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Type-S and Type-M errors
May 21, 2015 May 21, 2015 Alex Etz Article Review, Fun Science, Psychology, Statistics	Errors, hypothesis testing, null-hypothesis, Sampling Distribution, Significance, Statistics
An anonymous reader of the blog emailed me:
I wonder if you'd be ok to help me to understanding this Gelman's graph. I struggle to understand what is the plotted distribution and the exact meaning of the red area. Of course I read the related article, but it doesn't help me much.
Rather than write a long-winded email, I figured it will be easier to explain on the blog using some step by step illustrations. With the anonymous reader's permission I am sharing the question and this explanation for all to read. The graph in question is reproduced below. I will walk through my explanation by building up to this plot piecewise with the information we have about the specific situation referenced in the related paper. The paper, written by Andrew Gelman and John Carlin, illustrates the concepts of Type-M errors and Type-S errors. From the paper:
We frame our calculations not in terms of Type 1 and Type 2 errors but rather Type S (sign) and Type M (magnitude) errors, which relate to the probability that claims with confidence have the wrong sign or are far in magnitude from underlying effect sizes (p. 2)
So Gelman's graph is an attempt to illustrate these types of errors. I won't go into the details of the paper since you can read it yourself! I was asked to explain this graph though, which isn't in the paper, so we'll go through step by step building our own type-s/m graph in order to build an understanding. The key idea is this: if the underlying true population mean is small and sampling error is large, then experiments that achieve statistical significance must have exaggerated effect sizes and are likely to have the wrong sign. The graph in question:
A few technical details: Here Gelman is plotting a sampling distribution for a hypothetical experiment. If one were to repeatedly take a sample from a population, then each sample mean would be different from the true population mean by some amount due to random variation. When we run an experiment, we essentially pick a sample mean from this distribution at random. Picking at random, sample means tend to be near the true mean of the population, and the how much these random sample means vary follows a curve like this. The height of the curve represents the relative frequency for a sample mean in a series of random picks. Obtaining sample means far away from the true mean is relatively rare since the height of the curve is much lower the farther out we go from the population mean. The red shaded areas indicate values of sample means that achieve statistical significance (i.e., exceed some critical value).
The distribution's form is determined by two parameters: a location parameter and a scale parameter. The location parameter is simply the mean of the distribution (μ), and the scale parameter is the standard deviation of the distribution (σ). In this graph, Gelman defines the true population mean to be 2 based on his experience in this research area; the standard deviation is equal to the sampling error (standard error) of our procedure, which in this case is approximately 8.1 (estimated from empirical data; for more information see the paper, p. 6). The extent of variation in sample means is determined by the amount of sampling error present in our experiment. If measurements are noisy, or if the sample is small, or both, then sampling error goes up. This is reflected in a wider sampling distribution. If we can refine our measurements, or increase our sample size, then sampling error goes down and we see a narrower sampling distribution (smaller value of σ).
Let's build our own Type-S and Type-M graph
In Gelman's graph the mean of the population is 2, and this is indicated by the vertical blue line at the peak of the curve. Again, this hypothetical true value is determined by Gelman's experience with the topic area. The null hypothesis states that the true mean of the population is zero, and this is indicated by the red vertical line. The hypothetical sample mean from Gelman's paper is 17, which I've added as a small grey diamond near the x-axis. R code to make all figures is provided at the end of this post (except the gif).
If we assume that the true population mean is actually zero (indicated by the red vertical line), instead of 2, then the sampling distribution has a location parameter of 0 and a scale parameter of 8.1. This distribution is shown below. The diamond representing our sample mean corresponds to a fairly low height on the curve, indicating that it is relatively rare to obtain such a result under this sampling distribution.
Next we need to define cutoffs for statistically significant effects (the red shaded areas under the curve in Gelman's plot) using the null value combined with the sampling error of our procedure. Since this is a two-sided test using an alpha of 5%, we have one cutoff for significance at approximately -15.9 (i.e., 0 – [1.96 x 8.1]) and the other cutoff at approximately 15.9 (i.e., 0 + [1.96 x 8.1]). Under the null sampling distribution, the shaded areas are symmetrical. If we obtain a sample mean that lies beyond these cutoffs we declare our result statistically significant by conventional standards. As you can see, the diamond representing our sample mean of 17 is just beyond this cutoff and thus achieves statistical significance.
