It has been almost three years since the American Statistical Association (ASA) issued its statement on statistical significance. Ronald L. Wasserstein & Nicole A. Lazar, “The ASA’s Statement on p-Values: Context, Process, and Purpose,” 70 *The American Statistician* 129 (2016) [ASA Statement]. Before the ASA’s Statement, courts and lawyers from all sides routinely misunderstood, misstated, and misrepresented the meaning of statistical significance.^{1} These errors were pandemic despite the efforts of the Federal Judicial Center and the National Academies of Science to educate judges and lawyers, through their *Reference Manuals on Scientific Evidence* and seminars. The interesting question is whether the ASA’s Statement has improved, or will improve, the unfortunate situation.^{2}

**The ASA Statement on Testosterone**

“Ye blind guides, who strain out a gnat and swallow a camel!”

Matthew 23:24

To capture the state of the art, or the state of correct and flawed interpretations of the ASA Statement, reviewing a recent but now resolved, large so-called mass tort may be illustrative. Pharmaceutical products liability cases almost always turn on evidence from pharmaco-epidemiologic studies that compare the rate of an outcome of interest among patients taking a particular medication with the rate among similar, untreated patients. These studies compare the observed with the expected rates, and invariably assess the differences as either a “risk ratio,” or a “risk difference,” for both the magnitude of the difference and for “significance probability” of observing a rate at least as large as seen in the exposed group, given the assumptions that that the medication did not change the rate and that the data followed a given probability distribution. In these alleged “health effects” cases, claims and counterclaims of misuse of significance probability have been pervasive. After the ASA Statement was released, some lawyers began to modify their arguments to suggest that their adversaries’ arguments offend the ASA’s pronouncements.

One litigation that showcases the use and misuse of the ASA Statement arose from claims that AbbVie, Inc.’s transdermal testosterone medication (TRT) causes heart attacks, strokes, and venous thromboembolism. The FDA had reviewed the plaintiffs’ claims, made in a Public Citizen complaint, and resoundingly rejected the causal interpretation of two dubious observational studies, and an incomplete meta-analysis that used an off-beat composite end point.^{3} The Public Citizen petition probably did succeed in pushing the FDA to convene an Advisory Committee meeting, which again resulted in a rejection of the causal claims. The FDA did, however, modify the class labeling for TRT with respect to indication and a *possible* association with cardiovascular outcomes. And then the litigation came.

Notwithstanding the FDA’s determination that a causal association had not been shown, thousands of plaintiffs sued several companies, with most of the complaints falling on AbbVie, Inc., which had the largest presence in the market. The ASA Statement came up occasionally in pre-trial depositions, but became a major brouhaha, when AbbVie moved to exclude plaintiffs’ causation expert witnesses.^{4}

**The Defense’s Anticipatory Parry of the ASA Statement**

As AbbVie described the situation:

“Plaintiffs’ experts uniformly seek to abrogate the established methods and standards for determining … causal factors in favor of precisely the kind of subjective judgments that

Daubertwas designed to avoid. Tests for statistical significance are characterized as ‘misleading’ and rejected [by plaintiffs’ expert witnesses] in favor of non-statistical ‘estimates’, ‘clinical judgment’, and ‘gestalt’ views of the evidence.”^{5}

AbbVie’s brief in support of excluding plaintiffs’ expert witnesses barely mentioned the ASA Statement, but in a footnote, the defense anticipated the Plaintiffs’ opposition would be based on rejecting the importance of statistical significance testing and the claim that this rejection was somehow supported by the ASA Statement:

“The statistical community is currently debating whether scientists who lack expertise in statistics misunderstand p-values and overvalue significance testing. [citing ASA Statement] The fact that there is a debate among professional statisticians on this narrow issue does not validate Dr. Gerstman’s [plaintiffs’ expert witness’s] rejection of the importance of statistical significance testing, or undermine Defendants’ reliance on accepted methods for determining association and causation.”

