
important and unimportant effects. Even a very small advantage of one therapy over an-
other may be important. In theoretical work, a small effect predicted by a theory may be
important support for that theory. In summary, if care is taken in interpreting measures
of strength, statistics such as f and co2 are useful additions to the test statistics usually
computed.
8.6 POWER OF THE F TEST
Power calculations can play an important role in planning the experiment. Ideally, re-
searchers would have an estimate of f or w from previous studies; these could then be
used to determine the sample size needed to achieve a specific level of power. However,
such estimates may be unavailable or very wrong, or the required n may be too large to
be practical. In such cases, it is still helpful to estimate the power of the F test after the
POWER OF THE F TEST 213
data have been collected. We describe this post hoc use of power functions first. Then we
consider a priori applications in which the researcher uses the power function of the F to
decide on a sample size before running the experiment.
There are several situations in which it is important to have a sense of the power of the data
analysis. In one case, the researcher predicts effects of the independent variable and there
is a clear trend among the observed means that supports the researcher's prior hypothesis.
However, the result is not statistically significant. Therefore, either there really are no effects
of importance in the sampled population and the pattern of observed means is due to chance,
or the treatment population means do differ but the research had too little power to detect
this treatment variance. In this situation we may ask what power the F test had to reject
HO assuming the population effect size estimated from the data. If this power is low, the
research may be replicated with a larger n or with procedures or measures designed to
reduce error variance.
In a second case, the researcher predicts no effect and does not obtain a significant result.
In this situation, before claiming a successful prediction, the researcher should demonstrate
that power to detect the effects estimated from the data was high and therefore the failure to
achieve significance was because the independent variable had no, or trivially small, effects.
Finally, in a third situation, the experimenter predicts no effect but the F test produces a
significant result. Here, if power is very high to detect even very small effects, the effects
estimated from the data, although possibly real, may not be large enough to be of theoretical
or practical interest.
In all of these situations, estimates of / and co2 will be important in discussing the
results because they help provide a sense of the absolute and relative contributions of
the independent variable to the total variability. Calculations of power supplement these
statistics and place the results of our statistical text in a useful context.
The power of the F test (and of other tests; see the discussions in Chapters 4-6) depends
on several factors:
1. The significance level, a. As we reduce the rejection region, say from .05 to .01,
we lower the probability of rejecting false, as well as true, null hypotheses. In
other words, a reduced Type 1 error rate is accompanied by reduced power.
2. The values of a and n. Increases in either numerator or denominator degrees of
freedom yield increased power. Ordinarily, the value of a is determined by the
goals of the experiment; n is usually more arbitrarily selected, although con-
strained by practical concerns such as time, effort, and cost. In Subsection 8.6.2,
we consider how power calculations can, and should, influence decisions about
sample size.
3. The error variance, a2. The less noise in our data, the easier it will be to detect
treatment effects. Therefore, power increases with decreases in error variance.
This variance will be a function of the dependent variable, the subject population,
and the experimental design.
4. The variance of the treatment effects, o^. In the case of the F test, we will have
more power to reject the null hypothesis of equal treatment population means
when the differences among them are larger.
To determine the power of the F test, we need the numerical values of the four factors
just cited, and we then need to find some way of relating power to them. Assuming we
have run the experiment, we know df1 and d/2, and have decided on a, we see that MSs/A
provides an estimate of c2. An estimate of the variance of the treatment effects is provided
by d^, which was defined in Equation 8.14. To obtain a value of power, we calculate one of
several closely related indices that are based on the ratio o^/oV Some sources (including
the previous edition of this book) contain "nomographs," charts in which power is plotted
as a function of 4> = / x Jn, with different curves for different values of df 1 , d/2, and
a. These charts are awkward to use, and provide at best approximate results.3 A somewhat
better approximation may be obtained by using an estimate of / (defined by Equation 8.18)
together with tables provided by Cohen (1988). Of course, no chart or table can provide
power values for every possible combination of degrees of freedom or for every possible
value of a. The best solution to the problem of calculating power is to use a software
application that calculates power when the necessary information is input. Several statistical
packages such as SYSTAT, SAS, and SPSS4 will do this, at least for some tests. Furthermore,
a number of easy-to-use programs are freely available on the Internet. We used two of
these, GPOWER and the UCLA calculator, to obtain the power of the t test in Chapter 6.
Table 8.10 illustrates the application of both these programs to find the power of the F test,
using the data from the probability-learning experiment.
It is easy to become so focused on the process that we forget what the result means.
What the estimated power of .39 in Table 8.10 means is the following: Given the sample
size we ran, and assuming the MSs/A is a reasonable estimate of c2, there is a probability
of about .39 that we will reject the null hypothesis, if the effects in the population are of
the order of magnitude estimated from our data. Put somewhat differently, if the treatment
population means are about as different as our sample means suggest, we still have a .61
(1 — .39) probability of making a Type 2 error. Noting the low power, the p value that fell
only a little short of .05, and the medium-sized estimate of f (or w2), we find evidence that
the independent variable may have an effect. Although we cannot reject the null hypothesis,
there is some basis for believing that a more powerful replication of the experiment might
produce a statistically significant effect. If we replicate the experiment, we might attempt
to improve its sensitivity. One way to do this is to increase the number of participants.
Another way would be to increase the number of items on the test given to the participants.
This would decrease the variance of the test score, and thus decrease the error variance.
Decreasing the variance of the dependent variable is often possible, and often less expensive
in time and resources than running more subjects.
Ideally, we should have some idea of the size of the effect we want to detect before running
an experiment. Pilot data, or a review of related experiments, might suggest that effects will
be of a certain magnitude. It is not necessary to have a precise estimate of f usually, some
sense of whether the effect is small, medium, or large (using Cohen's 1988 guidelines)
will do. Certainly, a decision about sample size based on any estimate of / will be an
improvement over an arbitrary selection of n. In some cases, the n required to achieve a
certain level of power against a specified effect size will be unpractically large. We can then
calculate what power we have with the largest n available for our study. If that power is very
TABLE 8.10 CALCULATING THE POWER OF THE F TEST
Using GROWER
1. Under "Tests," select "F Tests (ANOVA)."
2. Fill in these values from the probability-learning experiment:
/ = .286
Alpha = .05
Total sample size = 48
Groups = 3
3. Click on "Calculate.", The results are:
Lambda = 3.9262
Critical F = 3.2043
Power = .3852.
Note that / was estimated previously (Subsection 8.5.2) by using Equation 8.2;
also, X = anf 2 = 48 x .2862.
Using the UCLA calculator
1 . Go to the UCLA calculator (see Table 8.4 or Chapter 6 for the URL).
2. Click on "Noncentral F CDF Calculator."
3. Fill in these values:
X value = 3.2043 (the critical value of F for a. = .05)
Probability = ?
Numerator df= 2
Denominator df= 45
Noncentrality parameter = 3.9262 (X)
4. Click on "Complete Me!"
5. The question mark is replaced by .6148; this is beta, the probability of a Type 2
error. Subtracting from 1, we have power = .3852.
Note. The X value can be obtained from Table D.5, which contains critical values of the central F distribution
for various combinations of a and degrees of freedom. If the degrees of freedom for your data set are not in Table D.5,
interpolation will usually provide a reasonable approximation to the required X value. Better still, the central F calculator
at the above Web site will provide the necessary result.
low, consideration should be given to ways of decreasing variability—perhaps a different
research design, or a different dependent variable.
Finding the required n is simple with GPOWER. Select "F Tests" and "A priori."
Indicate the value of /, the a level, the desired power, and the number of groups. If
power = .8, f = .25 (medium), ct = .05, and there are three groups, the total N is 159, or
n = 53 participants in each group. If the UCLA calculator is used, the process involves trial
and error. Having selected a trial N (the total number of participants, no), you must calculate
the denominator degrees of freedom, find the F needed for significance at the desired a
level (the "X Value"), and calculate the noncentrality parameter, A. Enter these together
with the numerator degrees of freedom and a question mark in the probability space. If
the probability that is returned is greater than one minus the desired power (remember, the
calculator returns (3), increase the n; this means recalculating the denominator degrees of
freedom, the critical F value, and X. If 3 is very small, you can decrease the n and make
the necessary adjustments in the variables needed for the calculator. The same parameters
are entered into the SPSS "Compute" module.
8.7 ASSUMPTIONS UNDERLYING THE F TEST
Although the critical values of F in Appendix Table C.5 are derived from the assumptions
presented previously, it does not follow that violations of the assumptions necessarily in-
validate the F test. For example, in view of our discussion of the central limit theorem and
the t test (in Chapter 6), we might guess that the ratio of mean squares will be distributed
approximately as F even when the populations are not normal. In this section, we look
at the role of assumptions more closely. We ask what the consequences of violations of
assumptions are and, in those cases in which there are undesirable consequences, what
alternatives to the standard analysis exist.
8.7.1 Validity of the Structural Model
It is important to bear in mind that the ANOVA for the one-factor design begins with the
assumption of Equation 8.9. That equation implies that only one factor systematically influ-
ences the data and that the residual variability (MSs/A) represents random error. However,
researchers sometimes ignore factors that have been manipulated but are not of interest
in themselves. If those factors contribute significant variability, the one-factor model is
not valid for the research design. Common examples arise when half of the subjects are
male and half are female, or when subject running is divided equally between two experi-
menters, or when the position of an object is counterbalanced in an experiment involving
a choice. Although the researcher may consider these variables irrelevant to the purpose
of the research, they may affect the outcome. If so, the MSs/A represents both error vari-
ance and variance caused by gender, experimenter, or position, but the variance caused
by these "irrelevant" variables will not contribute to MSA because—for example—there
will be an equal number of male and female subjects at each level of A. The analysis
based on the one-factor model then violates the principle that the numerator and denomi-
nator of the F ratio should have the same expectation when H0 is true. In such situations,
the denominator has a larger expectation because the irrelevant variable makes a contri-
bution. The result is a loss of power that can be considerable if the irrelevant variable
has a large effect. We say that the F test is negatively biased in this case. As a general
rale, the researcher should formulate a complete structural model, one which incorpo-
rates all systematically varied factors, even those thought to be irrelevant or uninteresting.
In the examples cited, this would mean viewing the study as one involving two factors,
A and gender (or experimenter, or position), and carrying out the analysis presented in
When only one observation is obtained from each participant, and participants are randomly
assigned to treatments or randomly sampled from distinct populations, the assumption that
the scores are independently distributed should be met. There are exceptions, however,
that are often unrecognized by researchers. For example, suppose we wished to compare
attitudes on some topic for male and female participants. Further suppose that before being
tested, the participants are involved in three-person discussions of the relevant topic. The
ASSUMPTIONS UNDERLYING THE F TEST 217
scores of individuals who were part of the same discussion group will tend to be posi-
tively correlated. If this failure of the independence assumption is ignored (and it often
is; see Anderson & Ager, 1978, for a review), there will be a positive bias—an inflation
of Type 1 error rate—in an F test of the gender effect. Why this is so, and the nature of
the proper analysis, is explained in Chapter 16. Another potential source of failure of the
independence assumption is the "bottom-of-the-barrel" problem. Researchers at universi-
ties often feel that as the semester progresses, the performance of volunteer participants in
experiments tends to become poorer because less motivated participants usually volunteer
for research credit late in the semester. Then scores obtained close in time will tend to be
correlated.
Autocorrelation plots, such as those in Fig. 8.4, provide one diagnostic tool. The
bars represent the average correlations of residuals (Yij — Y.j) that are various temporal
distances ("lags") apart. If any of the bars exceed the confidence limits indicated by the
two horizontal lines, the independence assumption may be invalid. In the upper panel of
Fig. 8.4, the correlation tends to be larger for scores nearer together in time, suggesting
some failure of the independence assumption. However, all the bars are within the CI and
the correlations are generally small. The bottom panel represents an artificial data set into
which we built a tendency for scores to become worse over time. As a result, scores near
each other had very high positive correlations and scores far apart tended to have a negative
correlation. Scatter diagrams of scores versus time of test may also be useful in checking for
trends over time (e.g., that performances were deteriorating or becoming more variable as
the semester progressed). Departures from a best-fitting line with slope of zero, or changes
in the spread of scores as a function of time, would suggest that scores were dependent on
when they were obtained.
Mathematical proofs (Scheffe, 1959) and computer-sampling studies (e.g., Donaldson,
1968; Lindquist, 1953, pp. 78-90) have shown that the Type 1 error probability associ-
ated with the F test is little affected by sampling from nonnormal populations unless the
samples are quite small and the departure from normality extremely marked. This reflects
the role played by the central limit theorem; the distribution of means and their differences
will tend to be normal as n increases even when the distribution of the parent populations
is not. The F test's robustness with respect to Type 1 error rate appears to hold even when
the independent variable is discretely distributed, as it is whenever rating data or response
frequencies are analyzed. Computer-sampling studies indicate that Type 1 error rates are
relatively unaffected when such measures are submitted to an ANOVA. With as few as
two rating points (Lunney, 1970) and two groups of 3 participants each (Bevan, Denton, &
Myers, 1974; Hsu & Feldt, 1969), in all but the most skewed distributions the proportion of
rejections of the null hypothesis fell within two standard deviations of the theoretical alpha.
Furthermore, the distance between the empirical and theoretical alpha decreased with more
rating points, larger samples, and more symmetric distributions.
Although the Type 1 error rate is relatively unaffected by nonnormality in most in-
stances, power may be a concern. When treatment distributions are not normal, there are
alternatives to the ANOVA that will be more powerful. Such nonparametric tests do not
require the assumption that the population distribution is normal. We consider two such
tests next.
Fig. 8.4 Autocorrelation (ACF) plots of two data sets (from SPSS).