But Gelman's graph assumes the population mean is actually 2, not zero. This is important because we can't actually have a sign error or a magnitude error if there isn't a true sign or magnitude. We can adjust the curve so that the peak is above 2 by shifting it over slightly to the right. The shaded areas begin in the same place on the x-axis as before (+/- 15.9), but notice that they have become asymmetrical. This is due to the fact that we shifted the entire distribution slightly to the right, shrinking the left shaded area and expanding the right shaded area.
And there we have our own beautiful type-s and type-m graph. Since the true population mean is small and positive, any sample mean falling in the left tail has the wrong sign and vastly overestimates the population mean (-15.9 vs. 2). Any sample mean falling in the right tail has the correct sign, but again vastly overestimates the population mean (15.9 vs. 2). Our sample mean falls squarely in the right shaded tail. Since the standard error of this procedure (8.1) is much larger than the true population mean (2), any statistically significant result must have a sample mean that is much larger in magnitude than the true population mean, and is quite likely to have the wrong sign.
In this case the left tail contains 24% of the total shaded area under the curve, so in repeated sampling a full 24% of significant results will be in the wrong tail (and thus be a sign error). If the true population mean were still positive but larger in magnitude then the shaded area in the left tail would become smaller and smaller, as it did when we shifted the true population mean from zero to 2, and thus sign errors would be less of a problem. As Gelman and Carlin summarize,
setting the true effect size to 2% and the standard error of measurement to 8.1%, the power comes out to 0.06, the Type S error probability is 24%, and the expected exaggeration factor is 9.7. Thus, it is quite likely that a study designed in this way would lead to an estimate that is in the wrong direction, and if "significant," it is likely to be a huge overestimate of the pattern in the population. (p. 6)
I hope I've explained this clearly enough for you, anonymous reader (and other readers, of course). Leave a comment below or tweet/email me if anything is unclear!
Here is a neat gif showing our progression! Thanks for reading 🙂
(I don't think this disclaimer is needed but here it goes: I don't think people should actually use repeated-sampling statistical inference. This is simply an explanation of the concept. Be a Bayesian!)
R code
#The first plot with the null value and the proposed true value
x <- seq(-35,35,.001) #set up for plotting the curve
y <- dnorm(x,0,8.1) #y values for plotting curve
plot(x=x,y=y, main="Type-S and Type-M error example", xlab="Estimated effect size",
ylab="Relative frequency", type="l", cex.lab=1.5, axes=F, col="white")
axis(1,at=seq(-30,30,10),pos=0) #make the axis nice
axis(2,at=seq(0,.05,.01),pos=-35,las=1) #make the axis nice
lines(c(0,0),c(0,.05),col="red",lwd=3) ##Add line at null value
lines(c(2,2),c(0,.05),col="blue",lwd=3) ##Add line at population mean
points(17, .001, pch=23, bg="grey",col="black",cex=1.5) ##Add sample mean
#######################################################################################################
##The second and third plots with the null sampling distribution and significance areas under the curve
plot(x,y, main="Type-S and Type-M error example", xlab="Estimated effect size",
ylab= "Relative frequency", type="l",cex.lab=1.5, las=1, lwd=3, axes = F)
axis(1,at=seq(-30,30,10),pos=0) #make the x axis nice
axis(2,at=seq(0,.05,.01),pos=-35,las=1) #make the y axis nice
lines(c(0,0),c(0,dnorm(0,0,8.1)),col="red",lwd=3) ##adds null line
lines(c(2,2),c(0,dnorm(2,0,8.1)),col="blue",lwd=3) ##adds true pop mean line
points(17, .001, pch=23, bg="grey",col="black",cex=1.5) ##adds sample mean
##Adds shaded area
cord.x <- c(-35, seq(-35,-15.9,.01),-15.9) ##set up for shading