^{6}

In its brief in support of excluding causation opinions, the defense took pains to define statistical significance, and managed to do so, painfully, or at least in ways that the ASA conferees would have found objectionable:

“Any association found must be tested for its statistical significance. Statistical significance testing measures the likelihood that the observed association could be due to chance variation among samples. Scientists evaluate whether an observed effect is due to chance using p-values and confidence intervals. The prevailing scientific convention requires that there be 95% probability that the observed association is not due to chance (expressed as a

p-value < 0.05) before reporting a result as “statistically significant. * * * This process guards against reporting false positive results by setting a ceiling for the probability that the observed positive association could be due to chance alone, assuming that no association was actually present.”^{7}

AbbVie’s brief proceeded to characterize the confidence interval as a tool of significance testing, again in a way that misstates the mathematical meaning and importance of the interval:

“The determination of statistical significance can be described equivalently in terms of the

calculated in connection with the association. A confidence interval indicates the level of uncertainty that exists around the measured value of the association (i.e., the OR or RR). A confidence interval defines the range of possible values for the actual OR or RR that are compatible with the sample data, at a specifiedconfidence interval, typically 95% under the prevailing scientific convention.confidence levelReference Manual, at 580 (Ex. 14) (“If a 95% confidence interval is specified, the range encompasses the results we would expect 95% of the time if samples for new studies were repeatedly drawn from the same population.”). * * * If the confidence interval crosses 1.0, this means there may be no difference between the treatment group and the control group, therefore the result is not considered statistically significant.”^{8}

Perhaps AbbVie’s counsel should be permitted a plea in mitigation by having cited to, and quoted from, the *Reference Manual on Scientific Evidence*’s chapter on epidemiology, which was also wide of the mark in its description of the confidence interval. Counsel would have been better served by the *Manual*’s more rigorous and accurate chapter on statistics. Even so, the above-quoted statements give an inappropriate interpretation of random error as a probability about the hypothesis being tested.^{9} Particularly dangerous, in terms of failing to advance AbbVie’s own objectives, was the characterization of the confidence interval as measuring * the* level of uncertainty, as though there were no other sources of uncertainty other than random error in the measurement of the risk ratio.

**The Plaintiffs’ Attack on Significance Testing**

The Plaintiffs, of course, filed an opposition brief that characterized the defense position as an attempt to:

“elevate statistical significance, as measured by confidence intervals and so-called p-values, to the status of an absolute requirement to the establishment of causation.”

^{10}

Tellingly, the plaintiffs’ brief fails to point to any modern-era example of a scientific determination of causation based upon epidemiologic evidence, in which the pertinent studies were not assessed for, and found to show, statistical significance.

After citing a few judicial opinions that underplayed the importance of statistical significance, the Plaintiffs’ opposition turned to the ASA Statement for what it perceived to be support for its loosey-goosey approach to causal inference.^{11} The Plaintiffs’ opposition brief quoted a series of propositions from the ASA Statement, without the ASA’s elaborations and elucidations, and without much in the way of explanation or commentary. At the very least, the Plaintiffs’ heavy reliance upon, despite their distortions of, the ASA Statement helped them to define key statistical concepts more carefully than had AbbVie in its opening brief.

The ASA Statement, however, was not immune from being misrepresented in the Plaintiffs’ opposition brief. Many of the quoted propositions were quite beside the points of the dispute over the validity and reliability of Plaintiffs’ expert witnesses’ conclusions of causation about testosterone and heart attacks, conclusions not reached or shared by the FDA, any consensus statement from medical organizations, or any serious published systematic review:

“P-values do not measure the probability that the studied hypothesis is true, … .”

^{12}

This proposition from the ASA Statement is true, but trivially true. (Of course, this ASA principle is relevant to the many judicial decisions that have managed to misstate what p-values measure.) The above-quoted proposition follows from the definition and meaning of the p-value; only someone who did not understand significance probability would confuse it with the probability of the truth of the studied hypothesis. P-values’ not measuring the probability of the null hypothesis, or any alternative hypothesis, is not a flaw in p-values, but arguably their strength.

