Muriel Bristol was a biologist who studied algae at the Rothamsted Experimental Station in England, after World War I. In addition to her knowledge of plant biology, Bristol claimed the ability to tell whether tea had been added to milk, or the tea poured first and then milk had been added. Bristol, as a scientist and a proper English woman, preferred the latter.
Ronald Fisher, who also worked at Rothamsted, expressed his skepticism over Dr. Bristol’s claim. Fisher set about to design a randomized experiment that would efficiently and effectively test her claim. Bristol was presented with eight cups of tea, four of which were prepared with milk added to tea, and four prepared with tea added to milk. Bristol, of course, was blinded to which was which, but was required to label each according to its manner of preparation. Fisher saw his randomized experiment as a 2 x 2 contingency table, from he could calculate the observed outcome (and ones more extreme if there were any more extreme outcomes) using the assumption of fixed marginal rates and the hypergeometric probability distribution. Fisher’s Exact Test was born at tea time.[1]
Fisher described the origins of his Exact Test in one of his early texts, but he neglected to report whether his experiment vindicated Bristol’s claim. According to David Salsburg, H. Fairfield Smith, one of Fisher’s colleagues, acknowledged that Bristol nailed Fisher’s Exact test, with all eight cups correctly identified. The test has gone on to become an important tool in the statistician’s armamentarium.
Fisher’s Exact, like any statistical test, has model assumptions and preconditions. For one thing, the test is designed for categorical data, with binary outcomes. The test allows us to evaluate whether two proportions are likely different by chance alone, by calculating the probability of the observed outcome, as well as more extreme outcomes.
The calculation of an exact attained significance probability, using Fisher’s approach, provides a one-sided p-value, with no unique solution to calculating a two-side attained significance probability. In discrimination cases, the one-sided p-value may well be more appropriate for the issue at hand. The Fisher’s Exact Test has thus played an important role in showing the judiciary that small sample size need not be an insuperable barrier to meaningful statistical analysis. In discrimination cases, the one-sided p-value provided by the test is not a particular problem.[2]
The difficulty of using Fisher’s Exact for small sample sizes is that the hypergeometric distribution, upon which the test is based, is highly asymmetric. The observed one-sided p-value does not measure the probability of a result equally extreme in the opposite direction. There are at least three ways to calculate the p-value:
- Double the one-sided p-value.
- Add the point probabilities from the opposite tail that are more extreme than the observed point probability.
- Use the mid-P value; that is, add all values more extreme (smaller) than the observed point probability from both sides of the distribution, PLUS ½ of the observed point probability.
Some software programs will proceed in one of these ways by default, but their doing so does guarantee the most accurate measure of two-tailed significance probability.
In the Lipitor MDL for diabetes litigation, Judge Gergel generally used sharp analyses to cut through the rancid fat of litigation claims, to get to the heart of the matter. By and large, he appears to have done a splendid job. In course of gatekeeping under Federal Rule of Evidence 702, however, Judge Gergel may have misunderstood the nature of Fisher’s Exact Test.
Nicholas Jewell is a well-credentialed statistician at the University of California. In the courtroom, Jewell is a well-known expert witness for the litigation industry. He is no novice at generating unreliable opinion testimony. See In re Zoloft Prods. Liab. Litig., No. 12–md–2342, 2015 WL 7776911 (E.D. Pa. Dec. 2, 2015) (excluding Jewell’s opinions as scientifically unwarranted and methodologically flawed). In re Zoloft Prod. Liab. Litig., MDL NO. 2342, 12-MD-2342, 2016 WL 1320799 (E.D. Pa. April 5, 2016) (granting summary judgment after excluding Dr. Jewell). See “The Education of Judge Rufe – The Zoloft MDL” (April 9, 2016).
In the Lipitor cases, some of Jewell’s opinions seemed outlandish indeed, and Judge Gergel generally excluded them. See In re Lipitor Marketing, Sales Practices and Prods. Liab. Litig., 145 F.Supp. 3d 573 (D.S.C. 2015), reconsideration den’d, 2016 WL 827067 (D.S.C. Feb. 29, 2016). As Judge Gergel explained, Jewell calculated a relative risk for abnormal blood glucose in a Lipitor group to be 3.0 (95% C.I., 0.9 to 9.6), using STATA software. Also using STATA, Jewell obtained an attained significance probability of 0.0654, based upon Fisher’s Exact Test. Lipitor Jewell at *7.
Judge Gergel did not report whether Jewell’s reported p-value of 0.0654, was one- or two-sided, but he did state that the attained probability “indicates a lack of statistical significance.” Id. & n. 15. The rest of His Honor’s discussion of the challenged opinion, however, makes clear that of 0.0654 must have been a two-sided value. If it had been a one-sided p-value, then there would have been no way of invoking the mid-p to generate a two-sided p-value below 5%. The mid-p will always be larger than the one-tailed exact p-value generated by Fisher’s Exact Test.