cord.y <- c(0,dnorm(seq(-35,-15.9,.01),0,8.1),0) ##set up for shading
polygon(cord.x,cord.y,col='red') ##shade left tail
cord.xx <- c(35, seq(35,15.9,-.01),15.9)
cord.yy <- c(0,dnorm(seq(35,15.9,-.01),0,8.1),0)
polygon(cord.xx,cord.yy,col='red') ##shade right tail
points(17, .001, pch=23, bg="grey",col="black",cex=1.5) ##replots the sample mean over the shading
##The fourth plot with the alternative sampling distribution and significance areas under the curve
ylab= "Relative frequency", type="l", cex.lab=1.5, las=1, lwd=3, axes = F)
axis(2,at=seq(0,.05,.01),pos=-35, las=1) #make the y axis nice
lines(c(0,0),c(0,dnorm(0,2,8.1)),col="red",lwd=3) ##add vertical line at null value
lines(c(2,2),c(0,dnorm(2,2,8.1)),col="blue",lwd=3) ##add vertical line at population mean
points(17, .001, pch=23, bg="grey",col="black", cex=1.5) ##replots sample mean over shading
view raw gistfile1.txt hosted with ❤ by GitHub
The One-Sided P-Value Paradox
April 20, 2015 September 4, 2015 Alex Etz Fun Science, Psychology, Statistics	probability, statistical evidence
Today on Twitter there was some chatting about one-sided p-values. Daniel Lakens thinks that by 2018 we'll see a renaissance of one-sided p-values due to the advent of preregistration. There was a great conversation that followed Daniel's tweet, so go click the link above and read it and we'll pick this back up once you do.
As you have seen, and is typical of discussions around p-values in general, the question of evidence arises. How do one-sided p-values relate to two-sided p-values as measures of statistical evidence? In this post I will argue that thinking through the logic of one-sided p-values highlights a true illogic of significance testing. This example is largely adapted from Royall's 1997 book.
The idea behind Fisher's significance tests goes something like this. We have a hypothesis that we wish to find evidence against. If the evidence is strong enough then we can reject this hypothesis. I will use the binomial example because it lends itself to good storytelling, but this works for any test.
Premise A: Say I wish to determine if my coin is unfair. That is, I want to reject the hypothesis, H1, that the probability of heads is equal to ½. This is a standard two-sided test. If I flip my coin a few times and observe x heads, I can reject H1 (at level α) if the probability of obtaining x or more heads is less than α/2. If my α is set to the standard level, .05, then I can reject H1 if Pr(x or more heads) ≤ .025. In this framework, I have strong evidence that the probability of heads is not equal to ½ if my p-value is lower than .025. That is, I can claim (at level α) that the probability of heads is either greater than ½ or less than ½ (proposition A).
Premise B: If I have some reason to think the coin might be biased one way or the other, say there is a kid on the block with a coin biased to come up heads more often than not, then I might want to use a one-sided test. In this test, the hypothesis to be rejected, H2, is that the probability of heads is less than or equal to ½. In this case I can reject H2 (at level α) if the probability of obtaining x or more heads is less than α. If my α is set to the standard level again, .05, then I can reject H2 if Pr(x or more heads) < .05. Now I have strong evidence that the probability of heads is not equal to ½, nor is it less than ½, if my p-value is less than .05. That is, I can claim (again at level α) that the probability of heads is greater than ½. (proposition B).
As you can see, proposition B is a stronger logical claim than proposition A. Saying that my car is faster than your car is making a stronger claim than saying that my car is either faster or slower than your car.
The paradox
If I obtain a result x, such that α/2 < Pr(x or more heads) < α, (e.g., .025 < p < .05), then I have strong evidence for the conclusion that the probability of heads is greater than ½ (see proposition B). But at the same time I do not have strong evidence for the conclusion that the probability of heads is > ½ or < ½ (see proposition A).
