Traditional, Frequentist Statistics Still Hegemonic

The Defense Fallacy

In civil actions, defendants, and their legal counsel sometimes argue that the absence of statistical significance across multiple studies requires a verdict of “no cause” for the defense. This argument is fallacious, as can be seen where there are many studies, say eight or nine, which all consistently find elevated risk ratios, but with p-values slightly higher than 5%. The probability that eight studies, free of bias, would consistently find an elevated risk ratio, regardless of the individual studies’ p-values, is itself very small. If the studies were amenable to meta-analysis, the summary estimate of the risk ratio would itself likely be highly statistically significant in this hypothetical.

The Plaintiffs’ Fallacy

The plaintiffs’ fallacy derives from instances, such as the hypothetical one above, in which statistical significance, taken as a property of individual studies, is lacking. Even though we can hypothesize such instances, plaintiffs fallaciously extrapolate from them to the conclusion that statistical significance, or any other measure of sampling estimate precision, is unnecessary to support a conclusion of causation.

In courtroom proceedings, epidemiologist Kenneth Rothman is frequently cited by plaintiffs as having shown or argued that statistical significance is unimportant. For instance, in the Zoloft multi-district birth defects litigation, plaintiffs argued in a motion for reconsideration of the exclusion of their epidemiologic witness that the trial court had failed to give appropriate weight to the Supreme Court’s decision in Matrixx Initiatives, Inc. v. Siracusano, 563 U.S. 27 (2011), as well as to the Third Circuit’s invocation of the so-called “Rothman” approach in a Bendectin birth defects case, DeLuca v. Merrell Dow Pharms., Inc., 911 F.2d 941 (3d Cir. 1990). According to the plaintiffs’ argument, their excluded epidemiologic witness, Dr. Anick Bérard, had used this approach in arriving at her novel conclusion that sertraline causes virtually every kind of birth defect.

The Zoloft plaintiffs did not call Rothman as a witness; nor did they even present an expert witness to explain what Rothman’s arguments were. Instead, the plaintiffs’ counsel, sneaked in some references and vague conclusions into their cross-examinations of defense expert witnesses, and submitted snippets from Rothman’s textbook, Modern Epidemiology.

If plaintiffs had called Dr. Rothman to testify, he would have probably insisted that statistical significance is not a criterion for causation. Such insistence is not as helpful to plaintiffs in cases such as Zoloft birth defects cases as their lawyers might have thought or hoped. Consider for instance the cases in which causal inferences are arrived at without formal statistical analysis. These instances are often not relevant to mass tort litigation that involve prevalent exposure and a prevalent outcome.

Rothman also would have likely insisted that consideration of random variation and bias are essential to the assessment of causation, and that many apparently or nominally statistically significant associations do not and cannot support valid inferences of causation. Furthermore, he might have been given the opportunity to explain that his criticisms of significance testing are as much directed to the creation of false positive as to false negative rates in observational epidemiology. In keeping with his publications, Rothman would have challenged strict significance testing with p-values as opposed to the use of sample statistical estimates in conjunction with confidence intervals. The irony of the Zoloft case and many other litigations was that the defense was not using significance testing in the way that Rothman had criticized; rather the plaintiffs were over-endorsing statistical significance that was nominal, plagued by multi-testing, and inconsistent.

Judge Rufe, who presided over the Zoloft MDL, pointed out that the Third Circuit in DeLuca had never affirmatively endorsed Professor Rothman’s “approach,” but had reversed and remanded the Bendectin case to the district court for a hearing under Rule 702:

by directing such an overall evaluation, however, we do not mean to reject at this point Merrell Dow’s contention that a showing of a .05 level of statistical significance should be a threshold requirement for any statistical analysis concluding that Bendectin is a teratogen regardless of the presence of other indicia of reliability. That contention will need to be addressed on remand. The root issue it poses is what risk of what type of error the judicial system is willing to tolerate. This is not an easy issue to resolve and one possible resolution is a conclusion that the system should not tolerate any expert opinion rooted in statistical analysis where the results of the underlying studies are not significant at a .05 level.”

2015 WL 314149, at *4 (quoting from DeLuca, 911 F.2d at 955). And in DeLuca, after remand, the district court excluded the DeLuca plaintiffs’ expert witnesses, and granted summary judgment, based upon the dubious methods employed by plaintiffs’ expert witnesses (including the infamous Dr. Done, and Shanna Swan), in cherry picking data, recalculating risk ratios in published studies, and ignoring bias and confounding in studies. On subsequent appeal, the Third Circuit affirmed the judgment for Merrell Dow. DeLuca v. Merrell Dow Pharma., Inc., 791 F. Supp. 1042 (3d Cir. 1992), aff’d, 6 F.3d 778 (3d Cir. 1993).

Judge Rufe similarly rebuffed the plaintiffs’ use of the Rothman approach, their reliance upon Matrixx, and their attempt to banish consideration of random error in the interpretation of epidemiologic studies. In re Zoloft (Sertraline Hydrochloride) Prods. Liab. Litig., MDL No. 2342; 12-md-2342, 2015 WL 314149 (E.D. Pa. Jan. 23, 2015) (Rufe, J.) (denying PSC’s motion for reconsideration). SeeZoloft MDL Relieves Matrixx Depression” (Feb. 4, 2015).