ASSUMPTIONS UNDERLYING THE FTEST 219
8.7.4 Tests Based on Ranks: The Kruskal-Wallis H
and the Rank-Transform F Tests
These tests are based on an analysis of ranks and are closely related. Both tests require that
all scores are ordered with a rank of 1 assigned to the lowest score in the data set and a rank
of N assigned to the largest, where N is the total number of scores in the data set. In case
of ties, the median rank is assigned. For example, if the 5 lowest scores are 1,4,1,9, and
9, they would receive ranks of 1, 2, 3,4.5, and 4.5 respectively. The Kruskal-Wallis H test
(Kruskal & Wallis, 1952) is available from the nonparametric menu of statistical packages
such as SPSS, SYSTAT, and SAS. Basically, it is the treatment sum of squares based on
the ranks, divided by the variance of N consecutive integers. Recall from Chapter 7 that
the ratio of a sum of squared deviations of normally distributed variables divided by the
population variance is distributed as x2 if the variables are normally distributed. If the HJ
are not very small—5 or more is the usually recommended cutoff—the means of the group
ranks will be approximately normally distributed and, under the null hypothesis, H will be
distributed as x2 on a — 1 df. A computing formula is
where nj is the number of scores in the jth group, TV is the total number of scores, and T.j
is the sum of ranks in the jth group.
The rank-transform F test (Conover & Iman, 1981) is conceptually even simpler. After
the scores are transformed, the usual one-way ANOVA is performed on the ranks and the
test statistic, FR, is evaluated on a—1 and N — a df. H and FR will generally result in
similar p values. This is not surprising given that they are related by the following equation:
Neither test requires the assumption that the treatment populations are normally dis-
tributed. However, the test of equal population means rests on the assumption that the
population distributions have identical shapes; that is, that they have the same values of
variance, skewness, and kurtosis. Under this constraint, the alternative to the null hypothe-
sis is referred to as the shift hypothesis, so called because it implies that the alternative is
that the treatment has shifted the distributions but not influenced them in any other way. If the
populations are not normally distributed but have the same shapes, then these tests will often
have more power than the F test. Furthermore, they are only slightly less powerful than the
F test when the distributions are normal. These points are illustrated in Table 8.11, which
presents the proportion of rejections for F and H in 1,000 computer-sampling experiments
with three different population distributions. The mixed-normal distribution is symmetric
but has heavier tails than the normal and the exponential is extremely skewed. When HO is
true (distance = 0), the empirical rejection rate is usually within two standard deviations
(or .014) of .05, the theoretical value of a. When the null hypothesis is false (distance = .4 or
.8), the F has a slight power advantage over the Kruskal-Wallis and rank-transform F tests
if the populations are normally distributed. However, for the two nonnormal distributions,
the nonparametric tests generally have more power.
There has been considerable confusion in journal articles and statistics texts about the
null hypothesis tested by H and FR. It has frequently been claimed that the Kruskal-Wallis
TABLE 8.11 EMPIRICAL REJECTION RATES FOR THE F AND KRUSKAl-WALLIS H TESTS
Distance Between Adjacent Population Means
0 .4 .8
n Test F H F H F H
5 Normal .054 .049 .167 .144 .504 .441
5 Mixed-normal .055 .055 .119 .122 .363 .357
5 Exponential .048 .047 .197 .243 .578 .616
10 Normal .048 .036 .308 .285 .876 .852
10 Mixed-normal .038 .040 .205 .234 .644 .720
10 Exponential .035 .047 .365 .528 .859 .936
H test should be used instead of the usual F test when there is heterogeneity of variance.
However, if the treatment populations do not have identical distributions, then H and FR
tests may reject the null hypothesis because of differences in the shapes of the distributions,
not their locations. On the basis of a computer-sampling study, Oshima and Algina (1992)
warn against using the Kruskal-Wallis test when the homogeneity of variance assumption
is suspect. A detailed discussion of what is tested by the Kruskal-Wallis and rank-transform
tests under various conditions is provided by Vargha and Delaney (1998). Our view is that
these tests can be useful when two conditions are met: first, there is clear evidence that the
data are not normally distributed; second, the treatment distributions are similar in shape.
Researchers entertaining the possibility of using either nonparametric alternative should first
view a normal probability plot of their data. As shown in Chapter 2, clear departures from
a straight line indicate a departure from normality. In addition, as discussed in Chapter 7,
tests of normality are available in SPSS and SAS.
If there is evidence of nonnormality, the next step would be to compare plots of the
data from the different treatment groups, using box plots, histograms, or dot-density plots.
Obvious differences in shape, or marked differences among variance, skewness, or kurtosis
measures, would suggest that the H or FR statistic is not appropriate for testing equality of
population means.5
When there are only two groups, the Kruskal-Wallis test is equivalent to the Wilcoxon
rank sum (or the equivalent Mann-Whitney U) test, and the rank-transform F test produces
the same result as a t test applied to the ranks. One drawback in using these rank-based
procedures is that although CIs on the median difference can be established, the process is
somewhat tedious and requires tables of the Mann-Whitney U test. Steps for calculating
the CI are outlined by Marascuilo and Serlin (1988, p. 242), and the necessary tables are in
the appendix to their book.
8.7.5 The Homogeneity of Variance Assumption
Variances may differ across conditions for one of several reasons. One possible cause of
heterogeneity of variance is an interaction of an experimental treatment with individual
characteristics. For example, a drug tested for its effects on depression may result in a
higher variance, but the same mean score, as a placebo. This would suggest that some
individuals had improved but others had been adversely affected by that drug. A second
possible reason for unequal variances is that some populations are more variable than others
ASSUMPTIONS INDERLY NG THE F TEST 221
on a particular task. For example, although boys may have higher average scores on some
measure of mathematical ability, they may also have a higher variance. Still another factor in
findings of heterogeneity of variance are floor, or ceiling, effects. Variability may be reduced
in one condition relative to another because of a lower, or upper, limit on performance
that is due to the measuring instrument. Finally, variances tend to be correlated, usually
positively, with means; the normal distribution is the sole exception in which the means and
variances are independently distributed. For all of these reasons, homogeneity of variance
is rarely present in the populations sampled in our research. In this section, we summarize
some consequences of the failure of this assumption; in the sections following we consider
alternatives to the standard F test.
When there are the same number of scores in all conditions, heterogeneous variances
usually will cause Type 1 error rates to be inflated. The inflation is usually less than .02 at
the .05 level, and less than .005 at the .01 level, provided the ratio of the largest to smallest
variance is no more than 4:1, and n is at least 5. Even larger ratios may not be a problem,
but this will depend on sample size, the number of groups, and the shape of the population
distributions. The results of computer simulations employing these factors are discussed in
articles by Clinch and Keselman (1982) and Tomarken and Serlin (1986).
When there are different numbers of scores in each condition, simulation studies clearly
demonstrate that heterogeneous variances are a problem. Sampling from sets of either 3 or
4 normally distributed populations, Tomarken and Serlin found that at a nominal .05 level,
the actual Type 1 error rate was as low as .022 when the group size was positively correlated
with the variance and as high as .167 when the correlation was negative. Sampling from
heavy-tailed and skewed distributions, and using variance ratios of largest to smallest as
high as 16:1, Lix and Keselman (1998) found that error rates were as high as .50 in some
conditions. The expected mean squares may clarify why heterogeneous variances are more
troublesome when ns are unequal. For simplicity, assume the null hypothesis is true. Then,
it can be shown that
where N = £j nj and 02 is the variance of the yth treatment population. If the nj all equal
n, then N = an, and both expressions reduce to the same quantity; under the null hypo-
thesis, E(M$A) = E(MSs/A) = Eja2/a the average population variance. This equality of
expectations does not mean that the ratio of mean squares will be distributed as F when
variances are heterogeneous; we have already noted that there may be inflation of Type 1
error rate even when ns are equal. Nevertheless, the equality does constrain the degree to
which the distribution of mean squares will vary from the theoretical F distribution.
The larger the nj, the smaller the contribution of the j th population variance to E(MSA)
in Equation 8.22, because N — nj decreases as HJ increases. The opposite is true for
E(MSs/A)', the larger the nj, the greater the contribution of the jth population variance
to the denominator because nj — 1 increases as nj does. Therefore, when large variances
are paired with large group sizes, they will increase the denominator relatively more than
the numerator; as a result, the true Type 1 error rate will be below the theoretical value
and power against false null hypotheses is likely to be reduced; there is negative bias. In
contrast, when large variances are paired with small group sizes, they tend to contribute
relatively more to the numerator of the F test than to its denominator. In this case, the bias
is positive; the Type 1 error rate is inflated.
There is evidence that extreme variance ratios do occur in the research literature
(Wilcox, 1987), and a host of simulation studies make clear that heterogeneity of vari-
ance can inflate Type 1 error rates and deflate power, depending on various factors such as
sample sizes and the type of distribution sampled. That leaves us with two questions. First,
for a given data set, how do we decide whether to abandon the standard ANOVA for some re-
medial procedure? Second, if we do decide that unequal variances are a threat to the validity
of the standard F test, what alternative should we use? We consider these questions next.
8.7.6 Detecting Heterogeneity of Variance
As always, we urge that researchers begin the data analysis by examining summary statistics
and plots of the data. SPSS's Explore module (click on "Analyze" followed by "Descriptive
Statistics") is very helpful in this respect. It provides descriptive statistics, tests of homo-
geneity of variance, and box plots. The box plot for the Beck depression data as a function
of educational level was presented in Fig. 8.3 and, as we noted there, differences among the
groups in shape and spread are quite evident. The range of variances in Table 8.7 strongly
suggests that the alpha level reported in Table 8.8 may not be the actual probability of a
Type 1 error. For confirmation of this, we may wish to test whether the variances are ho-
mogeneous. Several tests of homogeneity of variance have been proposed. Some of these
(Bartlett, 1937; Cochran, 1941; Hartley, 1950) have been shown not to be robust in the
face of data that are not normally distributed and therefore are not considered further. The
Box-Scheffe (1953, 1959) test provides Type 1 error rates that are close to the nominal
alpha but lacks power relative to other procedures (Games, Keselman, & Clinch, 1979). A
test proposed by Levene (1960) is available in several statistical packages such as SYSTAT,
SPSS, BMDP, and SAS. In this test, the absolute residual of each score from its group mean,
Yij — Y.J , is computed and these residuals are then submitted to the ordinary ANOVA.
Although these residuals do not directly represent the variance, they are an index of the
spread of scores. For the data summarized in Table 8.7, SPSS's Explore module reports the
value of this statistic as 7.722, which, on 3 and 124 df, is very significant (p = .000) and in-
dicates that the mean absolute residual varies significantly as a function of education level.
The SPSS output also reports three other "Levene statistics." The first of these, actually
developed by Brown and Forsythe (1974a), is similar to the Levene test except that the data
are absolute deviations from the group medians, rather than the means. This test should be
less sensitive to outliers. The second alternative to the Levene test is a further modification
of the Brown-Forsythe test, in which degrees of freedom are adjusted as a function of the
group variances. The third alternative is one in which an ANOVA is performed on residuals
from the trimmed group means. The trimmed means are obtained by removing the highest
and lowest 5% of the scores in each group, and then calculating the means. This provides
another way of reducing the impact of outliers. With respect to the depression scores, all
four methods produced p values of .005 or less, confirming our sense that the spread of
scores was indeed a function of the educational level.
How robust are these tests of the null hypothesis of equal variances? The Levene test
appears to have an unacceptably high Type 1 error rate when the population distributions
are skewed or bimodal. Empirical rejection probabilities reach as high as .25 with very
skewed distributions when the nominal Type 1 error rate is .05 (Church & Wike, 1976).
We recommend the Brown-Forsythe test based on deviations from the median. Sampling
studies indicate it has only a slightly inflated Type 1 error rate and good power relative to
various competitors even when ns are not equal and distributions depart markedly from the
normal (Games et al., 1979).
Once we conclude that the population variances are not equal, the next question is,
What shall we do about it? One possible response is to seek a transformation of the data that
yields homogeneity of variance on the scale resulting from the transformation. A second
possibility is to compute an alternative to the usual F test. We consider each of these
approaches next.
8.7.7 Transformations of the Data
Statisticians have frequently recommended transforming data, for example by raising all
scores to some power, or by taking the logarithm of all scores. Transformations have been
used to transform skewed distributions into more nearly normal distributions, to reduce
heterogeneity of variance, and to remedy a condition known as "nonadditivity" in designs
in which each participant is tested on several trials or under several treatment levels. We
delay discussion of this third purpose until Chapter 13. For now, we note that a transformation
that best achieves one purpose may not be equally suitable for other purposes, although it
is true that transformations that equate variances do tend to yield more normally distributed
scores. Our focus here is on transformations designed to achieve homogeneous variances.
Typically, a variance-stabilizing transformation can be found when there is a functional
relation between cell variances and cell means. Smith (1976) has presented the rationale un-
derlying the derivation of transformation based on the relation between variances and means,
as well as several transformations representing different functional relations. Emerson
and Stoto (1983) have described a general approach that will frequently produce more
nearly equal variances. The technique involves plotting the log of the H spread (or in-
terquartile range; see Chapter 2) as a function of the log of the median and then finding
the slope of the best-fitting straight line. Then, if Z is the transformed score, the power
transformation is
If the slope equals one, the power equals zero, and the appropriate transformation is
Z = log(7). SPSS provides a spread versus level plot of log(H spread) as a function
of log(median), and outputs the slope and power. Select "Analyze," Descriptive Statistics,"
and "Explore," and then choose the "Plots" option and, within that, "Power Transforma-
tions." We followed this procedure with data from Royer's (1999) study of arithmetic skills
in elementary school children. Submitting response times (RTs) on multiplication problems
to SPSS's Explore module, we found that the slope of the spread-versus-level plot was 2.227
and the recommended power was therefore —1.227. We rounded this, letting Z — Y-1 =
1 / Y, thus reexpressing RT as response speed, a measure that is easily understood. Table 8.12
presents the group means and variances on the original and new data scales. On the original
RT scale, the ratio of largest to smallest variance is almost 15:1; on the speed scale, that
ratio is only 1.4:1. To provide a further check on the adequacy of the transformation, we had
SPSS plot the spread-versus-level function for the speed data. The slope of this function
was —.144 and therefore the power to which each score should be raised is 1— slope =
TABLE 8.12 MEANS AND VARIANCES OF MULTIPLICATION RTs FROM
THE ROYER DATA
Mean or Variance 5 6 7 8
RT Mean 3.837 1.998 1.857 1.935
RT Variance 4.864 .612 .328 .519
Speed Mean .350 .560 .586 .583
Speed Variance .033 .028 .031 .038
1.144. But this is close to one, indicating that no improvement is likely to be obtained by a
further transformation.