I have defied the rules of logic. I have concluded the stronger proposition, probability of heads > ½, but I cannot conclude the weaker proposition, probability of heads > ½ or < ½. As Royall (1997, p. 77) would say, if the evidence justifies the conclusion that the probability of heads is greater than ½ then surely it justifies the weaker conclusion that the probability of heads is either > ½ or < ½.
Should we use one-sided p-values?
Go ahead, I can't stop you. But be aware that if you try to interpret p-values, either one- or two-sided, as measures of statistical (logical) evidence then you may find yourself in a p-value paradox.
References and further reading:
Royall, R. (1997). Statistical evidence: A likelihood paradigm (Vol. 71). CRC press. Chapter 3.7.
Edwards, Lindman, and Savage (1963) on why the p-value is still so dominant
February 9, 2015 Alex Etz Psychology, Statistics	hypothesis testing, NHST, null-hypothesis, probability, Psychology, Science Reform, Significance, Statistics
Below is an excerpt from Edwards, Lindman, and Savage (1963, pp. 236-7), on why p-value procedures continue to be dominant in the empirical sciences even after it has been repeatedly shown to be an incoherent and nonsensical statistic (note: those are my choice of words, the authors are very cordial in their commentary). The age of the article shows in numbers 1 and 2, but I think it is still valuable commentary; Numbers 3 and 4 are still highly relevant today.
From Edwards, Lindman, and Savage (1963, pp. 236-7):
If classical significance tests have rather frequently rejected true null hypotheses without real evidence, why have they survived so long and so dominated certain empirical sciences ? Four remarks seem to shed some light on this important and difficult question.
1. In principle, many of the rejections at the .05 level are based on values of the test statistic far beyond the borderline, and so correspond to almost unequivocal evidence [i.e., passing the interocular trauma test]. In practice, this argument loses much of its force. It has become customary to reject a null hypothesis at the highest significance level among the magic values, .05, .01, and .001, which the test statistic permits, rather than to choose a significance level in advance and reject all hypotheses whose test statistics fall beyond the criterion value specified by the chosen significance level. So a .05 level rejection today usually means that the test statistic was significant at the .05 level but not at the .01 level. Still, a test statistic which falls just short of the .01 level may correspond to much stronger evidence against a null hypothesis than one barely significant at the .05 level. …
2. Important rejections at the .05 or .01 levels based on test statistics which would not have been significant at higher levels are not common. Psychologists tend to run relatively large experiments, and to get very highly significant main effects. The place where .05 level rejections are most common is in testing interactions in analyses of variance—and few experimenters take those tests very seriously, unless several lines of evidence point to the same conclusions. [emphasis added]
3. Attempts to replicate a result are rather rare, so few null hypothesis rejections are subjected to an empirical check. When such a check is performed and fails, explanation of the anomaly almost always centers on experimental design, minor variations in technique, and so forth, rather than on the meaning of the statistical procedures used in the original study.
4. Classical procedures sometimes test null hypotheses that no one would believe for a moment, no matter what the data […] Testing an unbelievable null hypothesis amounts, in practice, to assigning an unreasonably large prior probability to a very small region of possible values of the true parameter. […]The frequent reluctance of empirical scientists to accept null hypotheses which their data do not classically reject suggests their appropriate skepticism about the original plausibility of these null hypotheses. [emphasis added]
Edwards, W., Lindman, H., & Savage, L. J. (1963). Bayesian statistical inference for psychological research. Psychological review, 70(3), 193-242.
Question: Why do we settle for 80% power? Answer: We're confused.
December 15, 2014 December 15, 2014 Alex Etz Psychology, Statistics	hypothesis testing, NHST, Psychology, Significance, Statistics
Coming back to the topic of my previous post, about how we must draw distinct conclusions from different hypothesis test procedures, I'd like to show an example of how these confusions might actually arise in practice. The following example comes from Royall's book (you really should read it), and questions why we settle for a power of only 80%. It's a question we've probably all thought about at some point. Isn't 80% power just as arbitrary as p-value thresholds? And why should we settle for such a large probability of error before we even start an experiment?
From Royall (1997, pp. 109-110):
Why is a power of only 0.80 OK?