Some Statisticians’ Errors

Recently, Dr. Rothman and three other epidemiologists set out to track the change, over time, from 1975 to 2014, of the use of various statistical methodologies. Andreas Stang, Markus Deckert, Charles Poole & Kenneth J. Rothman, “Statistical inference in abstracts of major medical and epidemiology journals 1975–2014: a systematic review,” 32 Eur. J. Epidem. 21 (2017) [cited below as Stang]. They made clear that their preferred methodological approach was to avoid the strictly dichotomous null hypothesis significance testing (NHST), which has evolved from Fisher’s significance testing and Neyman’s null hypothesis testing (NHT), in favor of the use of estimation with confidence intervals (CI). The authors conducted a meta-study, that is a study of studies, to track the trends in use of NHST, ST, NHT and CI reporting in the major bio-medical journals.

Unfortunately, the authors limited their data and analysis to abstracts, which makes their results very likely misleading and incomplete. Even when abstracts reported using so-called CI-only approaches, the authors may well have reasoned that point estimates with CIs that spanned no association were “non-significant.” Similarly, authors who found elevated risk ratios with very wide confidence intervals may well have properly acknowledged that their study did not provide credible evidence of an association. See W. Douglas Thompson, “Statistical criteria in the interpretation of epidemiologic data,” 77 Am. J. Public Health 191, 191 (1987) (discussing the over-interpretation of skimpy data).

Rothman and colleagues found that while a few epidemiologic journals had a rising prevalence of CI-only reports in abstracts, for many biomedical journals the NHST approach remained more common. Interestingly, at three of the major clinical medical journals, the Journal of the American Medical Association, the New England Journal of Medicine, and Lancet, the NHST has prevailed over the almost four decades of observation.

The clear implication of Rothman’s meta-study is that consideration of significance probability, whether or not treated as a dichotomous outcome, and whether or not treated as a p-value or a point estimate with a confidence interval, is absolutely critical to how biomedical research is conducted, analyzed, and reported. In Rothman’s words:

Despite the many cautions, NHST remains one of the most prevalent statistical procedures in the biomedical literature.”

Stang at 22. See also David Chavalarias, Joshua David Wallach, Alvin Ho Ting & John P. A. Ioannidis, “Evolution of Reporting P Values in the Biomedical Literature, 1990-2015,” 315 J. Am. Med. Ass’n 1141 (2016) (noting the absence of the use of Bayes’ factors, among other techniques).

There is one aspect to the Stang article that is almost Trump-like in its citing to an inappropriate, unknowledgable source and then treating its author as having meaningful knowledge of the subject. As part of their rhetorical goals, Stang and colleagues declare that:

there are some indications that it has begun to create a movement away from strict adherence to NHT, if not to ST as well. For instance, in the Matrixx decision in 2011, the U.S. Supreme Court unanimously ruled that admissible evidence of causality does not have to be statistically significant [12].”

Stang at 22. Whence comes this claim? Footnote 12 takes us to what could well be fake news of a legal holding, an article by a statistician about a legal case:

Joseph L. Gastwirth, “Statistical considerations support the Supreme Court’s decision in Matrixx Initiatives v. Siracusano, 52 Jurimetrics J. 155 (2012).

Citing a secondary source when the primary source is readily available, and what is at issue, seems like poor scholarship. Professor Gastwirth is a statistician, not a lawyer, and his exegesis of the Supreme Court’s decision is wildly off target. As any first year law student could discern, the Matrixx case could not have been about the admissibility of evidence because the case had been dismissed on the pleadings, and no evidence had ever been admitted or excluded. The only issue on appeal was the adequacy of the allegations, not the admissibility of evidence.

Although the Court managed to muddle its analysis by wandering off into dicta about causation, the holding of the case is that alleging causation was not required to plead a case of materiality for a securities fraud case. Having dispatched causality from the case, the Court had no serious business in setting the considerations for alleging in pleadings or proving at trial the elements of causation. Indeed, the Court made it clear that its frolic and detour into causation could not be taken seriously:

We need not consider whether the expert testimony was properly admitted in those cases [cited earlier in the opinion], and we do not attempt to define here what constitutes reliable evidence of causation.”

Matrixx Initiatives, Inc. v. Siracusano, 563 U.S. 27, 131 S.Ct. 1309, 1319 (2011).

The word “admissible” or “admissibility” never appear in the Court’s opinion, and the above quote explains that the admissibility was not considered. Laughably, the Court went on to cite three cases as examples of supposed causation opinions in the absence of statistical significance. Two of the three were specific causation, differential etiology cases that involved known general causation. The third case involved a claim of birth defects from contraceptive jelly, when the plaintiffs’ expert witnesses actually relied upon statistically significant (but thoroughly flawed and invalid) associations.1

When it comes to statistical testing the legal world would be much improved if lawyers actually and carefully read statistics authors, and if statisticians and scientists actually read court opinions.

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