The choice of a dependent variable is often arbitrary and some transformation of the
original scale may be preferable if the distribution on the new scale has desirable properties,
such as normality or homogeneity of variance, that are lacking in the original data. One
potential problem is that values on the new scale may be less easily interpreted than on
the original scale. For example, the percentage of correct answers on a test (y) is easily
understood and communicated, but this is less true, for example, of the arc sine transfor-
mation (sin -1 ^/y, the angle whose sine is the square root of y), often recommended to
stabilize the variances of percentage scores. Another potential problem is that, although
variance-stabilizing transformations will usually maintain the ordering of the group means,
the relative distances among means may change, creating problems when the effects of
the manipulated factors are interpreted. Suppose a researcher has predicted that response
time (RT) will vary as a linear function of the levels of the independent variable. There is
an excellent chance that a test of linearity on a transformed scale will fail to support the
prediction because the means on the new scale are likely to fall on a curve. Transformations
only make sense when predictions are not dependent on the measurement scale. When the
measuring scale is arbitrary, the researcher may find it helpful to seek a transformation,
assessing the success of the transformation by plotting a spread-versus-level function as we
described in our example of the multiplication RTs and speeds. Keep in mind that although
it is perfectly reasonable to try several transformations, assessing each by plotting spread
against level, it is not appropriate to conduct a significance test after each transformation,
settling for the data scale that yields the largest F value. Such a procedure is bound to
increase the probability of a Type 1 error if the population means do not differ.
Often the researcher will not wish to transform the data, either because of the difficulty
of interpreting effects (or lack of effects) on the new scale, or because a strong theory
dictates the dependent variable. In other instances, it may be impossible to find a variance-
stabilizing transformation. Fortunately, there are other solutions that, although they also are
not always satisfactory, can often solve the heterogeneity problem. We turn now to consider
modifications of the standard F test.
8.7.8 Alternative Tests When Variances Are not Equal
Several alternatives to the standard F test of the equality of the a population means have
been proposed (Alexander & Govern, 1994; Brown & Forsythe, 1974b; James, 1951, 1954;
Welch, 1951), but no one test is best under all conditions. Each has strengths and weaknesses,
depending on the shapes of the treatment distributions, the relative sample sizes, and the
ratio of variances. Detailed information about the influences of these factors on alternative
tests of the null hypothesis is provided in articles by Coombs, Algina, and Oltman (1996),
Grissom (2000), and Lix, Keselman, and Keselman (1996). These, particularly the Grissom
article, will in turn provide many additional references.
When the data are normally distributed and ns are equal, most of the procedures are
reasonably robust with respect to Type 1 error rate; however, the standard F is slightly more
powerful if the population variances are equal. When the variances are not equal, which test
is best depends on the degree of skew and kurtosis, whether outliers are present, the degree
of heterogeneity of variance, the relation between group sizes and group variances, and the
total N (Clinch & Keselman, 1982; Coombs et al., 1996; Lix et al., 1996; Tomarken &
Serlin, 1986). Although there is rarely a clear-cut choice for any given data set, we illustrate
the application of the Welch test, Fw, to the data summarized in Table 8.7. Fw performs
well relative to various competitors except when the data are highly skewed (skew > 2.0) or
group sizes are less than 10 (Tomarken & Serlin, 1986; Lix et al., 1996). Furthermore, the
test is available in BMDP and SAS/IML. Table 8.13 presents the necessary formulas and
illustrates their application to the statistics of Table 8.7. The resulting p value is considerably
higher than the .016 we obtained by using the standard F calculations. The discrepancy can
TABLE 8.13 FORMULAS FOR THE WELCH (F w ) TEST AND APPLICATION TO THE
STATISTICS OF TABLE 8.7
The Welch test
Substituting Table 8.7 values in the preceding equations
Wj .550 5.528 3.752 1.488
Y.j 6.903 3.674 3.331 4.847
Then, u = 11.318, F. = 3.871, A = 2.594, B = 1.024, and F = 2.533;
also, df1 = 3, df2 = 1.018 = 55, and p = .066.
226 8/BETWEEN-SUBJECTS DESIGNS: ONE FACTOR
be accounted for by noting that the correlation between nj and s2 is negative —.59. There
are only 19 participants in the group having only a high school education (HS), whereas
the other groups all have at least 33 participants. That smaller group size is paired with the
largest group variance. As we noted earlier in the chapter, when larger groups have smaller
variances, they have too much weight in the denominator of the F test, with a resulting
inflated probability of Type 1 errors. The Welch test has compensated for this by taking the
inequalities in group sizes and variances into account.
In summary, we recommend the following with respect to alternative tests:
1. Look at summary statistics, including skew and kurtosis measures, and plot the
data. Box plots are widely available and will often provide a sufficient basis for
deciding whether heterogeneity of variance is a problem. Visual inspection will
often be sufficient, but tests of homogeneity such as the Brown-Forsythe test can
provide confirmation (Brown & Forsythe, 1974a).
2. If the ns are equal and the ratio of largest to smallest variance is no more than 4:1,
the ordinary F test will have a Type 1 error rate close to its nominal value and
will have slightly more power than alternative tests against false null hypotheses.
However, when in doubt, it is probably best to consider an alternative. Even when
all the assumptions of the F test are met, the evidence suggests that the power of
the F will exceed that of competitors such as Fw by less than .04.
3. Although there is no one test that combines honest Type 1 error rates with higher
power than its competitors, we recommend the Welch test. It is available in sta-
tistical packages and performs reasonably well except when the distributions are
very skewed and the ns are less than 10. For n < 6, the Brown-Forsythe test of
the means (Brown & Forsythe, 1974b) appears to have Type 1 error rates closer to
the nominal value and power roughly equal to that of Fw. Another possibility is
to apply a test such as the Welch or Brown-Forsythe test to means based on data
from which the highest and lowest 20% have been trimmed. Computer-sampling
results for this procedure are promising (Lix & Keselman, 1998; also see Wilcox,
1997, p. 134, for a description of the procedure). The main drawback is the com-
putational complexity, but Wilcox has made a program available at a Web site
linked to the Academic Press Web site: (http://www.apnet.com).
No one test will be best in every situation. The relative merits of various procedures will
depend on a complex combination of factors. Researchers with heterogeneous variances
should turn to articles that have reported the effects of various combinations of factors
on error rates and power (e.g., Coombs et al., 1996; Lix et al., 1996; Lix & Keselman,
1998; Tomarken & Serlin, 1986). A test may then be selected on the basis of consideration
of the relative Type 1 error rates and power under conditions similar to the researcher's
own study. In some cases, there will be no test that clearly meets the researcher's needs. In
such instances, information about the distributions in the various conditions, including plots
of the data, should be presented in research reports. Indeed, such information should be
routinely presented regardless of whether significance test results are presented. Whatever
approach is taken, decisions about the appropriate null hypothesis test should never be made
by running several tests and selecting the one that yields the lowest p value; this procedure
is likely to inflate the Type 1 error rate.
Most studies involve more than a single factor. There may be several treatment variables
or, as in the Seasons data, participants may be categorized in several ways. Furthermore,
in many studies, participants are tested under several conditions or on several trials. Nev-
ertheless, important concepts and issues have been raised in this context, and these will
require consideration regardless of the design of the research. We should always consider
the appropriate structural model for our data, and we should always be concerned about
whether assumptions have been met and, if not, what the consequences are likely to be
and what remedies are available. In understanding our data and in assessing the validity
of assumptions, we should be guided by summary statistics and data plots readily avail-
able from many computer packages. Normal probability plots, box plots, dot-density plots,
stem-and-leaf plots, and spread-versus-level plots can all be helpful.
We have focused on tests of the omnibus null hypothesis in this chapter because they
are routinely conducted and routinely presented in research reports, and they provide an
overview of the effects of the independent variable. However, we believe that the omnibus
null hypothesis is rarely, if ever, of primary interest to researchers. Usually, either because
of reasons based on theory or prior knowledge, or because of the observed pattern of
cell means, we are most interested in estimating and testing contrasts of means. As we
demonstrated in Chapter 6, the t distribution provides the basis for CIs and tests of contrasts,
and these calculations are straightforward. Nevertheless, inferences about contrasts raise
certain issues, some of which are as much philosophical as statistical. We turn to these in
treatment populations completely randomized design
between-subjects designs within-subjects (repeated-measures)
structural model fixed-effect variables
random-effect variables omnibus null hypothesis
dot-density plot between-groups mean square
treatment effect error component
total sum of squares (SStot) between-groups sum of squares (SSA)
within-groups sum of squares (SS S/A } definitional formulas
raw-score formulas n2
shrunken R2 expected mean squares (EMS)
error term co2
noncentral F distribution noncentrality parameter, X
negative bias of the F test positive bias
autocorrelation plots robustness
nonparametric tests Kruskal-Wallis H test
rank-transform F test shift hypothesis
trimmed means power transformation
spread-versus-level plot Welch test, Fw
8.1 A data set has three groups of 5 scores each. Because the scores involve decimal
values, each score is multiplied by 100.
(a) How will the mean squares change (relative to an analysis on the original data set)?
(b) Should the F ratio change?
(c) In general, what happens to a variance when every score is multiplied by a
constant?
(d) Suppose we just added a constant, say 10, to all 15 scores. How would that effect
the mean squares and F ratio?
(e) Suppose we added 5 to all scores in the first group, 10 to all scores in the second
group, and 15 to all scores in the third group. Should MSA change? Should
MSs/A ? Explain.
8.2 Following are summary statistics from a three-group experiment. Present the ANOVA
table when (a) n1 = n2 = n3 = 10 and (b) n1 — 6, n^ = 8, and n3 = 10 (T.j is the
total, or sum of scores, for group7).
AI A2 A3
T.j = 30 48 70
s] = 3.2 4.1 5.7
8.3 The data are:
A, 27 18 16 33 24
A2 23 33 26 19 38
(a) Perform the ANOVA.
(b) Next, do a t test. How are the results of parts (a) and (b) related?
0 8.4 The file Ex8_4 (Exercise folder) on the CD contains three groups of 15 scores.
(a) Explore the data; examine statistics and graphs relevant to assessing the normality
and homogeneity of variance assumptions.
(b) Calculate the F and Kruskal-Wallis H tests for these data and comment on the
outcome, relating your discussion to your answer to part (a).
9 8.5 (a) Calculate treatment and residual effects for the scores in Ex8_4. What should the
average treatment effect be? The average residual? Are the results as expected?
(b) Multiply the sum of the squared treatment effects by n and compare the result
with the between-groups sum of squares from Ex8_4, part (b). Also sum the
squared residuals and compare the result with the within-groups sum of squares
from Ex8_4, part (b).
8.6 The following are the results of two experiments, each with three levels of the
independent variable.
SV df MS SV df MS
A 2 80 A 2 42.5
S/A 21 5 S/A 12 5
(a) (i) For each of the two tables, calculate F and estimates of w^.
(ii) What does a comparison of the two sets of results suggest about the effect of
the change in n upon these three quantities?
(b) Calculate T^for each table. How does n effect the value of i]2A ?
(c) Suppose F = 1. (i) What must the value of a>2A be? (ii) What must the value of
T)^ be (as a function of a and n)?
(d) Comment on the relative merits of the various statistics calculated as indices of
the importance of A.
8.7 The results of an ANOVA of a data set based on three groups of 10 scores each is as
A 2 192 96 3.2
S/A 27 810 30
(a) Is there a significant A effect if a = .05?
(b) Estimate Cohen's / for these results.
(c) Assuming this is a good estimate of the true effect of A, what power did the
experiment have?
(d) How many participants would be required to have power = .8 to detect a medium-
sized effect? Use Cohen's guidelines (Subsection 8.5.1 for w2 and Subsection
8.5.2 for /) for effect size.
8.8 According to a mathematical model for the experiment in Exercise 8.7, the predicted
scores are 10 in Condition 1, 14 in Condition 2, and 18 in Condition 3. If the theory
is correct, what sample size would be needed to achieve .8 power to reject the null
hypothesis of no difference among the means? Assume that the error mean square is
a good estimate of the population variance, and a = .05.
8.9 In this chapter, we examined descriptive statistics for the probability-learning data
of Myers et al. (1983; see Table 8.1). Although we decided that heterogeneity of
variance was not a problem, we should test this assumption.
(a) Perform the Brown-Forsythe test of equality of spread.
(b) We also concluded that nonnormality was not a problem. However, we did not
examine normal probability plots for each condition. Do so.
(c) Perform the Kruskal-Wallis H test of the probability-learning data. Does it lead
to a different conclusion than the F test result reported in this chapter?
• 8.10 Open the Sayhlth file in the Seasons folder of your CD. This file contains Sayhlth
scores (self-ratings of health) of 1-4 (excellent to fair; 3 participants with poor
ratings in the Seasons file are not included). The four categories will be the in-
dependent variable in this exercise and the Beck_D score will be the dependent
variable in the following analyses. The Beck_D score is an average of the four sea-
sonal Beck depression scores and is available only for those subjects whose scores
were recorded in all four seasons. The distribution of BeckJD scores tends to be
skewed, and, as in most nonnormal distributions, heterogeneity of variance is often a
(a) Explore the data, using any statistics and plots you think are relevant, and com-
ment on the relative locations, shapes, and variabilities of the scores in the four
(b) Using the four Sayhlth categories, plot the spread versus level; as stated in Chap-
ter 8, this is the log of the H spread plotted against the log of the median. Several
statistical software packages make this plot available. Find the best-fit regression
line for this plot and transform the Beck_D scores by raising them to the power,
1 — slope.
(c) Explore the distribution of the transformed scores at each Sayhlth category. Has
the transformation had any effect on the shape of the distributions or on their
variances? Test for homogeneity of variance.