We begin with a mild peculiarity — why is it that the Type I error rate α is ordinarily required to be 0.05 or 0.01, but a Type II error rate as large as 0.20 is regularly adopted? This often occurs when the sample size for a clinical trial is being determined. In trials that compare a new treatment to an old one, the 'null' hypothesis usually states that the new treatment is not better than the old, while the alternative states that it is. The specific alternative value chosen might be suggested by pilot studies or uncontrolled trials that preceded the experiment that is now being planned, and the sample size is determined [by calculating power] with α = 0.05 and β = 0.20. Why is such a large value of β acceptable? Why the severe asymmetry in favor of α? Sometimes, of course, a Type I error would be much more costly than a Type II error would be (e.g. if the new treatment is much more expensive, or if it entails greater discomfort). But sometimes the opposite is true, and we never see studies proposed with α = 0.20 and β = 0.05. No one is satisfied to report that 'the new treatment is statistically significantly better than the old (p ≤ 0.20)'.
Often the sample-size calculation is first made with β = α = 0.05. But in that case experimenters are usually quite disappointed to see what large values of n are required, especially in trials with binomial (success/failure) outcomes. They next set their sights a bit lower, with α = 0.05 and β = 0.10, and find that n is still 'too large'. Finally they settle for α = 0.05 and β = 0.20.
Why do they not adjust α and settle for α = 0.20 and β = 0.05? Why is small α a non-negotiable demand, while small β is only a flexible desideratum? A large α would seem to be scientifically unacceptable, indicating a lack of rigor, while a large β is merely undesirable, an unfortunate but sometimes unavoidable consequence of the fact that observations are expensive or that subjects eligible for the trial are hard to find and recruit. We might have to live with a large β, but good science seems to demand that α be small.
What is happening is that the formal Neyman-Pearson machinery is being used, but it is being given a rejection-trial interpretation (Emphasis added). The quantities α and β are not just the respective probabilities of choosing one hypothesis when the other is true; if they were, then calling the first hypothesis H2 and the second H1 would reverse the roles of α and β, and α = 0.20, β = 0.05 would be just as satisfactory for the problem in its new formulation as α = 0.05 and β = 0.20 were in the old one. The asymmetry arises because the quantity α is being used in the dual roles that it plays in rejection trials — it is both the probability of rejecting a hypothesis when that hypothesis is true and the measure of strength of the evidence needed to justify rejection. Good science demands small α because small α is supposed to mean strong evidence. On the other hand, the Type II error probability β is being interpreted simply as the probability of failing to find strong evidence against H1 when the alternative H2 is true (Emphasis added. Recall Fisher's quote about the impossibility of making Type II errors since we never accept the null.) … When observations are expensive or difficult to obtain we might indeed have to live with a large probability of failure to find strong evidence. In fact, when the expense or difficulty is extreme, we often decide not to do the experiment at all, thereby accpeting values of α = 0 and β = [1].
— End excerpt.
So there we have our confusion, which I alluded to in the previous post. We are imposing rejection-trial reasoning onto the Neyman-Pearson decision framework. We accept a huge β because we interpret our results as a mere failure (to produce strong enough evidence) to reject the null, when really our results imply a decision to accept the 'null'. Remember, with NP we are always forced to choose between two hypotheses — we can never abstain from this choice because the respective rejection regions for H1 and H2 encompass the entire sample space by definition; that is, any result obtained must fall into one of the rejection regions we've defined. We can adjust either α or β (before starting the experiment) as we see fit, based on the relative costs of these errors. Since neither hypothesis is inherently special, adjusting α is as justified as adjusting β and neither has any bearing on the strength of evidence from our experiment.
And surely it doesn't matter which hypothesis is defined as the null, because then we would just switch the respective α and β — that is, H1 and H2 can be reversed without any penalty in the NP framework. Who cares which hypothesis gets the label 1 or 2?
But imagine the outrage (and snarky blog posts) if we tried swapping out the null hypothesis with our pet hypothesis in a rejection trial. Would anybody buy it if we tried to accept our pet hypothesis simply based on a failure to reject it? Of course not, because that would be absurd. Failing to find strong evidence against a single hypothesis has no logical implication that we have found evidence for that hypothesis. Fisher was right about this one. And this is yet another reason NP procedures and rejection trials don't mix.
