One of the reasons that, in 2016, the American Statistical Association (ASA) issued, for the first time in its history, a consensus statement on p-values, was the persistent and sometimes deliberate misstatements and misrepresentations about the meaning of the p-value. Indeed, of the six principles articulated by the ASA, several were little more than definitional, designed to clear away misunderstandings. Notably, “Principle Two” addresses one persistent misunderstanding and states:
“P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.
Researchers often wish to turn a p-value into a statement about the truth of a null hypothesis, or about the probability that random chance produced the observed data. The p-value is neither. It is a statement about data in relation to a specified hypothetical explanation, and is not a statement about the explanation itself.”[1]
The ASA consensus statement followed on the heels of an important published article, written by seven important authors in the fields of statistics and epidemiology.[2] One statistician,[3] who frequently shows up as an expert witness for multi-district litigation plaintiffs, described the article’s authors as the “A-Team” of statistics. In any event, the seven prominent thought leaders identified common statistical misunderstandings, including the belief that:
“2. The P value for the null hypothesis is the probability that chance alone produced the observed association; for example, if the P value for the null hypothesis is 0.08, there is an 8% probability that chance alone produced the association. No!”[4]
This is all basic statistics.
Frederick Schauer is the David and Mary Harrison Distinguished Professor of Law at the University of Virginia. Schauer has had contributed prolifically to legal scholarship, and his publications are often well written and thoughtful analyses. Schauer’s recent book, The Proof: Uses of Evidence in Law, Politics, and Everything Else, published by the Harvard University Press is a contribution to the literature of “legal epistemology,” and the foundations of evidence that lie beneath many of our everyday and courtroom approaches to resolving disputes.[5] Schauer’s book might be a useful addition to an undergraduate’s reading list for a course in practical epistemology, or for a law school course on evidence. The language of The Proof is clear and lively, but at times wanders into objectionable and biased political correctness. For example, Schauer channels Naomi Oreskes and her critique of manufacturing industry in his own discussion of “manufactured evidence,”[6] but studiously avoids any number of examples of explicit manufacturing of fraudulent evidence in litigation by the lawsuit industry.[7] Perhaps the most serious omission in this book on evidence is its failure to discuss the relative quality and hierarchy of evidence in science, medicine, and in policy. Readers will not find any mention of the methodology of systematic reviews or meta-analyses in Schauer’s work.
At the end of his chapter on burdens of proof, Schauer adds “A Long Footnote on Statistical Significance,” in which he expresses surprise that the subject of statistical significance is controversial. Schauer might well have brushed up on the statistical concepts he wanted to discuss.
Schauer’s treatment of statistical significance is both distinctly unbalanced, as well as misstated. In an endnote,[8] Schauer cites some of the controversialists who have criticized significance tests, but none of the statisticians who have defended their use.[9]
As for conceptual accuracy, after giving a serviceable definition of the p-value, Schauer immediately goes astray:
“And this likelihood is conventionally described in terms of a p-value, where the p-value is the probability that positive results—rejection of the “null hypothesis” that there is no connection between the examined variables—were produced by chance.”[10]
And again, for emphasis, Schauer tells us:
“A p-value of greater than .05 – a greater than 5 percent probability that the same results would have been the result of chance – has been understood to mean that the results are not statistically significant.”[11]
And then once more for emphasis, in the context of an emotionally laden hypothetical about an experimental drug “cures” a dread, incurable disease, p = 0.20, Schauer tells us that he suspects most people would want to take the medication:
“recognizing that an 80 percent likelihood that the rejection of ineffectiveness was still good enough, at least if there were no other alternatives.”
Schauer wants to connect his discussion of statistical significance to degrees or varying strengths of evidence, but his discursion into statistical significance largely conflates precision with strength. Evidence can be statistically robust but not be very strong. If we imagine a very large randomized clinical trial that found that a medication lowered systolic blood pressure by 1mm of mercury, p < 0.05, we would not consider that finding to constitute strong evidence for therapeutic benefit. If the observation of lowering blood pressure by 1mm came from an observational study, p < 0.05, the finding might not even qualify as evidence in the views of sophisticated cardiovascular physicians and researchers.