(d) Next let's try a different transformation. Calculate log(Beck_D +1) and discuss
the effects of this transformation.
(e) What might be the advantges of transforming data to a scale on which they are
normally distributed with homogeneous variances?
8.11 Continuing with the Sayhlth file, perform the following steps.
(a) Using the four Sayhlth categories as your independent variable, do separate
ANOVAs of the BeckJD data for men and for women.
(b) Calculate Cohen's / for each sex and compare the effect sizes (see Equation
8.19).
8.12 The Sayhlth file also categorizes individuals by employment category: 1 = employed
full time; 2 = employed part time; and 3 = not employed.
(a) Explore the Beck_D data in each Employ category, looking at relevant graphs
and statistics. Comment on the validity of the ANOVA assumptions.
(b) In Exercise 8.10, we considered transformations of the BeckJD data, one of
which appeared to provide results more in accord with the ANOVA model. Use
that transformation and again explore the data. Are the results more in accord
with the ANOVA model?
(c) Perform ANOVAs on the BeckJD scores and the transformed scores as a function
of employment status. Discuss the results.
(d) Calculate Cohen's / for both the original and the transformed data. How would
you characterize the effect sizes?
Partitioning the Total Variability in the One-Factor Design
The following developments involve two indices of summation: i indexes a value from 1
to n within each group, where n is the number of individuals in a group; j indexes a value
from 1 to a, where a is the number of groups. Appendix A at the back of the book provides
an explanation of the use of this notation, using several examples.
Squaring both sides of Equation 8.1 yields
Summing over i and 7, and applying the rules of Appendix A on summation, we have
Rearranging terms, we can show that the rightmost (cross-product) term equals zero:
The last result follows because the sum of deviations of scores about their mean is zero.
Raw-Score Formulas for Sums of Squares
The raw-score formula for SStot is obtained by squaring each score, summing, and subtract-
ing a"correction term," C, where C = anY 2.. = (£ a =1 En=\ Y i j ) 2 / a n , and
C corrects for the fact that we are subtracting deviations about the grand mean.
Raw-score formulas for most designs can be readily obtained from the relation between
degrees of freedom and SS. Let Tj stand for the sum of the n scores in the j'th group, and
let T stand for the sum of all an scores. Now consider the SSs/A-
1. Expand the corresponding degrees of freedom: a(n — 1) = an — a. If the group
sizes are not equal, dfs/A = N — a, where N — E j n j -
2. Each degree of freedom in the expanded term corresponds to a squared quantity.
Therefore, we now have
Note that each squared quantity must have subscripts that correspond exactly to
the indices of summation to its left. Any indices that are not subscripted according
to this rule are replaced by dots, indicating that summation has taken place for that
index prior to squaring. The designation T.I indicates that we have summed the rij
scores in group j and then squared the total.
3. Divide each squared quantity by the number of values on which it is based. Because
YIJ is a single score, it is divided by one. However, T..J is a sum of nj scores, so
Following these rules, because dfA = a— 1,
where C = T 2 / N .
EXAMPLE 1 Applying these formulas to the data of Table 8.1, we have
Contrasts Among Means
In Chapter 8, we tested the effects of educational level on male depression scores in the Sea-
sons data set. That is, we tested the null hypothesis that the population means of male depres-
sion scores at each educational level are identical. Such tests of the omnibus null hypothesis
are conducted in part because tradition demands it, and in part because there is a belief that
such a preliminary screening is required before we can address the questions that really in-
terest us. A significant F indicates that not all of the population means are equal. However, it
does not reveal which population means differ. Presumably, we are interested in testing, and
constructing confidence intervals (CIs) for, such differences as those between the HS (high
school education only) and C (some college) means, or between the HS and the combined
C, B (bachelor's degree), and GS (graduate school) means. The calculations for such tests
and CIs are straightforward; we have already illustrated them as examples of applications of
the t statistic in Chapter 6. However, evaluating several contrasts raises a number of issues.
The first issue we confront is that the risk of a Type 1 error increases as we conduct
more significance tests. To understand why this is so, assume that three contrasts among
population means are tested. Further assume that the null hypothesis is true in all three cases
so that any rejection is a Type 1 error. Finally, assume that the three significance tests are
independent; the result of any one test does not change the probability of the result of any
of the other tests. There are eight possible patterns of test results. For example, there might
be a significant result on the first test followed by nonsignificant results on the second and
third test. Assuming HO to be true and that the test results are independent, we find that the
probability of this is .05 x .95 x .95. Of the eight possible patterns of results, seven have at
least one rejection; that is, at least one Type 1 error. In fact, the probability of at least one
Type 1 error is one minus the probability that no test results in a rejection; this is 1 — .953, or
.143. This is notably higher than the .05 we associate with an individual test. Furthermore,
the more tests that are performed, the higher this error probability becomes. Although the
conditions we set in this example—all null hypotheses true and all tests independent of
234 9 / CONTRASTS AMONG MEANS
each other—will not usually hold in our studies, the general principle does. As the number
of tests increases, there is also an increase in the probability of at least one Type 1 error;
this probability is referred to as the familywise error rate, or FWE. There have been many
proposals for controlling the error rate associated with a family of tests, and a major purpose
of this chapter is to sort through these and make recommendations.
The second issue is, What is a family? In a one-factor design such as those considered in
Chapter 8, the answer seems obvious; it is the set of all tests conducted. However, suppose
we conducted two different sets of three tests each. For example, the contrasts among
educational levels might be performed on the depression data and on one other measure,
perhaps anxiety scores. Is there a single family consisting of six tests, or two families of
three tests each? The answer will influence the inferences that are drawn because a larger
value of t will be needed for significance when the FWE is based on six, as opposed to
three, tests. We discuss this issue in Section 9.4.
A third issue centers on the distinction between planned contrasts, those determined
before the data were collected, and post hoc contrasts, those based on "data snooping,"
testing differences that look interesting after the data have been collected. Should this
distinction influence the process by which we arrive at our inferences, and, if so, how? We
will argue that the distinction is important, and, in separate sections for planned and post
hoc contrasts, we discuss procedures for controlling the FWE.
The results of tests conducted on the Myers et al. (1983) probability-learning experiment
data in Chapters 6 and 8 raise yet a fourth issue. In Chapter 6, a t test on these data revealed
that the mean of the HE (high explanatory) condition was significantly higher than the mean
of the combined S (standard) and LE (low explanatory) conditions. However, the F test
in Chapter 8 failed to reject the omnibus null hypothesis that all three population means
are equal; the p was .062. Because the omnibus F was not significant, some investigators
would argue that no further tests should have been carried out. These individuals view the
omnibus F test as a gatekeeper and therefore would not proceed to tests of contrasts in
the absence of a significant result in the test of the omnibus null hypothesis. However, if
the FWE is maintained at or below its nominal level, not only is there no need for a prior
test of the omnibus null hypothesis before testing contrasts, but requiring such a test before
proceeding further will lead to a reduction of power.
The plan for this chapter is as follows. We first define some basic terms and present
examples of contrasts. Following this, we review calculations of t statistics for testing
hypotheses about contrasts, noting possible modifications under conditions of unequal ns
or unequal variances. We then address the issues raised previously in this introduction. We
illustrate methods for maintaining the FWE in several different situations—when a subset
of all possible contrasts has been planned, when all possible contrasts based on pairs of
means are made, when each of several treatment means is contrasted with the mean of a
control group, and when contrasts are selected for testing after an inspection of the data.
9.2 DEFINITIONS AND EXAMPLES OF CONTRASTS
We define a contrast, or comparison, of population means as a linear combination of the
means. We denote contrasts by the Greek letter, psi (W):
CALCULATIONS OF THE f STATISTIC FOR TESTING HYPOTHESES ABOUT CONTRASTS 235
where at least one Wj is not zero and EjWj =0.
Some examples are
For example, 1^3 contrasts the average of the last three treatment population means with the
first population mean. The point estimate of i); is
For the contrasts in Equation 9.2, the point estimates are
Each of these estimates a different contrast of the four population means and therefore
provides the basis for testing a different null hypothesis.
Because any set of weights that sum to zero is allowed under the definition of a con-
trast, there are an infinite number of possible contrasts. In practice, researchers are usually
interested in contrasts between the means of two treatment populations; v|/1 is an example.
Such contrasts are often referred to as pairwise comparisons. Less often, we are interested
in more complex contrasts in which at least one of the two means is itself an average of
several means; the remaining three contrasts in Equation 9.4 are all examples of this. If we
have four treatments, there will be six pairwise comparisons and 19 additional contrasts in-
volving subsets of means, such as w2,w3, and ^4. Such an abundance of testable hypotheses
may tempt us to test them all or, what is essentially the same thing, to look at the various
differences between sets of means, selecting the largest one and testing it. As we indicated
earlier, this may cause a greatly inflated Type 1 error rate. We present methods for control-
ling error rates even for large families of tests, but we must be aware that these methods
involve a cost. As we increase the number of contrasts we test while holding the FWE at
some reasonable level (ordinarily .05 or .10), power is lost. In view of this, we should think
hard about which hypotheses are of interest before we collect the data. We need to focus
both the research design and the power of our significance tests on those questions that are
of most interest to us.
9.3 CALCULATIONS OF THE f STATISTIC FOR TESTING
HYPOTHESES ABOUT CONTRASTS
In Chapter 6, we presented calculations for the t statistic for contrasts.1 In this section we
review those calculations with another example, and we discuss the selection of weights
when ns are unequal, and alternative calculations when variances are unequal. Consider the
response speeds on a multiplication test for the fifth- to eighth-grade students in Royer's
study (Royer et al., 1999). Suppose we had initially hypothesized that between fifth and
sixth grade, there was a clear change in speed, but little, if any, further improvement
in subsequent grades. This suggests that the fifth-grade mean will differ from the mean
of the combined sixth through eighth graders, and that there will be no pairwise differ-
ences among the higher three grades.2 This in turn suggests the following set of four null
hypotheses:
Table 9.1 presents the means and variances of the four groups, and the group sizes, and it
illustrates the calculation of the t statistic for the first contrast. Note that the weights for v1 \
(Wj0 have been multiplied by 3 so that all weights are now integers. This does not change the
value of the t statistic because both the numerator and the denominator are increased by a
TABLE 9.1 THE t TEST OF H01 : w1 = (1/3)(|A6 + X7 + u8) -u5 = 0
Variance 0.033 0.028 0.031 0.038
n 23 26 21 20 N =90
MSS/A = (22/86)(.033) + (25/86)(.028) + (20/86)(.031) + (19/86)(.038) = .032
and dfs/A = 90 - 4 = 86. To test H01 , we calculate
In this example,
TABLE 9.2a CONTRAST COEFFICIENTS AND TESTS FOR MEAN MULTIPLICATION
Contrast Coefficients
Contrast 5 6 7 8
1 -3 1 1 1
2 0 1 _1 0
3 0 1 0 -1
TABLE 9.2b
CONTRAST TESTS
Value of Sig.
Contrast Contrast Std. Error t df (2-tailed)
SPEEDMRT Assume equal variances 1 .680140 .129976 5.233 86 .000
2 2.62E-02 5.25E-02 -.500 86 .619
4 2.64E-03 5.59E-02 .047 86 .962
Does not assume equal 1 .680140 .131143 5.186 38.3 .000
variances 2 2.62E-02 5.05E-02 -.519 41.7 .606
3 2.36E-02 5.44E-02 -.434 37.3 .667
4 2.64E-03 5.83E-02 .045 38.2 .964
Note: Table 9.2 output is from SPSS.
factor of 3. It does simplify the calculations and makes rounding errors less likely. However,
one caution is in order: Confidence limits based on these weights will also be increased by
a factor of 3. To get back on the original speed scale, the limits are divided by 3. Note that
when we compare CIs, we must be sure they are all on the same scale. This can be achieved
by always transforming back to the original scale, or by adopting the convention that the
negative contrast weights should add to — 1 and the positive weights to +1.
Tables 9.2a and 9.2b present the SPSS output for all four contrasts. There are two sets of
results depending on whether homogeneity of variance is assumed. We discuss calculations
when the assumption of equal variances is violated in Subsection 9.3.2. The "Value of
Contrast" column in Table 9.2b contains the values of the ur s, and the "Std. Error" column
is for s^. (The value of the t for ty \ is slightly different from ours because we rounded the
group means.)
9.3.1 Weighting Means When us Are Unequal
In the multiplication speed data, the four grades had different numbers of students. How-
ever, there is no reason to view the four sampled populations as unequal in size. If the
sixth- through eighth-grade populations are equal in size, then they should receive equal
weight when their mean is contrasted against the mean of the fifth-grade population. There-
fore, the weights used in testing H01 (—3,1,1,1) are appropriate. Equal weighting of means
that are averaged on one side of a contrast usually will also be appropriate in the anal-
ysis of data from any true experiment in which the independent variable is manipulated.
In contrast, in many observational studies, differences in group ns reflect differences in
population size. A case in point is the Seasons study, which we cited previously. In Chap-
ter 8, we tested the omnibus null hypothesis that mean depression scores were equal for
four populations defined by their education level. The four groups were male participants
with only a high school education (HS), some college experience (C), a bachelor's de-
gree (B), or graduate school experience (GS). Table 9.3a presents group sizes, means, and
variances.