However, when we are using concepts of power and Type II errors, we are working with NP procedures which are completely symmetrical and have no concept of strength of evidence per se. Failure to reject the null hypothesis has the exact same meaning as accepting the null hypothesis — they are simply different ways to say the same thing. If what you want is to measure evidence, fine; I think we should be measuring evidence in any case. But then you don't have a relevant concept of power, as Fisher has reiterated time and time again. If you want to use power to help plan experiments (as seems to be recommended just about everywhere you look) then you must cast aside your intuitions about interpreting observations from that experiment as evidence. You must reject the rejection trial and reject notions of statistical evidence.
Or don't, but then you're swimming in a sea of confusion.
Royall, R. (1997). Statistical evidence: a likelihood paradigm (Vol. 71). CRC press.
Are all significance tests made of the same stuff?
December 11, 2014 September 4, 2015 Alex Etz Psychology, Statistics	hypothesis testing, NHST, null-hypothesis, probability, Psychology, Statistics
No! If you are like most of the sane researchers out there, you don't spend your days and nights worrying about the nuances of different statistical concepts. Especially ones as traditional as these. But there is one concept that I think we should all be aware of: P-values mean very different things to different people. Richard Royall (1997, p. 76-7) provides a smattering of different possible interpretations and fleshes out the arguments for why these mixed interpretations are problematic (much of this post comes from his book):
In the testing process the null hypothesis either is rejected or is not rejected. If the null hypothesis is not rejected, we will say that the data on which the test is based do not provide sufficient evidence to cause rejection. (Daniel, 1991, p. 192)
A nonsignificant result does not prove that the null hypothesis is correct — merely that it is tenable — our data do not give adequate grounds for rejecting it. (Snedecor and Cochran, 1980, p. 66)
The verdict does not depend on how much more readily some other hypothesis would explain the data. We do not even start to take that question seriously until we have rejected the null hypothesis. …..The statistical significance level is a statement about evidence… If it is small enough, say p = 0.001, we infer that the result is not readily explained as a chance outcome if the null hypothesis is true and we start to look for an alternative explanation with considerable assurance. (Murphy, 1985, p. 120)
If [the p-value] is small, we have two explanations — a rare event has happened, or the assumed distribution is wrong. This is the essence of the significance test argument. Not to reject the null hypothesis … means only that it is accepted for the moment on a provisional basis. (Watson, 1983)
Test of hypothesis. A procedure whereby the truth or falseness of the tested hypothesis is investigated by examining a value of the test statistic computed from a sample and then deciding to reject or accept the tested hypothesis according to whether the value falls into the critical region or acceptance region, respectively. (Remington and Schork, 1970, p. 200)
Although a 'significant' departure provides some degree of evidence against a null hypothesis, it is important to realize that a 'nonsignificant' departure does not provide positive evidence in favour of that hypothesis. The situation is rather that we have failed to find strong evidence against the null hypothesis. (Armitage and Berry, 1987, p. 96)
If that value [of the test statistic] is in the region of rejection, the decision is to reject H0; if that value is outside the region of rejection, the decision is that H0 cannot be rejected at the chosen level of significance … The reasoning behind this decision process is very simple. If the probability associated with the occurance under the null hypothesis of a particular value in the sampling distribution is very small, we may explain the actual occurrence of that value in two ways; first we may explain it by deciding that the null hypothesis is false or, second, we may explain it by deciding that a rare and unlikely event has occurred. (Siegel and Castellan, 1988, Chapter 2)
These all mix and match three distinct viewpoints with regard to hypothesis tests: 1) Neyman-Pearson decision procedures, 2) Fisher's p-value significance tests, and 3) Fisher's rejection trials (I think 2 and 3 are sufficiently different to be considered separately). Mixing and matching them is inappropriate, as will be shown below. Unfortunately, they all use the same terms so this can get confusing! I'll do my best to keep things simple.