The court noted that Dr. Jewell had testified that he believed that STATA generated this confidence interval by “flip[ping]” the Taylor series approximation. The STATA website notes that it calculates confidence intervals for odds ratios (which are different from the relative risk that Jewell testified he computed), by inverting the Fisher exact test.[3] Id. at *7 & n. 17. Of course, this description suggests that the confidence interval is not based upon exact methods.
STATA does not provide a mid p-value calculation, and so Jewell used an on-line calculator, to obtain a mid p-value of 0.04, which he declared statistically significant. The court took Jewell to task for using the mid p-value as though it were a different analysis or test. Id. at *8. Because the mid-p value will always be larger than the one-sided exact p-value from Fisher’s Exact Test, the court’s explanation does not really make sense:
“Instead, Dr. Jewell turned to the mid-p test, which would ‘[a]lmost surely’ produce a lower p-value than the Fisher exact test.”
Id. at *8. The mid-p test, however, is not different from the Fisher’s exact; rather it is simply a way of dealing with the asymmetrical distribution that underlies the Fisher’s exact, to arrive at a two-tailed p-value that more accurately captures the rate of Type I error.
The MDL court acknowledged that the mid-p approach, was not inherently unreliable, but questioned Jewell’s inconsistent, selective use of the approach for only one test.[4] Jewell certainly did not help the plaintiffs’ cause and his standing by having discarding the analyses that were not incorporated into his report, thus leaving the MDL court to guess at how much selection went on in his process of generating his opinions.. Id. at *9 & n. 19.
None of Jewell’s other calculated p-values involved the mid-p approach, but the court’s criticism begs the question whether the other p-values came from a Fisher’s Exact Test with small sample size, or other highly asymmetrical distribution. Id. at *8. Although Jewell had shown himself willing to engage in other dubious, result-oriented analyses, Jewell’s use of the mid-p for this one comparison may have been within acceptable bounds after all.
The court also noted that Jewell had obtained the “exact p-value and that this p-value was not significant.” Id. The court’s notation here, however, does not report the important detail whether that exact, unreported p-value was merely the doubled of the one-sided p-value given by the Fisher’s Exact Test. As the STATA website, cited by the MDL court, explains:
“The test naturally gives a one-sided p-value, and there are at least four different ways to convert it to a two-sided p-value (Agresti 2002, 93). One way, not implemented in Stata, is to double the one-sided p-value; doubling is simple but can result in p-values larger than one.”
Wesley Eddings, “Fisher’s exact test two-sided idiosyncrasy” (Jan. 2009) (citing Alan Agresti, Categorical Data Analysis 93 (2d ed. 2002)).
On plaintiffs’ motion for reconsideration, the MDL court reaffirmed its findings with respect to Jewell’s use of the mid-p. Lipitor Jewell Reconsidered at *3. In doing so, the court insisted that the one instance in which Jewell used the mid-p stood in stark contrast to all the other instances in which he had used Fisher’s Exact Test. The court then cited to the record to identify 21 other instances in which Jewell used a p-value rather than a mid-p value. The court, however, did not provide the crucial detail whether these 21 other instances actually involved small-sample applications of Fisher’s Exact Test. As result-oriented as Jewell can be, it seems safe to assume that not all his statistical analyses involved Fisher’s Exact Test, with its attendant ambiguity for how to calculate a two-tailed p-value.
[1] Sir Ronald A. Fisher, The Design of Experiments at chapter 2 (1935); see also Stephen Senn, “Tea for three: Of infusions and inferences and milk in first,” Significance 30 (Dec. 2012); David Salsburg, The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century (2002).
[2] See, e.g., Dendy v. Washington Hosp. Ctr., 431 F. Supp. 873 (D.D.C. 1977) (denying preliminary injunction), rev’d, 581 F.2d 99 (D.C. Cir. 1978) (reversing denial of relief, and remanding for reconsideration). See also National Academies of Science, Reference Manual on Scientific Evidence 255 n.108 (3d ed. 2011) (“Well-known small sample techniques [for testing significance and calculating p-values] include the sign test and Fisher’s exact test.”).
[3] See Wesley Eddings, “Fisher’s exact test two-sided idiosyncrasy” (Jan. 2009), available at <http://www.stata.com/support/faqs/statistics/fishers-exact-test/>, last visited April 19, 2016 (“Stata’s exact confidence interval for the odds ratio inverts Fisher’s exact test.”). This article by Eddings contains a nice discussion of why the Fisher’s Exact Test attained significance probability disagrees with the calculated confidence interval. Eddings points out the asymmetry of the hypergeometric distribution, which complicates arriving at an exact p-value for a two-sided test.
[4] See Barber v. United Airlines, Inc., 17 Fed. Appx. 433, 437 (7th Cir. 2001) (“Because in formulating his opinion Dr. Hynes cherry-picked the facts he considered to render an expert opinion, the district court correctly barred his testimony because such a selective use of facts fails to satisfy the scientific method and Daubert.”).