Earlier in the chapter, Schauer points to instances in which substantial evidence for a conclusion is downplayed because it is not “conclusive,” or “definitive.” He is obviously keen to emphasize that evidence that is not “conclusive” may still be useful in some circumstances. In this context, Schauer yet again misstates the meaning of significance probability, when he tells us that:
“[j]ust as inconclusive or even weak evidence may still be evidence, and may still be useful evidence for some purposes, so too might conclusions – rejections of the null hypothesis – that are more than 5 percent likely to have been produced by chance still be valuable, depending on what follows from those conclusions.”[12]
And while Schauer is right that weak evidence may still be evidence, he seems loathe to admit that weak evidence may be pretty crummy support for a conclusion. Take, for instance, a fair coin. We have an expected value on ten flips of five heads and five tails. We flip the coin ten times, but we observe six heads and four tails. Do we now have “evidence” that the expected value and the expected outcome are wrong? Not really. The probability of observing the expected outcome on the binomial model that most people would endorse for the thought experiment is 24.6%. The probability of not observing the expected value in ten flips is three times greater. If we look at an epidemiologic study, with a sizable number of participants, the “expected value” of 1.0, embodied in the null hypothesis, is an outcome that we would rarely expect to see, even if the null hypothesis is correct. Schauer seems to have missed this basic lesson of probability and statistics.
Perhaps even more disturbing is that Schauer fails to distinguish the other determinants of study validity and the warrants for inferring a conclusion at any level of certainty. There is a distinct danger that his comments about p-values will be taken to apply to various study designs, descriptive, observational, and experimental. And there is a further danger that incorrect definitions of the p-value and statistical significance probabilities will be used to misrepresent p-values as relating to posterior probabilities. Surely, a distinguished professor of law, at a top law school, in a book published by a prestigious publisher (Belknap Press) can do better. The message for legal practitioners is clear. If you need to define or discuss statistical concepts in a brief, seek out a good textbook on statistics. Do not rely upon other lawyers, even distinguished law professors, or judges, for accurate working definitions.
[1] Ronald L. Wasserstein & Nicole A. Lazar, “The ASA’s Statement on p-Values: Context, Process, and Purpose,” 70 The Am. Statistician 129, 131 (2016).
[2] Sander Greenland, Stephen J. Senn, Kenneth J. Rothman, John B. Carlin, Charles Poole, Steven N. Goodman, and Douglas G. Altman, “Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations,” 31 European J. Epidemiol. 337 (2016).[cited as “Seven Sachems”]
[3] Martin T. Wells.
[4] Seven Sachems at 340 (emphasis added).
[5] Frederick Schauer, The Proof: Uses of Evidence in Law, Politics, and Everything Else (2022). [Schauer] One nit: Schauer cites a paper by A. Philip Dawid, “Statistical Risk,” 194 Synthese 3445 (2017). The title of the paper is “On individual risk.”
[6] Naomi Oreskes & Erik M. Conway, Merchants of Doubt: How a Handful of Scientists Obscured the Truth on Issues from Tobacco Smoke to Climate Change (2010).
[7] See, e.g., In re Silica Prods. Liab. Litig., 398 F.Supp. 2d 563 (S.D.Tex. 2005); Transcript of Daubert Hearing at 23 (Feb. 17, 2005) (“great red flags of fraud”).
[8] See Schauer endnote 44 to Chapter 3, “The Burden of Proof,” citing Valentin Amrhein, Sander Greenland, and Blake McShane, “Scientists Rise Up against Statistical Significance,” www .nature .com (March 20, 2019), which in turn commented upon Blakey B. McShane, David Gal, Andrew Gelman, Christian Robert, and Jennifer L. Tackett, “Abandon Statistical Significance,” 73 American Statistician 235 (2019).
[9] Yoav Benjamini, Richard D. DeVeaux, Bradly Efron, Scott Evans, Mark Glickman, Barry Braubard, Xuming He, Xiao Li Meng, Nancy Reid, Stephen M. Stigler, Stephen B. Vardeman, Christopher K. Wikle, Tommy Wright, Linda J. Young, and Karen Kafadar, “The ASA President’s Task Force Statement on Statistical Significance and Replicability,” 15 Annals of Applied Statistics 1084 (2021); see also “A Proclamation from the Task Force on Statistical Significance” (June 21, 2021).
[10] Schauer at 55. To be sure, Schauer, in endnote 43 to Chapter 3, disclaims any identification of p-values or measures of statistical significance with posterior probabilities or probabilistic measures of the burden of proof. Nonetheless, in the text, he proceeds to do exactly what he disclaimed in the endnote.
[11] Schauer at 55.
[12] Schauer at 56.