Assume that one question of interest was whether the mean depression scores differed
between men with a graduate school education and all other men. We might weight the
TABLE 9.3a SUMMARY STATISTICS FOR DEPRESSION SCORES
AS A FUNCTION OF EDUCATIONAL LEVEL
Educational Level n Mean Variance
HS 19 6.903 34.541
C 33 3.674 5.970
B 37 3.331 9.861
GS 39 4.847 26.218
TABLE 9.3b SPSS OUTPUT FOR DEPRESSION SCORES AS A
FUNCTION OF EDUCATIONAL LEVEL
Contrast HS C B GS
2 19 33 37 -89
D-SCORE Assume equal variances 1 -.6329 2.4385 -.260 124 .796
2 -55.7348 71.3989 -.781 124 .437
Does not assume equal 1 -.6329 2.8837 -.219 60.134 .827
variances 2 -55.7348 80.8897 -.689 55.239 .494
Note. Table 9.3b is output from SPSS.
means (from HS to GS) 1, 1, 1, and —3; in this way we can test
This weights the HS mean the same as the C and B means, despite the fact that the relative
frequencies suggest that the HS population is considerably smaller than the others. Alterna-
tively, we can assume that the four populations vary in size such that weights proportional to
their size would be WHS. Wc, WB, and WgS. Then the mean of the first three populations—call
it u<GS ("less than graduate school")—would be a weighted average; that is,
and the contrast of interest is
Because the t test of a contrast is not affected by multiplying all weights by a constant, we
can simplify things by multiplying the expression by WHS + we + WB, giving us
and we test
Unless we know the actual sizes of the populations, we now need estimates of the ws.
In many situations, the most reasonable and simplest estimate will be the group sizes.
Table 9.3b presents output from SPSS when the weights are 1, 1, 1, and —3 and when
they are based on the group sizes provided in Table 9.3a. Results are reported when equal
variances are assumed and when they are not; we consider the latter case shortly. For now,
note that, in the second contrast, the ws have been replaced by their corresponding ns.
Although neither contrast is statistically significant, there are clear differences between the
two t values, and between their p values.
Equal weighting and weighting by frequency will not always yield very different results;
the distribution of group sizes is the critical factor. We originally intended to contrast the
HS mean with the mean of the three groups having at least some college education but soon
found that the two sets of weights (—3,1,1,1 versus —109, 33, 37, 39) yielded very similar
results. The reason for this becomes evident if we divide the latter set of weights by 36; we
get -3.028, .917, 1.028, and 1.083—not very different from -3, 1, 1, and 1.
Bear in mind that the issue of weights arises only with contrasts in which one or both
subsets are based on at least two means. When pairwise comparisons are tested (by far the
most common situation), the weights will always be 1 and — 1 for the two means involved
in the comparison, and 0 for all other means.
9.3.2 Testing Contrasts When Variances Are not Equal
Suppose we wish to contrast the depression mean for the HS group with the mean of
individuals with more than a high school education. We use the weights 3, — 1, — 1, and — 1.
Before continuing our calculations, we note in Tables 9.3 that the variances of depression
scores are very different at the four educational levels. As we noted in both Chapters 6
TABLE 9.4a WELCH'S t1 TEST OF H01 : Ws = (l/3)(M-c + ub + JIGS) = 0.
From Table 9.3, $ = (3)(YHs) - (Yc + YB + YGS) = 8.857
The denominator of t' is
Then t' = 8.857/4.181 = 2.1 18. From Equation 9.6, the degrees of freedom are
TABLE 9.4b CONTRAST COEFFICIENTS AND TESTS FOR
DEPRESSION SCORES
1 3 -1 -1 -1
D .SCORE Assume equal variances 1 8.8572 3.1169 2.842 124 .005
Does not assume equal 1 8.8572 4.1810 2.118 20.5 .047
Note. Output is printed from SPSS.
and 8, when the homogeneity of variance assumption is violated, the error rates associated
with the standard t and F tests may be badly distorted. The t' solution proposed by Welch
(1947) was presented as an alternative to the usual t and F tests. A general form of the t'
statistic can be applied even when more than two groups are involved in the contrast. The
THE PROPER UNIT FOR THE CONTROL OF TYPE 1 ERROR 241
test statistic is
and the degrees of freedom are
Table 9.4a illustrates the calculations for the contrast of the HS mean with the mean of
the combined other three educational groups. The calculations are somewhat tedious, but
fortunately, as Table 9.4b illustrates, the same result can be obtained from most statistical
software packages. These provide results under both the assumption of equal population
variances (in which case the standard error of the mean, SEM, is based on the MSs/A} and
under the more general assumption that the variances are not equal.
There are two situations in which the MSs/A is not an appropriate error term. In one case,
the contrast to be tested involves conditions for which the assumption of equal variances
is suspect. Here, t' should be calculated. In other instances, the variances corresponding
to the conditions involved in the contrast are very similar but different from the vari-
ances of those conditions not included in the contrast (that is, having zero weight). The
standard t is appropriate here but the denominator should be based only on the pool of
the variances corresponding to the included conditions. For example, if there are three
groups with variances 10, 11, and 5, and the means of the first two groups are to be
compared, the variance of 5 should not be included in the denominator because the es-
timate of the standard error (SE) of the contrast will be too small, and the Type 1 error
rate will be inflated. In summary, when there is reason to doubt that all population vari-
ances are equal, only those corresponding to means involved in the contrast should be
included. If the equal-variance assumption is at all suspect even for those variances, cal-
culate t'. Finally, as usual, we warn that such decisions should be made a priori; basing
decisions about which result to report on the output of both analyses invalidates the inference
9.4 THE PROPER UNIT FOR THE CONTROL OF TYPE 1 ERROR
As we argued in the introduction to this chapter, the probability of a Type 1 error increases
with the number of significance tests. Therefore, if the probability of each significance
test is set without regard to how many tests might be conducted, the error rate for the
entire collection of tests may rise to an unacceptable level. Statisticians and researchers are
generally agreed that the proper unit for control of the Type 1 error rate is not the individual
test but a set of contrasts called a family. Before we address the question of how to limit the
Type 1 error rate for the family, we more closely consider possible definitions of a family
of contrasts.
We begin by distinguishing between the error rate per contrast (EC)—the probability
that a single contrast results in a Type 1 error—and the FWE—the probability that a set, or
family, of contrasts will contain at least one Type 1 error. For a family of K independent
tests,
The probability of a Type 1 error on a single test is the EC. Therefore,
If a family consists of six independent tests, each conducted at EC = .05, substitution
in Equation 9.7 results in FWE = 1 - (1 -.05)6 = .265, considerably greater than .05. If
the six tests are not independent, the FWE is less than .265 but more than .05, although the
exact value is difficult to calculate. In any event, the larger the family, the more the FWE
exceeds the EC.
One extreme view we might take with respect to the discrepancy between the EC
and the FWE is simply to ignore the FWE. We could choose a value for the EC without
considering the total number of contrasts tested, and we could test each contrast by using
that p value, perhaps .05, as the criterion for significance. The problem with this approach
is that even if there are no differences among the treatment population means, we would
have a good chance of finding "significant" results if we performed a large enough number
of tests, because the FWE would then be large. Publishing such findings could result in
wasted effort spent investigating and attempting to replicate effects that did not exist. In
view of this, we want the EC to be such that the FWE for a family of contrasts will be
kept within acceptable bounds. The exact criterion for significance for the test of any one
contrast will depend on how large the family is. If we want to keep the FWE constant as
family size increases, the EC will have to decrease. This line of reasoning requires that we
decide exactly what we mean by a family of contrasts.
An investigator working in a research area over a period of years might perform hun-
dreds of experiments and test thousands of hypotheses. If we considered these thousands of
tests to form a single family and set the FWE equal to .05, the EC would be infinitesimally
small. Although this ultraconservative approach would result in a very low Type 1 error
rate, the Type 2 error rate would soar to unacceptable levels. The experimenter could be
confident that significant results revealed real effects but would miss finding many real
effects. Because lowering the EC results in a reduction of power, the definition of family
must be based on a compromise between concerns about Type 1 and Type 2 errors.
When we conduct an experiment with one independent variable and one dependent
variable, the definition of the family seems straightforward. If we have four levels of the
independent variable, and plan three specific contrasts, the family is of size 3. If we test
every possible pairwise contrast, the family is of size 6. If we scan the data, and select the
largest possible contrast, the family size is infinite because there are an infinite number
of sets of weights possible. The situation is different if we have several independent or
dependent variables in a single study. For example, consider the study of seasonal variation
in psychological and physical measures that we have referenced several times previously. We
might conduct pairwise comparisons among four educational levels on depression means
PLANNED VERSUS POST HOC CONTRASTS 243
and on anxiety means. We might do this separately for male and for female participants. We
have six pairwise comparisons for one measure for one sex. Because there are two measures
and two sexes, there are a total of 24 comparisons. Do we control the FWE for a family of
24 comparisons, or for each of four families of 6 comparisons? We recommend the second
option—viewing each combination of factor level and dependent variable as the basis for
defining the family. This seems a reasonable compromise between Type 1 and Type 2 errors.
Furthermore, it provides a natural basis for comparison with other studies. Suppose another
investigator carries out a study of the effects of educational level on depression scores in
female participants and also tests all six possible pairwise comparisons. Any comparison
of the FWEs of the two studies should be based on the same definition of family. Because
the second study used only female participants, and only the depression measure, such a
comparison is possible only if we have also defined the family in this way, rather than in
terms of all 24 comparisons we made.
9.5 PLANNED VERSUS POST HOC CONTRASTS
In Chapter 8, we analyzed depression scores as a function of educational level; the educa-
tional groups were HS, C, B, and GS. We tested the omnibus null hypothesis that the four
population means were equal. With respect to contrasts, consider two scenarios:
1. After inspecting the data, we find that the mean depression score for the HS group
is higher than that for the other three groups. Accordingly, we test the contrast of
the HS group mean against the combined mean of the other three groups; that is,
we test H0: |xHs = (1/3XM-C + M-B + M-HS)-
2. Before conducting the experiment, we predict on the basis of theory or previous
research that the HS mean will differ from that of individuals having at least some
college experience. On this basis, we test the same null hypothesis as in Scenario 1.
Assuming the result of the t test is significant at the .05 level, we should have more confidence
that the difference exists in the population in the second scenario than in the first. If the null
hypothesis is true and, without looking at the data, we select a single contrast to be tested
at a = .05, the probability of a Type 1 error is .05. However, if we first examine the data by
looking for a difference, and then we test the largest effect we observe, it is as if we have
carried out many significance tests. The EC is .05 but the FWE, the probability of at least
one Type 1 error, is considerably higher. The first scenario represents a situation in which
tests of contrasts are post hoc (or a posteriori), whereas the second scenario represents a
situation in which the contrasts are planned (or a priori).
A simple analogy may make the distinction clearer. Assume that I bet you at even odds
that the fourth toss of a coin will be a head. Assuming the coin is a fair one, so is the bet; the
probability of a head on the fourth toss is exactly .5. This is similar to a planned (predicted)
contrast. Now assume instead that I bet you at even odds that there will be a head somewhere
in the sequence of four tosses. Either refuse the bet or ask for (much) better odds because
the probability of at least one head in four tosses is considerably more than .5. This situation
is similar to a post hoc test of a contrast.
Before considering ways of controlling the FWE in each of these situations, we wish to
raise several points. First, plan ahead. If there is only one planned contrast, the EC and FWE
will be the same because the family is of size 1. Although we usually plan more than one
contrast, typically the family of planned contrasts will be fairly small. Consequently, the EC
will not be greatly reduced relative to the FWE. As we noted earlier, the larger the family,
the lower the EC, and the lower the power of each individual test. For example, although
there are methods for controlling the FWE for the family of all pairwise comparisons, if
we test only a few that were decided on before we looked at the data, we will have greater
power and narrower CIs for each test.
The second point is that, although contrasts should be planned, often the pattern of
means is unexpected but suggests something interesting, although unanticipated. It is not
only legitimate to inspect one's data and carry out tests of contrasts that look interesting, it
should be obligatory. However, the criterion for significance of such post hoc tests should
be such that the FWE is held to a reasonable level.
A third consideration is the nature of that reasonable level. What should the FWE be?
Traditionally, the FWE has been set at the .05 level. That seems reasonable for small families
of planned contrasts. However, Scheffe (1959), in a classic text on ANOVA, suggested that
for post hoc tests, the FWE could be set at .10. The effective size of a family of post hoc
contrasts is determined not by the number of contrasts actually tested but by those that might
conceivably have been tested had the data suggested it was worth doing so. This family is
large and the power to test contrasts will often be extremely low if the FWE is set at .05
in this situation. Bear in mind that the EC and FWE are different concepts, and there is no
reason that traditional EC levels of significance need be applied to the FWE. Even with the
FWE at .10, the EC will generally be quite low.
9.6 CONTROLLING THE FWE FOR FAMILIES OF K PLANNED
Here, we discuss some very general methods for dealing with planned contrasts. All use
the t statistic presented in Table 9.1, or Welch's t' (Subsection 9.3.2) if variances are
unequal. However, they differ in their criteria for significance, and in whether they allow
the construction of CIs. These methods are very versatile. Although we illustrate their use
with multiple t tests, it should be emphasized that they can be used with any statistical
tests, parametric or nonparametric. Furthermore, if the summary statistics and plots of the
data indicate that the population variances are not equal, then these methods can be used
with Welch's t' test. Most important, it is not necessary that the omnibus F be significant
prior to testing planned contrasts, provided the FWE is controlled. In fact, power is lost by
requiring a significant F before carrying out planned tests with a procedure that controls
the FWE. What is critical is that the contrasts are decided on before the data are collected
and a method for evaluating significance of the tests is used that maintains the FWE at or
below a reasonable level, presumably .05 or .10.
9.6.1 Methods Based on the Bonferroni Inequality
Equation 9.7 describes the relation between the FWE and EC when the K tests are in-
dependent. Because this condition rarely holds, a more general statement of the relation
CONTROLLING THE FWE FOR FAMILIES OF K PLANNED CONTRASTS 245
In other words, the FWE is equal to or less than the term on the right, with the inequality
holding when the tests are not independent. Furthermore, if K tests are conducted with error
rates EC1, EC 2 ,..., ECK, then
where EQ is the probability of a Type 1 error for the &th contrast. The relationship expressed
in Equation 9.9 is known as the Bonferroni inequality, and it is the basis for several
procedures for testing planned contrasts. From the inequality, it follows that if each of the
K contrasts that make up the family is tested at EC = FWE/K, the probability of a Type 1
error for the family cannot exceed the FWE. If, for example, the family contains five planned
contrasts, the FWE will not be larger than .05 if each contrast in the family is tested at the
.01 level.