“A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.”

^{13}

Again, true, true, and immaterial. The existence of other importance metrics, such as the magnitude of an association or correlation, hardly detracts from the importance of assessing the random error in an observed statistic. The need to assess clinical or practical significance of an association or correlation also does not detract from the importance of the assessed random error in a measured statistic.

“By itself, a p-value does not provide a good measure of evidence regarding a model or hypothesis.”

^{14}

The Plaintiffs’ opposition attempted to spin the above ASA statement as a criticism of p-values involves an *elenchi ignoratio*. Once again, the p-value assumes a probability model and a null hypothesis, and so it cannot provide a “measure” or the model or hypothesis’ probability.

The Plaintiffs’ final harrumph on the ASA Statement was their claim that the ASA Statement’s conclusion was “especially significant” to the testosterone litigation:

“Good statistical practice, as an essential component of good scientific practice, emphasizes principles of good study design and conduct, a variety of numerical and graphical summaries of data, understanding of the phenomenon under study, interpretation of results in context, complete reporting and proper logical and quantitative understanding of what data summaries mean. No single index should substitute for scientific reasoning.”

^{15}

The existence of other important criteria in the evaluation and synthesis of a complex body of studies does not erase or supersede the importance of assessing stochastic error in the epidemiologic studies. Plaintiffs’ Opposition Brief asserted that the Defense had attempted to:

“to substitute the single index, the p-value, for scientific reasoning in the reports of Plaintiffs’ experts should be rejected.”

^{16}

Some of the defense’s opening brief could indeed be read as reducing causal inference to the determination of statistical significance. A sympathetic reading of the entire AbbVie brief, however, shows that it had criticized the threats to validity in the observational epidemiologic studies, as well as some of the clinical trials, and other rampant flaws in the Plaintiffs’ expert witnesses’ reasoning. The Plaintiffs’ citations to the ASA Statement’s “negative” propositions about p-values (to emphasize what they are not) appeared to be the stuffing of a strawman, used to divert attention from other failings of their own claims and proffered analyses. In other words, the substance of the Rule 702 application had much more to do with data quality and study validity than statistical significance.

What did the trial court make of this back and forth about statistical significance and the ASA Statement? For the most part, the trial court denied both sides’ challenges to proffered expert witness testimony on causation and statistical issues. In sorting the controversy over the ASA Statement, the trial court apparently misunderstood key statistical concepts and paid little attention to the threats to validity other than random variability in study results.^{17} The trial court summarized the controversy as follows:

“In arguing that the scientific literature does not support a finding that TRT is associated with the alleged injuries, AbbVie emphasize [sic] the importance of considering the statistical significance of study results. Though experts for both AbbVie and plaintiffs agree that statistical significance is a widely accepted concept in the field of statistics and that there is a conventional method for determining the statistical significance of a study’s findings, the parties and their experts disagree about the conclusions one may permissibly draw from a study result that is deemed to possess or lack statistical significance according to conventional methods of making that determination.”

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Of course, there was never a controversy presented to the court about drawing a conclusion from “a study.” By the time the briefs were filed, both sides had multiple observational studies, clinical trials, and meta-analyses to synthesize into opinions for or against causal claims.

Ironically, AbbVie might claim to have prevailed in having the trial court adopt its misleading definitions of p-values and confidence intervals:

“Statisticians test for statistical significance to determine the likelihood that a study’s findings are due to chance. *** According to conventional statistical practice, such a result *** would be considered statistically significant if there is a 95% probability, also expressed as a “p-value” of <0.05, that the observed association is not the product of chance. If, however, the p-value were greater than 0.05, the observed association would not be regarded as statistically significant, according to prevailing conventions, because there is a greater than 5% probability that the association observed was the result of chance.”

