Statistical Approaches to Measurement Invariance
Roger E. Millsap
Routledge
This book reviews the statistical procedures used to detect measurement bias. Measurement bias is examined from a general latent variable perspective so as to accommodate different forms of testing in a variety of contexts including cognitive or clinical variables, attitudes, personality dimensions, or emotional states. Measurement models that underlie psychometric practice are described, including their strengths and limitations. Practical strategies and examples for dealing with bias detection are provided throughout.

The book begins with an introduction to the general topic, followed by a review of the measurement models used in psychometric theory. Emphasis is placed on latent variable models, with introductions to classical test theory, factor analysis, and item response theory, and the controversies associated with each, being provided. Measurement invariance and bias in the context of multiple populations is defined in chapter 3 followed by chapter 4 that describes the common factor model for continuous measures in multiple populations and its use in the investigation of factorial invariance. Identification problems in confirmatory factor analysis are examined along with estimation and fit evaluation and an example using WAIS-R data. The factor analysis model for discrete measures in multiple populations with an emphasis on the specification, identification, estimation, and fit evaluation issues is addressed in the next chapter. An MMPI item data example is provided. Chapter 6 reviews both dichotomous and polytomous item response scales emphasizing estimation methods and model fit evaluation. The use of models in item response theory in evaluating invariance across multiple populations is then described, including an example that uses data from a large-scale achievement test. Chapter 8 examines item bias evaluation methods that use observed scores to match individuals and provides an example that applies item response theory to data introduced earlier in the book. The book concludes with the implications of measurement bias for the use of tests in prediction in educational or employment settings.

A valuable supplement for advanced courses on psychometrics, testing, measurement, assessment, latent variable modeling, and/or quantitative methods taught in departments of psychology and education, researchers faced with considering bias in measurement will also value this book. 

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"Over the last 20 years, I have worked on aspects of the bias detection problem in psychology, always returning to the problem eventually in spite of other research obligations and interests. One reason for my fascination with the detection problem is that it remains only partially solved. Fundamentally, while we have detection methods that work under limited circumstances, we have no completely general method for detecting bias in psychological measures that works across all of the many measurement conditions encountered in psychology. Instead, we have a collection of approaches that apply under specific conditions and assumptions, and that may not function well when these conditions or assumptions are violated. A further problem is that it is usually hard to verify some of the assumptions that are needed in practice for many of the detection methods. It is undeniable, however, that substantial progress has been made in the last 20 years in the development of new methods for bias detection and in enhancements to existing methods.
I have written this book with several goals in mind. The first goal is to acquaint the reader with the broad set of statistical procedures currently used to approach the problem of detecting measurement bias. A second goal, which follows upon the first, is to provide the necessary background material so that readers can place the many detection procedures in context. Toward that end, the book devotes considerable space to describing the measurement models that underlie psychometric practice. I have also included some theoretical results that subsume many types of detection procedures, showing their strengths and limitations. Finally, I hope that the book will inspire more researchers to become involved in bias detection research. The problem of detecting bias in psychological measures is a difficult one, and it represents a significant challenge to measurement in psychology. In addition to the detection problem itself, the measurement bias problem has implications for the use of tests in many domains as predictors or as tools for making decisions about people. The last chapter of the book explores these implications for the use of tests in prediction and for the related problem of bias in prediction.
The intended audience for this book consists of researchers and students who wish to understand the array of available procedures for detecting bias in psychological measurement. The problem of measurement bias arises in many research settings, ranging from applied research in employment or educational testing to the use of tests or questionnaires in experimental studies with multiple groups. I hope that the book will serve as a useful reference for researchers in all of these settings. The book is intended to also serve as a resource for students who want to learn more about psychometrics, particularly in relation to the problem of measurement bias as a challenge to the validity of measurement. To benefit from the book, a student should have an understanding of basic probability and statistical theory, along with some knowledge of regression."

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"Millsap provides extensive background, is technically rigorous, and illustrates the approaches with interesting psychological examples. The book is well written and deserves to become the classic reference in the field. ... [The] book provides a broad and very thorough exposition of the most important topics associated with the statistical study of measurement invariance. It is a timely book on an important topic. ... The book is a must read for psychometricians and for researchers who compare test scores across groups. ... [Readers] will be equipped with invaluable knowledge of the nature of group differences in psychological measurement." - Jelte Wicherts, Ph.D., University of Amsterdam, Netherlands, in PsycCRITIQUES

"Measurement invariance is a key concept in psychological assessment. Millsap has provided the most readable account yet of this difficult topic, combining clear prose, technical details, and compelling examples. A "must have" for quantitative expert and practicing scientist alike."  – Keith F. Widaman, University of California at Davis, USA

"Roger Millsap is a leading authority on the problem of measurement invariance and has written an extraordinary book on this critically important topic. This book is a "must read" by anyone working on the development of measurements for national and international surveys."  – David Kaplan, University of Wisconsin – Madison, USA Member, OECD/PISA Questionnaire Expert Group

"This comprehensive treatment of measurement invariance is sure to become the standard reference work on the topic. With thorough coverage of observed and latent variable models for prediction and assessment, Millsap's book is packed with lucid discussions of the foundational role of measurement invariance in situations that require the comparison of measured attributes. All persons in the biobehavioral sciences and business who use test data when making decisions would benefit by reading this book." – Niels Waller, University of Minnesota, USA

"A substantial contribution to the field, this book offers a comprehensive treatment of the statistical methods used to detect measurement bias. With an emphasis on latent variable models, it introduces us to many measurement perspectives and places the need for detecting bias into a larger societal context, one that attempts to foster social justice through accurate and unbiased measurement in the fields of psychology, education, and public policy. There is little doubt this book will become a classic in the field."  – Howard T. Everson, City University of New York, USA

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About the Author

Roger E. Millsap is a Professor in the Department of Psychology and a faculty member in the Doctoral Program in Quantitative Psychology at Arizona State University. He received his Ph.D. in Psychology in 1983 from the University of California-Berkeley. Dr. Millsap’s research interests include psychometrics, latent variable models, and multivariate statistics. He has published more than 60 papers in professional journals and co-edited the _Sage Handbook of Quantitative Methods in Psychology_ with Alberto Maydeu-Olivares in 2009. Dr. Millsap is a Past-President of the Psychometric Society, of Division 5 of the American Psychological Association, and of the Society of Multivariate Experimental Psychology. He is a Past –Editor of _Multivariate Behavioral Research_ and is the current Executive Editor of _Psychometrika_.