To illustrate methods based on the Bonferroni inequality, we reconsider the SPSS
output in Table 9.2. The significance values reported by SPSS are the ECs. In this instance,
it appears that we reach the same conclusion whether our criterion is the EC or the FWE. The
first contrast has such a small p value that it should be significant by any criterion, whereas
the last three contrasts are clearly not significant by any reasonable standard. Nevertheless,
we consider how each of two methods controls the FWE for this set of tests.
The Dunn-Bonferroni method follows from Equation 9.9. (Dunn, 1961). If there are
K contrasts, the FWE will not exceed a nominal value if the EC is set at that value divided
by K. For example, assuming a FWE of .05, we test the four contrasts in Table 9.2 at the
.0125 (.05/4) alpha level. To conduct the test, calculate
where the MSe (error mean square) will be MSs/A in the one-factor design and the error
degrees of freedom (dfe) will be N — a, where N = E j n j - If the exact p value is avail-
able, as for example in the SPSS output of Table 9.2, to control the FWE we compare it
with FWE/K, and reject H0 only for those contrasts such that p < FWE/K. Alternatively,
Appendix Table C.I provides critical values of t as a function of the FWE, dfe, and K.
Note that these values assume that the alternative hypothesis is two tailed. To illustrate the
use of the table, suppose we have five tests and FWE = .05. Further assume that dfe = 20.
According to Table C.7, we require t = 2.845 for significance. If our tests were one tailed,
2.845 would correspond to FWE = .025. To maintain the FWE at .05 for a set of one-tailed
tests, the critical value of t is 2.528.
For dfe not listed in the table, the critical value of t can be readily obtained from
statistical packages such as SAS, SYSTAT, or SPSS. For example, with a data file containing
a p value and the degrees of freedom, SPSS's IDF.T function will yield the corresponding
value of t (to use this function, first click on "Transform," and then on "Compute"). The
CDF calculator for the Student distribution at the UCLA Statistics Department Web site
(see Chapter 6 or 8 for the URL) allows us to input any two of these three: X (the value
of ?), the degrees of freedom, or the p value, with a question mark for the third variable.
We enter 86 for the degrees of freedom and, assuming the test is two tailed, .00625 0/2 x
.0125) for the probability; the result is 2.551. If programs such as these are not available,
an approximation based on the normal probability table can be calculated:
We see that za, the z exceeded by .00625, is slightly less than 2.50, approximately 2.4975.
Substituting this value for za, we again have ta = 2.551.
With this value of ta and the SEs of the four contrasts, we can compute CIs. The equation
for the CI is exactly the same as in Chapter 6; the difference is that the critical t is based on
the FWE of .0125 (two tailed), not the EC of .05. For ty i, the limits are
Substituting the critical / and values from Table 9.2, we find that the limits are
Because we used weights of —3, 1,1, and 1, we divide the limits by 3 to return to the
original data scale. The limits on the difference between the mean speeds of the fifth-grade
students and the mean of the other three grades are .116 and .337.
When CIs are based on the FWE, as in this example, they are interpreted somewhat
differently than when they are based on the EC. It may help to understand the distinction if
we assume many random replications of an experiment. In each replication, a set of four CIs,
one for each of the planned contrasts, is calculated and the critical value of t is 2.551. We
expect that, in .95 of the replications, each of the four intervals will contain the estimated
population contrast. In other words, the probability is .95 that the CIs for all members of
the family of contrasts will contain their corresponding u values. Because of this property,
intervals based on the FWE are referred to as simultaneous CIs.
Division of the FWE by K provides an approximate value of the EC. However, an exact
solution to Equation 9.7 is possible. This solution, originally proposed by Sidak (1967), is
obtained by solving for EC in Equation 9.7:
In the Dunn-Sidak method, H0k: wk =0 is rejected if pk < 1 - (1 - FWE)1/K. Because
1 — (1 — FWE)1/K > FWE/K, this method has more power and a narrower CI than the
original Dunn-Bonferroni procedure. However, the difference is very small, as can be seen
in Table 9.5; this table presents values of the EC for the original Dunn-Bonferroni test
(ECO and for the Dunn-Sidak revision (EC2).
A feature of the Bonferroni-based methods is that they allow the possibility of weighting
contrasts unequally (Myers, 1979; Rosenthal & Rubin, 1983). From Equation 9.9, it follows
that the FWE does not have to be divided into K equal parts. Suppose tests of three contrasts
are planned but one is of greater interest than the other two. Instead of a division of the
FWE by 3 in the Dunn-Bonferroni procedure, the most important contrast might be tested
at the .025 level, and the other two each at .0125. The FWE is still less than or equal to .05,
but power is increased in the test of the most interesting contrast at the expense of the other
two. An important qualification is that the determination of the division of the FWE should
be made before the data are collected.
TESTING ALL PAIRWISE CONTRASTS 247
TABLE 9.5 DUNN-BONFERRONI AND SIDAK VALUES OF EC
WHEN FWE = .05 OR. 10
FWE = .05 FWE -.10
K EC, EC2 EC, EC2
2 .025000 .025321 .050000 .051317
10 .005000 .005116 .010000 .010481
Note. EC, = FWE/K and EC2 = 1 - (1- FWE)1/K.
9.6.2 Hochberg's Sequential Method
The Dunn-Bonferroni and Dunn-Sidak procedures are simultaneous methods. A further
increment in power is provided by sequential methods (also referred to as stepwise, or mul-
tistage), in which tests of contrasts involve several stages. The simplest of these are Holm's
sequentially rejective method (Holm, 1979), and Hochberg's step-up method (Hochberg,
1988). We describe only the latter method here because it is the more powerful of the two.
The K contrasts are rank ordered according to their p values, with p\ being the smallest
and px being the largest. In the example of Table 9.2, the contrasts would be ordered as
v1 ,V2, ^3, and u4 If pK < FWE, all K null hypotheses are rejected. If not, consider the
next largest p value, PK-I- If PK-I < FWE/2, then reject the null hypothesis correspond-
ing to the K — 1 contrast and all remaining contrasts. If this test is not significant, then test
whether PK-I < FWE/3, PK-I < FWE/4, and so on. Using the example of Table 9.2, and
assuming FWE = .05, we first compare .962 with .05. This fails and we compare the next
largest p value, .659, against .025. This fails and .619 is compared with .0167. The only
significant comparison is that of .000 with .0125, and we can reject only HQ: v|; \ = 0. The
power of this procedure can be increased at the cost of greater complexity (Hommel, 1988;
Rom, 1990).
The choice between the Bonferroni-based methods and the Hochberg procedure should
depend on whether the researcher wants CIs. If only tests of hypotheses are required, use the
Hochberg method because it provides a more powerful test while still maintaining the FWE
at the targeted level. When CIs are to be calculated, the Dunn-Bonferroni or Dunn-Sidak
methods should be used because readily interpretable CIs are not available for sequential
9.7 TESTING ALL PAIRWISE CONTRASTS
It is quite common for researchers to decide to contrast all possible pairs of means. The
methods used in the preceding section are applicable, but there are other procedures that
provide slightly greater power. Of these, the ones we present are based on q, the studentized
range statistic. This statistic is the range of a set of observations from a normally distributed
population, divided by the estimated standard deviation of the population. If the observations
are group means,
where Fmax and Fmin are the largest and smallest means in a set of a ordered means and .sy,
the SEM, is
assuming homogeneity of variance and equal ns. Critical values of q can be found in
Appendix Table C.9 as a function of the FWE, a (the number of means), and the degrees
of freedom associated with the error mean square.
If we were to carry out a t test of the difference between the same two means, assuming
equal variances and group sizes in a one-factor design, the statistic would be
This relation can be useful in comparing the results of procedures based on t (such as the
Dunn-Bonferroni method) with those based on q; it has also been the basis for dealing with
unequal ns and unequal variances.
In the following sections, we consider several procedures for testing all (a) pairwise
contrasts. It is not necessary to conduct a preliminary test of the omnibus null hypothesis
prior to using any of these methods, because each method maintains the FWE at or below its
nominal value. In fact, requiring rejection of the omnibus null hypothesis before applying
a procedure that controls the FWE for the set of (2) comparisons is likely to increase the
probability of failing to detect true differences between means.
9.7.1 Tukey's HSD Test
Tukey's BSD (honestly significant difference; Tukey; 1953) test controls the FWE for the
set of all possible pairwise comparisons. We illustrate the test by using the means and
standard deviations for the fifth- through eighth-grade multiplication speed scores in the
Royer study. These were presented in Table 9.1. However, because Equation 9.14 requires
that the ns be equal, assume that there are 16 students in each of the four grades. (We will
shortly redo the calculations with the true ns, using a version of the Tukey test modified to
deal with unequal ns.) Table 9.6 presents the necessary steps in testing all pairwise contrasts
and in constructing simultaneous CIs, assuming equal ns and equal variances. The results
are also readily obtained from many statistical software packages.
In many studies, only a few of the possible pairwise comparisons will be of interest.
In such cases, the Dunn-Bonferroni procedure may provide more power and narrower CIs.
The ratio of interval widths of the Tukey to the Dunn-Bonferroni method is simply the ratio
TABLE 9.6 TUKEY'S HSD TEST AND CIs
1. Order the means in Table 9.1 from smallest to largest:
Grade: 5 6 8 7
Mean: .350 .560 .583 .586
We assume the means are based on 16 scores each.
2. Find the value of q required for significance when FWE = .05, a = 4, and df= a(n — 1) = 60.
That value is 3.74.
3. Calculate the SEM by using the values of the variance in Table 9.1, but assuming n = 16.
Averaging the variances (assuming equal n), MSs/A = -0325, and
4. A value of q can now be calculated for each of the six possible pairwise contrasts by dividing
each of the six differences between means by .045, and comparing the result with the critical
value, 3.74. For example, the difference between the fifth- and seventh-grade means is .236 and
therefore q = .236/.045 = 5.24. Because this is larger than 3.74, we conclude that these two
means differ significantly. We can therefore proceed to test the next largest difference. If we
proceed in this way, successively testing each smaller difference, we stop testing when we
encounter a nonsignificant difference.
Alternatively, we can find a critical difference between the means against which the
observed differences can be compared. In this example,
Compare the largest difference with dcrit, rejecting the null hypothesis if the observed difference
exceeds .169. Continue testing in the order of decreasing differences until a difference does not
exceed d crit . This procedure is only applicable when the ns are equal.
5. The equivalent test can be carried out by using the t distribution. Calculate the usual t statistic
and compare with q.05.4.60 A/2, or 2.645. To carry out the test by using the differences, calculate
6. Simultaneous CIs can also be constructed by using Tukey's method. If the FWE is .05, then the
probability is .95 that all possible pairwise contrasts are contained within intervals of the form
w ± q.o5 a,df SY or, equivalently, w ± (q. 05 , a.df V2)s$ . For example, The .95 confidence limits
on the difference in mean speeds of the fifth- and sixth-grade populations are
.21 ± (2.645)(.064) = .041, .379
of the critical values of their respective test statistics,
where K is the number of comparisons and is equal to (2); for example, if a (the number of
groups) = 4, then K = 6. Dunn (1961) has evaluated this ratio for various combinations of
K , a , d f e (the degrees of freedom associated with the error term for the omnibus F test), and
the FWE. When all possible pairwise comparisons are tested, the Tukey procedure will have
the narrower interval (and the more powerful test); that is, the ratio of intervals will be less
TABLE 9.7 CRITICAL VALUES OF f FOR THE D-B METHOD AS K VARIES
t Ratio (D-B/HSD)
K EC dfe = 20 dfe = 60 dfe = 20 dfe = 60
6 .0083 2.927 2.728 1.045 1.032
5 .01 2.845 2.660 1.016 1.006
2 .025 2.423 2.299 0.865 0.869
Note. Critical values are made with the assumption that a = 4, df = 20,
df = 60, and FWE = .05. D-B - Dunn-Bonferroni. The ratio (D-B/HSD) divides
the critical t for the D-B method by that for the Tukey HSD method. A ratio less than
1 indicates that the D-B method requires a smaller t for significance and therefore is
more powerful and has a narrower CI.
than 1. For fixed a, the advantage of the Tukey procedure declines as the FWE decreases,
the degrees of freedom increases, or K decreases. Table 9.7 illustrates that if only a subset
of all possible pairwise comparisons is planned and tested, a point is reached at which the
Dunn-Bonferroni procedure is more powerful. More precisely, if only four comparisons are
tested, the Dunn-Bonferroni method requires a smaller value of t for significance than does
Tukey's method. Therefore, it has the narrower CI (when K < 4) and greater power; the
advantage of the Dunn-Bonferroni method increases if even fewer comparisons are made.
Although the crossover point at which the Dunn-Bonferroni method has the advantage is
a function of K, it usually does not require K to be very small. For example, if a = 5, the
crossover point occurs when 7 of the possible 10 contrasts are planned if dfe = 60, and
when 6 are planned if dfe = 20.
9.7.2 When ns Are Unequal: The Tukey-Kramer Test
In the Royer study, the number of students in the fifth through eighth grades varied, as
indicated in Table 9.1. A modification of Tukey's HSD test suggested by Kramer (1956)
may be applied in such situations. The standard t statistic is calculated and compared with
qFWE,A,df/V2. SPSS provides an alternative approach. The s$ and p values, as well as the
.95 confidence limits, can be obtained from SPSS; the output for all pairwise comparisons
is presented in Table 9.8. The Tukey-Kramer t equals the mean difference (ijr) value divided
by the Std. (standard) Error (s^).