1. Neyman-Pearson (NP) decision procedure:
Neyman describes it thusly:
The problem of testing a statistical hypothesis occurs when circumstances force us to make a choice between two courses of action: either take step A or take step B… (Neyman 1950, p. 258)
…any rule R prescribing that we take action A when the sample point … falls within a specified category of points, and that we take action B in all other cases, is a test of a statistical hypothesis. (Neyman 1950, p. 258)
The terms 'accepting' and 'rejecting' a statistical hypothesis are very convenient and well established. It is important, however, to keep their exact meaning in mind and to discard various additional implications which may be suggested by intuition. Thus, to accept a hypothesis H means only to take action A rather than action B. This does not mean that we necessarily believe that the hypothesis H is true. Also if the application … 'rejects' H, this means only that the rule prescribes action B and does not imply that we believe that H is false. (Neyman 1950, p. 259)
So what do we take from this? NP testing is about making a decision to choose H0 or H1, not about shedding light on the truth of any one hypothesis or another. We calculate a test statistic, see where it lies with regard to our predefined rejection regions, and make the corresponding decision. We can assure that we are not often wrong by defining Type I and Type II error probabilities (α and β) to be used in our decision procedure. According to this framework, a good test is one that minimizes these long-run error probabilities. It is important to note that this procedure cannot tell us anything about the truth of hypotheses and does not provide us with a measure of evidence of any kind, only a decision to be made according to our criteria. This procedure is notably symmetric — that is, we can either choose H0 or H1.
Test results would look like this:
α and β were prespecified -based on relevant costs associated with the different errors- for this situation at yadda yadda yadda. The test statistic (say, t=2.5) falls inside the rejection region for H0 defined as t>2.0 so we reject H0 and accept H1." (Alternatively, you might see "p < α = x so we reject H0. The exact value of p is irrelevant, it is either inside or outside of the rejection region defined by α. Obtaining a p = .04 is effectively equivalent to p = .001 for this procedure, as is obtaining a result very much larger than the critical t above.)
2. Fisher's p-value significance tests
Fisher's first procedure is only ever concerned with one hypothesis- that being the null. This procedure is not concerned with making decisions (and when in science do we actually ever do that anyway?) but with measuring evidence against the hypothesis. We want to evaluate 'the strength of evidence against the hypothesis' (Fisher, 1958, p.80) by evaluating how rare our particular result (or even bigger results) would be if there were really no effect in the study. Our objective here is to calculate a single number that Fisher called the level of significance, or the p-value. Smaller p is more evidence against the hypothesis than larger p. Increasing levels of significance* are often represented** by more asterisks*** in tables or graphs. More asterisks mean lower p-values, and presumably more evidence against the null.
What is the rationale behind this test? There are only two possible interpretations of our low p: either a rare event has occurred, or the underlying hypothesis is false. Fisher doesn't think the former is reasonable, so we should assume the latter (Bakan, 1966).
Note that this procedure is directly trying to measure the truth value of a hypothesis. Lower ps indicate more evidence against the hypothesis. This is based on the Law of Improbability, that is,
Law of Improbability: If hypothesis A implies that the probability that a random variable X takes on the value x is quite small, say p(x), then the observation X = x is evidence against A, and the smaller p(x), the stronger the evidence. (Royall, 1997, p. 65)
In a future post I will attempt to show why this law is not a valid indicator of evidence. For the purpose of this post we just need to understand the logic behind this test and that it is fundamentally different from NP procedures. This test alone does not provide any guidance with regard to taking action or making a decision, it is intended as a measure of evidence against a hypothesis.
The present results obtain a t value of 2.5, which corresponds to an observed p = .01**. This level of significance is very small and indicates quite strong evidence against the hypothesis of no difference.