If the necessary software is lacking, the studentized range table (Appendix Table C.9)
can be used. In the example of the speed data, a = 4 and dfe = 86. However, there is no
entry in Table C.9 for these degrees of freedom; the closest are for df= 60 and 120. To
obtain the critical q value for df = 86, we can use a nonlinear interpolation method proposed
by Harter, Clemm, and Guthrie (1959). We first find <7.o5,4.,6o = 3.74 and <?.o5,4.,i20 = 3.69
from the table, and the reciprocals of 86 (.0116), 60 (.0167), and 120 (.0083). The critical
value for dfe = 86 is then given by
TABLE 9.8 SPSS OUTPUT FOR TUKEY-KRAMER TESTS OF THE DATA OF TABLE 9.1: MULTIPLE
Multiple Comparisons
Dependent Variable: SPEEDMRT
Tukey HSD
Mean 95% Confidence Interval
(I) GRADE (J) GRADE (I-J) Std. Error Sig. Lower Bound Upper Bound
5 6 -.210099* 5.12E-02 .001 -.344329 -7.5869E-02
7 -.236340* 5.40E-02 .000 -.377872 -9.4808E-02
6 5 .210099* 5.12E-02 .001 7.58695E-02 .344329
7 -2.624E-02 5.25E-02 .959 -.163820 .111340
6 2.624E-02 5.25E-02 .959 -.111340 .163820
8 2.638E-03 5.59E-02 1.000 -.143872 .149149
7 -2.638E-03 5.59E-02 1.000 -.149149 .143872
Note. The dependent variable is SPEEDMRT.
*The mean difference is significant at the .05 level.
Then, assuming FWE = .05, we find qwE,a,df/V2 = 3.710/1.414 = 2.623. To test the
difference between the fifth- and sixth-grade means, we calculate the t statistic of Equation
6.17 (with Spooled replaced by MS s/A) ):
This clearly exceeds 2.623 and is therefore significant. Note that this is the ratio based on
the Table 9.8 output.
9.7.3 When Variances Are Unequal
In Chapter 8, in our analysis of the effects of education level on mean depression scores of
male subjects in the Seasons study, we found that the variances were quite heterogeneous.
Several methods have been proposed to deal with this problem. Most use Welch's t' (Equa-
tion 9.5) and df (Equation 9.6) but differ in the criterion against which t' is evaluated. In
the Games-HoweH test, t' is compared with qpwE, a,dfe /V2 (Games & Howell, 1976). The
procedure is illustrated in Table 9.9, using the depression data.
Investigations of the error rates associated with the Games-Howell test indicate that if
the variances are fairly homogeneous and the group sizes are less than 50, the FWE may
TABLE 9.9 GAMES-HOWELL PROCEDURE FOR TESTING ALL PAIRWISE COMPARISONS WHEN
VARIANCES ARE NOT EQUAL
Applying Equations 9.5 and 9.6 to the fifth- and sixth-grade speed statistics of Table 9.1, we have
To obtain the critical value of t corresponding to 45 df, interpolate in Appendix Table C.3 between
df- 40 and df- 60, with a = 4, FWE = .05. The critical q value is 3.773. Then
Because 4.227 > 2.668, we reject H0: (0,5 = fX6. In similar fashion, values of t' and df can be
calculated for each of the remaining five pairwise comparisons and evaluated against 2.668.
sometimes be as high as .07 when the nominal probability is .05 (Dunnett, 1980; Games,
Keselman, & Rogan, 1981). Even when the variances are not homogeneous, with AZS less
than 6, the FWE may be inflated. However, these same studies indicate that under most
other conditions the FWE is quite close to the nominal level. Furthermore, the test is more
powerful than any of the several competitors that have been proposed and, as might therefore
be expected, has narrower CIs for each comparison. If the researcher is concerned about
the possible inflation of the FWE, Dunnett's T3 test (Dunnett, 1980) appears to be the most
powerful of several alternatives that maintain the FWE at less than or equal to the nominal
value. The test requires tables of the studentized maximum modulus distribution. Miller
has described the procedure and has provided tables of the distribution (Miller, 1981,
pp. 70-75). Statistical packages provide a welcome shortcut to the analytic labor involved
in these tests. Table 9.10 presents SPSS output for the Games-Howell and Dunnett T3
methods applied to the depression data. Note that the p values (in the "Sig." Column) are
consistently lower for the Games-Howell procedure.
9.7.4 The Fisher-Hayter Test
Although the tests considered so far do not require a preliminary test of the omnibus null
hypothesis, there are procedures for controlling the FWE that include such a test as a first
stage. In Fisher's LSD (least significant difference) procedure (Fisher, 1935), the first stage
is the omnibus F test. Pairwise comparisons are tested by the usual t test at the .05 level in
a second stage, but only if the first stage yields a significant result. The LSD test has been
shown to have an inflated FWE for a > 3. However, Hayter's modification of the LSD test
maintains the FWE at or below its nominal level (Hayter 1986). In the Fisher-Hayter test,
the first stage consists of the omnibus F test. If this test yields a significant result, then all
pairwise comparisons may be tested. Standard t statistics are formed and tested against the
criterion qFWE, a-1,df e /V2. Note that in Appendix Table C.9, the column corresponding to
a — 1 means provides the critical value. Table 9.11 provides an example, again using the
Royer multiplication speed data.
As with other sequential testing methods, power is gained relative to simultaneous tests
but at the loss of the ability to construct simultaneous CIs. Therefore, the choice between
TABLE 9.10 RESULTS OF THE DUNNETT T3 AND GAMES-HOWELL TESTS OF ALL PAIRWISE
Dependent Variable: D.MEAN
(I) (J) (I-J) Std. Error Sig. Lower Bound Upper Bound
Dunnett T3 HS C 3.2288 1.2031 .170 -.8314 7.2890
B 3.5724 1.1791 .115 -.5579 7.7028
GS 2.0562 1.1688 .723 - 2.3523 6.4646
C HS -3.2288 1.2031 .170 -7.2890 .8314
B .3436 1.0003 .996 - 1.4671 2.1543
GS -1.1727 .9881 .745 -3.6871 1.3418
B HS -3.5724 1.1791 .115 -7.7028 .5579
C -.3436 1.0003 .996 -2.1543 1.4671
GS - 1.5163 .9587 .534 -4.1427 1.1101
GS HS -2.0562 1.1688 .723 -6.4646 2.3523
C 1.1727 .9881 .745 -1.3418 3.6871
B 1.5163 .9587 .534 -1.1101 4.1427
Games-Howell HS C 3.2288 1.2031 .133 -.7023 7.1600
GS 2.0562 1.1688 .568 -2.2214 6.3337
B .3436 1.0003 .956 -1.4188 2.1061
HS HS -2.0562 1.1688 .568 -6.3337 2.2214
C 1.1727 .9881 .586 - 1.2727 3.6180
B 1.5163 .9587 .406 - 1.0400 4.0726
Note: Output is from SPSS.
the Fisher-Hayter test and the Tukey (or Tukey-Kramer) test depends on whether such CIs
are desired. Another consideration is whether variances are assumed to be homogenous.
The Fisher-Hayter test was derived under that assumption, and therefore a test such as
the Games-Howell or the Dunnett T3 test should be used if homogeneity of variance is in
doubt.3 Another issue is the choice of the omnibus test used in the first stage of testing.
Although the F test is usually referenced with respect to the Fisher-Hayter method, the q
(or q/V2) statistic applied to the largest difference between means provides an alternative
test of the omnibus null hypothesis. Seaman, Levin, and Serlin (1991) found that there was
little difference in the power of the Fisher-Hayter method as a function of whether F or
q was used in the first stage. Which procedure had the advantage depended on the pattern
TABLE 9.11 EXAMPLE OF THE FISHER-HAYTER TEST APPLIED TO THE STATISTICS OF TABLE 9. 1
1. In the first stage, the ANOVA is performed. Because the result (see the SPSS output below) is
significant, we can proceed to the second stage, testing all pairwise contrasts.
SPEEDMRT
Sum of
Squares df Mean Square F Sig.
Between Groups 0.880 3 .293 9.153 .000
Within Groups 2.755 86 3.203E-02
Total 3.634 89
2. Assuming homogeneous variances, we now perform the pairwise tests. To obtain the critical
value of q in Appendix Table C.9, we interpolate between the reciprocals of 60 and 120 in the
column corresponding to three means (one less than the total number of groups). The value of
q.05.3,86 is 3.376 and, dividing by \/2, we find that the critical value of t is 2.387. The t statisti
may now be calculated for each contrast, and the null hypothesis corresponding to any given
contrast will be rejected if the corresponding t exceeds 2.387.
Note. The ANOVA output is from SPSS.
of the means. However, it should be noted that the q test of the largest difference between
means is appropriate for the first stage only if the ns are equal; otherwise, the largest raw
difference may not correspond to the largest standardized effect.
9.7.5 Pairwise Comparisons: Summing Up
Seaman et al. (1991) simulated tests of all pairwise comparisons under conditions of equal
ns and equal variances. With respect to the methods we have presented, averaging over
ns of 5, 10, and 19, the Tukey HSD test generally had a power advantage of 2-3% over
the Dunn-Bonferroni method. The Fisher-Hayter method had a power advantage over the
Tukey method that varied from about 2% to as much as 9%, with the most typical difference
being about 6%. The advantage of the Fisher-Hayter method decreased as a increased from
3 to 5, and as n decreased from 19 to 10. In summary, if (a) all pairwise contrasts are to be
tested, (b) the power to detect differences is the primary consideration, and (c) homogeneity
of variance can reasonably be assumed, then the Fisher-Hayter procedure is the method of
choice. If only some subset of pairwise comparisons are of interest, the Dunn-Bonferroni
procedure may prove more powerful than the alternatives. If CIs are desired, use the Tukey
procedure, or the Dunn-Bonferroni one if only a few of the possible comparisons are of
interest. If there is any evidence of unequal variances, use the Games-Howell test, or the
Dunnett T3 test if n < 6.
Seaman et al. compared 23 different methods (including variations such as whether
F or q was used in the first stage of the Fisher-Hayter method), and even this set of
methods is not exhaustive. We have excluded from the current discussion some methods
COMPARING a - 1 TREATMENT MEANS WITH A CONTROL: DUNNETT'S TEST 255
that yield FWEs that often are considerably in excess of the nominal value. Among these
are Fisher's LSD test, the Student-Newman-Keuls test (Keuls, 1952; Newman, 1939),
and Duncan's multiple range test (Duncan, 1955). We have also excluded several proce-
dures that maintain the FWE at or below its nominal level and have slightly more power
than the Fisher-Hayter method under some combinations of number of groups and group
size. On the basis of various sampling studies, the very slight power advantage of these
methods (usually 1-2%) does not warrant the added complexity they usually entail. The
omitted methods include proposals by Peritz (1970), Ramsay (1978, 1981), Shaffer (1979,
1986), and Welsch (1977). Descriptions of these methods, together with results of sam-
pling experiments, may be found in the article by Seaman et al. (1991). Other discus-
sions of several tests of pairwise comparisons may be found in review articles by Zwick
(1993) and Shaffer (1995), and in Toothaker's (1993) monograph on multiple comparison
9.8 COMPARING a - 1 TREATMENT MEANS WITH A
CONTROL: DUNNETT'S TEST
Dunnett (1955, 1964) proposed a test for studies in which the researcher plans to contrast
each of several treatments with a control. If these are the only comparisons of interest,
methods that control the FWE for a family consisting of all pairwise comparisons will be
overly conservative; power will be lost and simultaneous CIs will be wider than necessary.
The Dunn-Bonferroni procedure with K = a — 1 will be an improvement but will still offer
less power and wider intervals than the Dunnett test.
Assuming that the group sizes are equal and that variances are homogeneous, we see
that the test is quite simple. To test the difference between the mean of the jth group and
the control group mean, compute the usual t statistic for each of the a — 1 contrasts:
where YC is the mean of the control group and the error mean square (MS s/A ) is the
average of all a within-group variances. This t statistic is evaluated against the critical
values of d FWE,a,dfe in Appendix Table C.8 where a is the number of means including the
control and dfe is the number of degrees of freedom associated with the ANOVA error term.
Dunnett (1964) has provided tables that allow comparisons when the standard error of YC
differs from that of the treatment group means. If the treatment group ns are not equal,
replace 2/n in Equation 9.16 by 1 / H J + 1/n c , and use the Dunn-Bonferroni procedure
with K = a — 1. If the treatment group variances are heterogeneous, use Welch's t' and
again use the Dunn-Bonferroni criterion.
Of course, the Dunn-Bonferroni method can be used even when the assumptions un-
derlying the Dunnett test are met. However, the Dunnett test has a power advantage in those
circumstances. For example, suppose we want to compare the sixth-, seventh-, and eighth-
grade mean speeds with the mean speed of the fifth graders. Further assume the means and
standard deviations in Table 9.1 but assume that the ns are all 16, as in Table 9.6. If the FWE
is .05, for a contrast to be significant by use of the Dunnett criterion, the t statistic must
exceed 2.43. Using the Bonferroni criterion, we see that the EC is .05/3, or .0167. and the
256 9/ CONTRASTS AMONG MEANS
t statistic must exceed 2.46. The difference is small but nonetheless the power advantage
lies with the Dunnett procedure. The CIs exhibit a similar relation. Let s$ = .064, the value
calculated in Table 9.6. Then the Dunnett confidence limits for any contrast with the fifth-
grade mean are
and the Dunn-Bonferroni limits are
The interval widths for the Dunnett and the Dunn-Bonferroni methods are .311 and .315,
respectively. Again, the difference is not large but the Dunnett method does have a slight
advantage.
9.9 CONTROLLING THE FAMILYWISE ERROR RATE
FOR POST HOC CONTRASTS
Sometimes observed patterns in the data suggest one or more effects that were not an-
ticipated. When the corresponding null hypotheses are tested to determine whether these
effects are significant, we should be quite conservative in evaluating the result. In testing
contrasts "after the fact" we are, in effect, investigating the family of all possible outcomes.
There are an infinite number of possible contrasts and therefore the methods we present are
quite conservative because they control for the probability of at least one Type 1 error in a
very large set of possible contrasts.
9.9.1 Scheffe's Method
Assuming that the populations are normally distributed and have equal variances, Scheffe's
method maintains the FWE at its nominal level when the family consists of all possible
contrasts associated with a source of variance (Scheffe, 1959). The test statistic is the t of
Equation 9.10 and it is evaluated against
where df1 and d/2 are the numerator and denominator degrees of freedom associated with
the omnibus F test.
Using the fifth- to eighth-grade multiplication speeds as an example, assume that we
did not anticipate the pattern of means in Table 9.1 but, after viewing the data, realized that
the sixth through eighth grades had very similar means, each higher than the fifth-grade
mean. We might wish to test whether the mean of the fifth-grade times differed significantly
from that of the three combined sixth- through eighth-grade times; that is, we might test
We calculated the t statistic in Table 9.1 as 5.212. Because the test was not planned, we
use the Scheffe method to test the preceding null hypothesis. With FWE = .05, f = 3, and
dfe = 86, the critical value of F is 2.71 (by interpolation in Appendix Table C.5, or by
CONTROLLING THE FAMILYWISE ERROR RATE FOR POST HOC CONTRASTS 257
using the UCLA calculator or software such as SYSTAT, SPSS, or SAS). Therefore,
We reject H0 if t > S or t < —S; because 5.264 > 2.851, the contrast is significant.
CIs can also be calculated. The probability is 1 — FWE that the values of all possible
contrasts in a family are simultaneously contained within intervals bounded by
From Table 9.1, we find that for the contrast in our example, $ = .679 and SQ, = .130. The
original weights were all multiplied by 3 to yield integer weights. Therefore, the confidence
limits on 31x5 — (|JL6 + IL7 + j u 8 ) are
However, we want limits on the original scale and we therefore divide these values by 3,
yielding .103 and .350. In Subsection 9.6.1, we treated this contrast as planned and found
the Dunn-Bonferroni limits, .116 and .337. The Scheffe interval is wider than the Dunn-
Bonferroni interval, revealing the price we pay in precision of estimation and power when
contrasts are not planned. A good strategy is to plan all those contrasts that might conceivably
be of interest, and then use the Dunn-Bonferroni or Fisher-Hayter method. Although the
power of these methods decreases as K, the number of planned contrasts, increases, a rather
large number of comparisons must be planned before the Scheffe criterion requires a smaller
value of t for significance. For a more detailed comparison of the Dunn-Bonferroni and the
Scheffe methods, consult the article by Perlmutter and Myers (1973).
Experimenters who have used both the standard ANOVA (analysis of variance) tests and
the Scheffe procedure have sometimes been surprised to find the omnibus null hypothesis
rejected by the ANOVA test but no contrasts significant by the Scheffe criterion. The
source of this apparent contradiction is that the overall F test has exactly the same power
as the maximum possible contrast tested by the Scheffe procedure. That contrast may
be of little interest and may not therefore have been tested. It could be something like
(1 l/37)u1 + (26/37)u2 - (17/45)u3 - (28/45)u4. In summary, although rejection of the
omnibus null hypothesis indicates that at least one contrast is significant by the Scheffe
criterion, there is no guarantee that any obvious or interesting contrast will be significant.
As with all the tests we have considered so far (except the Fisher-Hayter test), there is
no logical necessity that the Scheffe tests of contrasts be preceded by a significant omnibus
F. In contrast, even the largest possible contrast will not be significant if the overall F test
is not. There seems little point in expending energy on a series of post hoc Scheffe tests
unless first determining, by the F test of the omnibus null hypothesis, whether there is any
possibility that a contrast of interest might be significant.
9.9.2 The Brown-Forsythe Method
Brown and Forsythe (1974b) proposed that Welch's t' and df` (Equations 9.5 and 9.6)
be used with a criterion similar to Scheffe's S when the assumption of homogeneity of
variance is in question. The only difference is that the critical value of S against which t'
is evaluated is based on df`. As an example, reconsider the Seasons data. Let us contrast
the mean depression scores of individuals with only a high school education (HS) with the
mean of individuals with more than a high school education (some college, C; bachelor's
degree, B; and graduate school, GS). We test
Turning back to Table 9.4, we find dfe (df') = 20.527, or approximately 21. With df1 =
a - 1, or 3, and FWE = .05, the critical value of F is 3.07. Therefore,
and, because t' = 2.118 (see Table 9.4), the result is not significant. The .95 limits on the
simultaneous CI of the contrast were calculated after all coefficients were multiplied by 3;
those limits are
Dividing by 3 to return to the original scale, we have —1.277, 7.182. The interval provides
an alternative test of the null hypothesis; the fact that it contains zero informs us that the
null hypothesis cannot be rejected.
9.10 THE SUM OF SQUARES ASSOCIATED
WITH A CONTRAST
The t statistic used throughout this chapter provides one approach to testing hypotheses
about contrasts. An alternative, but equivalent, test is based on components of SSA, the sums
of squares for the A source of variance. We develop this approach now in order to emphasize
the continuity between the ANOVA and test of contrasts, and to enhance our understanding
of contrasts. Furthermore, the relation between contrasts and sums of squares will prove
useful in the developments of the next chapter on trend analysis. A central idea is that any
contrast of the group means corresponds to a component of the SS A , the sum of squares
for the variable, A. That component sum of squares, SS^,, will be distributed on 1 df and
the contrast can be tested by dividing it by MS S / A - Because, as we showed in Chapter 7,
F1 dfc =tdfc, the F statistic formed in this way will equal t2. This relation leads us directly
to a formula for SS:,.:
THE SUM OF SQUARES ASSOCIATED WITH A CONTRAST 259
TABLE 9.12 CONTRASTS OF THE MEANS IN TABLE 9.1 , ASSUMING n = 1 6
The contrasts are
The ANOVA table is
A 3 0.621 .207 6.37
o1 1 0.085 .085 2.62
42 1 0.183 .183 5.63
4-3 1 0.353 .353 10.86
S/A 60 1.95 .033
To illustrate and to further develop the relation between contrasts and the ANOVA,
reconsider the multiplication speed scores in the Royer study. The means (from Table 9.1)
are .350, .560, .586, and .583 for the fifth through eighth grades, and to simplify calculations
we again assume n = 16 in all grades. Table 9.12 presents three contrasts and the ANOVA
table containing all relevant sums of squares, degrees of freedom, mean squares, and F
tests. The contrasts represent (a) the difference between the mean for the eighth grade and
the mean for the other three grades, (b) the difference between the seventh-grade mean and
the mean of the combined fifth and sixth grades, and (c) the difference between the fifth-
and sixth-grade means. An important point is that the SS$ s in Table 9.12 sum to .621, the
SSA; that is,
We can think of the treatment sum of squares as a pie and each contrast as a piece of the
pie. In the example of Table 9.1, the pieces are nonoverlapping, and together they account
for the whole pie. Here, the contrast of the fifth- and sixth-grade means (1^3) is the biggest
piece, accounting for about 57% (.353/.621) of the variability among the treatment means.
As with any pie, the SSA can be divided into pieces in many ways. For example, let
Applying Equation 9.19, we find the corresponding sums of squares for these contrasts to
be .421, .181, and .019; these values again sum to .621.
Will every set of contrasts result in sums of squares that add to SSA? Not at all; in
each of the preceding examples, the contrasts making up the set had a particular property
that resulted in their accounting for different portions of the variability, for nonoverlapping
pieces of the pie. When this is the case, the contrasts are said to be orthogonal. In addition,
the number of orthogonal contrasts in each set, a — 1, was equal to the degrees of freedom
of the treatment sum of squares. The sums of squares of dfA orthogonal contrasts will always
add to $SA .
Before defining orthogonality more precisely, let's consider an example in which it
does not occur. Let's test
If the mean of the fifth-grade population (u5) differs from that of the sixth-grade population
(u6), there is a good chance that it will also differ from the combined mean of u 6 ,u 7 , and (Jtg,
because that mean contains x5. In other words, there is a positive relation between the two
contrasts. This lack of independence between the two contrasts is called nonorthogonality.
It becomes evident in our example when we calculate the sums of squares corresponding
to the two contrasts. The sums of squares for the tests of H01 and H02 are .353 and .615,
respectively. The sum is clearly greater than SSA, .621. The two pieces of the pie overlap.
We do not have to add the sums of squares to determine whether two contrasts are, or
are not, orthogonal. Consider two contrasts, i|>p and tyq, such that
If there are n scores at all levels of A, the criterion for orthogonality is
For example, we know that the first two contrasts in Equation 9.20 are orthogonal because
If the ns vary across treatment conditions, the criterion for orthogonality becomes
Several points about orthogonality deserve emphasis. First, a set of a — 1 orthogonal con-
trasts can be thought of as asking a — 1 logically independent questions that collectively
"use up" all the degrees of freedom and variability associated with the independent variable.
Note that the variability can be partitioned in different ways, so that it is possible to find
different sets of orthogonal contrasts. Also, whether or not two contrasts are orthogonal
depends on the contrast weights, not on the values of the means being contrasted. One way
of thinking about this is that orthogonality depends on what questions are addressed by the
contrasts, not on what the answers turn out to be. The second point is that we choose to
test contrasts because they are of substantive interest, whether or not they are orthogonal to
one another. For example, researchers commonly test pairwise comparisons; these are not
orthogonal, but they are often of interest and should be tested when they are.
Table 9.13 summarizes the conditions under which each of the methods described in the
preceding sections is appropriate. The methods that are listed do not exhaust all the possi-
bilities. We have excluded some methods because they allow the FWE to exceed its nominal
TABLE 9.13 SUMMARY OF METHODS FOR CONTROLLING FWEs
Contrast Cl Group Size (n) Pop. Variances 0 Test Stat. Method
Planned Yes NR Equal t D-B or D-S
Yes NR Not equal t' D-B or D-S
No NR Equal t Hochberg
No NR Not equal t' Hochberg
vs. Control Yes Equal Equal t Dunnett
Yes Not equal Equal t D-B or D-S
All pairwise Yes Equal Equal q or t Tukey
Yes Not equal Equal q or t T-K
Yes NR Not equal t' G~H or T3
No NR Equal F or q, then F-H
q or t
General post hoc Yes NR Equal t Scheffe
Yes NR Not equal t' Scheffe
Note. NR = not relevant; D-B = Dunn-Bonferroni; D-S = Dunn-Sidak; T-K - Tukey-Kramer; G-H = Games-
Howell; T3 = Dunnett's T3; F-H = Fisher-Hayter; t = t statistic with SE based on MS S / A ; t' = Welch's t using df: see
Equations 9.5 and 9.6.
For population variances: Equal = the population variances are assumed to be homogeneous; Not equal = homogeneity
of variance assumption is suspect.
level, and others because they provide a very small gain in power at the cost of added com-
plexity. The methods listed in Table 9.13 are easy to use and are available in several software
packages such as SPSS and SAS. Further discussions of these procedures and descriptions
of other methods may be found in the previously cited review article by Shaffer (1995).
Seaman et al. (1991) also describe many methods for controlling FWEs for the family of
all pairwise comparisons and present the results of a large-scale computer-sampling study
of error rates and power.
In closing, we emphasize two points. First, among the tests we have presented, only
the Fisher-Hayter procedure requires that the omnibus null hypothesis be rejected before
contrasts are tested. Such a requirement causes a loss of power in using the Bonferroni-
based, or the Tukey, or the Dunnett methods. In the case of Scheffe's test, the preliminary
F is useful in possibly saving unnecessary labor because if it is not significant, no test
of a contrast will have a significant outcome; however, the omnibus test is not logically
necessary. In other instances, for example, when several contrasts have been planned, a
significant outcome of the omnibus test suggests that something is going on and if the
planned contrasts are not significant, the investigator may wish to do some data snooping,
using the Scheffe method. However, if the omnibus null hypothesis is not rejected, planned
contrasts can still be tested provided the FWE is maintained at or below the nominal value.
The second point, which cannot be emphasized too strongly, is the importance of planning
contrasts. A procedure such as the Dunn-Bonferroni may be considerably more powerful
than the" Tukey procedure if only a few of the pairwise comparisons are of any possible
interest. In addition, close consideration to the questions that are of primary, or perhaps
sole, interest may lead to research designs that are more closely focused on those questions.
262 9/CONTRASTS AMONG MEANS
omnibus null hypothesis planned contrasts
post hoc contrasts family
familywise error rate (FWE) error rate per contrast (EC)
Bonferroni inequality Dunn-Bonferroni method
simultaneous CIs Dunn-Sidak method
simultaneous methods sequential methods
Hochberg's sequential method the studentized range statistic (q)
Tukey's HSD test Tukey-Kramer test
Games-Howell test Fisher-Hayter test
Dunnett's test Scheffe's method
Brown-Forsythe method orthogonal contrasts
nonorthogonality
9.1 There are five treatment conditions in a problem-solving study, each with n = 20.
Two groups, F1 and F2, are given instructions designed to facilitate problem solving.
The third group is a control group given neutral instructions. The fourth and fifth
groups, I1 and 72, are given instructions designed to interfere with problem solving.
The data are as follows:
Fl F2 C /1 12
Y.i 14.6 14.9 13.8 11.8 11.7
S2 3 4 5 4 4
Test each of the following hypotheses with a = .05. State H0 and H1.
(a) The average of the facilitation group population means is greater than the mean
of the control population.
(b) The average of the interference population means is different from the mean of
the control population.
(c) The average of the facilitation means is not the same as the average of the inter-
ference means.
9.2 (a) Assume that all three tests in Exercise 9.1 were planned prior to data collection.
Using the Dunn-Bonferroni method, construct .90 simultaneous CIs for the three
contrasts. Reevaluate whether the null hypotheses in parts (b) and (c) should be
rejected, using the Dunn-Bonferroni criterion with FWE = .10.
(b) Assume the contrasts were chosen after the means were viewed. Use the Scheffe
method to construct simultaneous CIs. Reevaluate whether the null hypotheses
in parts (b) and (c) of Exercise 9.1 should be rejected with FWE = .10.
9.3 The following group means are each based on 10 scores:
A, A2 A3
(a) Calculate 55^-
(b) Calculate the sum of squares for each of the following contrasts: (i) 41, =
F.i -F. 2 ;(ii)$21 =F.i -U/2)(F.2+F.3);(iii)4f 3 =F.1 -F.3.
What should be true of the relation between SSA and the sums of squares for $ j
and vj> 2 ? Why?
(c) We can remove the effect associated