3. Fisher's rejection trials
This is a strange twist on both of the other procedures above, taking elements from each to form a rejection trial. This test is a decision procedure, much like NP procedures, but with only one explicitly defined hypothesis, a la p-value significance tests. The test is most like what psychologists actually use today, framed as two possible decisions, again like NP, but now they are framed in terms of only one hypothesis. Rejection regions are back too, defined as a region of values that have small probability under H0 (i.e., defined by a small α). It is framed as a problem of logic, specifically,
…a process analogous to testing a proposition in formal logic via the argument known as modus tollens, or 'denying the consequent': if A implies B, then not-B implies not-A. We can test A by determining whether B is true. If B is false, then we conclude that A is false. But, on the other hand, if B is found to be true we cannot conclude that A is true. That is, A can be proven false by such a test but it cannot be proven true — either we disprove A or we fail to disprove it…. When B is found to be true, so that A survives the test, this result, although not proving A, does seem intuitively to be evidence supporting A. (Royall, 1997, p. 72)
An important caveat is that these tests are probabilistic in nature, so the logical implications aren't quite right. Nevertheless, rejection trials are what Fisher referred to when he famously said,
Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis… The notion of an error of the so-called 'second kind,' due to accepting the null hypothesis 'when it is false' … has no meaning with reference to simple tests of significance. (Fisher, 1966)
So there is a major difference from NP — With rejection trials you have a single hypothesis (as opposed to 2) combined with decision rules of "reject the H0 or do not reject H0" (as opposed to reject H0/H1 or accept H0/H1). With rejection trials we are back to making a decision. This test is asymmetric (as opposed to NP which is symmetric) — that is, we can only ever reject H0, never accept it.
While we are making decisions with rejection trials, the decisions have a different meaning than that of NP procedures. In this framework, deciding to reject H0 implies the hypothesis is "inconsistent with the data" or that the data "provide sufficient evidence to cause rejection" of the hypothesis (Royall, 1997, p.74). So rejection trials are intended to be both decision procedures and measures of evidence. Test statistics that fall into smaller α regions are considered stronger evidence, much the same way that a smaller p-value indicates more evidence against the hypothesis. For NP procedures α is simply a property of the test, and choosing a lower one has no evidential meaning per se (although see Mayo, 1996 for a 4th significance procedure — severity testing).
The present results obtain a t = 2.5, p = .01, which is sufficiently strong evidence against H0 to warrant its rejection.
What is the takeaway?
If you aren't aware of the difference between the three types of hypothesis testing procedures, you'll find yourself jumbling them all up (Gigerenzer, 2004). If you aren't careful, you may end up thinking you have a measure of evidence when you actually have a guide to action.
Funny enough, I don't endorse any of them. I contend that p-values never measure evidence (in either p-value procedures or rejection trials) and NP procedures lead to absurdities that I can't in good faith accept while simultaneously endorsing them.
Why write 2000 words clarifying the nuanced differences between three procedures I think are patently worthless? Well, did you see what I said at the top referring to sane researchers?
A future post is coming that will explicate the criticisms of each procedure, many of the points again coming from Royall's book.
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Bakan, D. (1966). The test of significance in psychological research.Psychological bulletin, 66(6), 423.
Daniel, W. W. (1991). Hypothesis testing. Biostatistics: a foundation for analysis in the health sciences, 5, 191.
Fisher, R. A. (1958).Statistical methods for research workers (13th ed.). New York: Hafner.
Fisher, R. A. (1966). The design of experiments (8th edn.) Oliver and Boyd.
Gigerenzer, G. (2004). Mindless statistics. The Journal of Socio-Economics,33(5), 587-606.
Mayo, D. G. (1996). Error and the growth of experimental knowledge. University of Chicago Press.
Murphy, E. A. (1985). A companion to medical statistics. Johns Hopkins University Press.
Neyman, J. (1950). First course in probability and statistic. Published by Henry Holt, 1950.,1.
Remington, R. D., & Schork, M. A. (1970). Statistics with applications to the biological and health sciences.
Siegel, S. C., & Castellan, J. NJ (1988). Nonparametric statistics for the behavioural sciences. New York, McGraw-Hill.
Snedecor, G. W. WG Cochran. 1980. Statistical Methods. Iowa State Univ. Press, Ames.
Watson, G. S. (1983). Hypothesis testing. Encyclopedia of Statistics in Quality and Reliability.
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