^{19}

The MDL court similarly appeared to accept AbbVie’s dubious description of the confidence interval:

“A confidence interval consists of a range of values. For a 95% confidence interval, one would expect future studies sampling the same population to produce values within the range 95% of the time. So if the confidence interval ranged from 1.2 to 3.0, the association would be considered statistically significant, because one would expect, with 95% confidence, that future studies would report a ratio above 1.0 – indeed, above 1.2.”

^{20}

The court’s opinion clearly evidences the danger in stating the importance of statistical significance without placing equal emphasis on the need to exclude bias and confounding. Having found an observational study and one meta-analysis of clinical trial safety outcomes that were statistically significant, the trial court held that any dispute over the probativeness of the studies was for the jury to assess.

Some but not all of AbbVie’s brief might have encouraged this lax attitude by failing to emphasize study validity at the same time as emphasizing the importance of statistical significance. In any event, trial court continued with its précis of the plaintiffs’ argument that:

“a study reporting a confidence interval ranging from 0.9 to 3.5, for example, should certainly not be understood as evidence that there is no association and may actually be understood as evidence in favor of an association, when considered in light of other evidence. Thus, according to plaintiffs’ experts, even studies that do not show a statistically significant association between TRT and the alleged injuries may plausibly bolster their opinions that TRT is capable of causing such injuries.”

^{21}

Of course, a single study that reported a risk ratio greater than 1.0, with a confidence interval 0.9 to 3.5 might be reasonably incorporated into a meta-analysis that in turn could support, or not support a causal inference. In the TRT litigation, however, the well-conducted, most up-to-date meta-analyses did not report statistically significant elevated rates of cardiovascular events among users of TRT. The court’s insistence that a study with a confidence interval 0.9 to 3.5 cannot be interpreted as evidence of no association is, of course, correct. Equally correct would be to say that the interval shows that the study failed to show an association. The trial court never grappled with the reality that the best conducted meta-analyses failed to show statistically significant increases in the rates of cardiovascular events.

The American Statistical Association and its members would likely have been deeply disappointed by how both parties used the ASA Statement for their litigation objectives. AbbVie’s suggestion that the ASA Statement reflects a debate about “whether scientists who lack expertise in statistics misunderstand p-values and overvalue significance testing” would appear to have no support in the Statement itself or any other commentary to come out of the meeting leading up to the Statement. The Plaintiffs’ argument that p-values properly understood are unimportant and misleading similarly finds no support in the ASA Statement. Conveniently, the Plaintiffs’ brief ignored the Statement’s insistence upon transparency in pre-specification of analyses and outcomes, and in handling of multiple comparisons:

“P-values and related analyses should not be reported selectively. Conducting multiple analyses of the data and reporting only those with certain p-values (typically those passing a significance threshold) renders the reported p-values essentially uninterpretable. Cherrypicking promising findings, also known by such terms as data dredging, significance chasing, significance questing, selective inference, and ‘p-hacking’, leads to a spurious excess of statistically significant results in the published literature and should be vigorously avoided.”

^{22}

Most if not all of the plaintiffs’ expert witnesses’ reliance materials would have been eliminated under this principle set forth by the ASA Statement.

1 *See, e.g*., *In re Ephedra Prods. Liab. Litig*., 393 F.Supp. 2d 181, 191 (S.D.N.Y. 2005). *See also* “__Confidence in Intervals and Diffidence in the Court__s” (March 4, 2012); “__Scientific illiteracy among the judiciary__” (Feb. 29, 2012).

2 “The American Statistical Association Statement on Significance Testing Goes to Court – Part I” (Nov. 13, 2018).

3Letter of Janet Woodcock, Director of FDA’s Center for Drug Evaluation and Research, to Sidney Wolfe, Director of Public Citizen’s Health Research Group (July 16, 2014) (denying citizen petition for “black box” warning).

4 Defendants’ (AbbVie, Inc.’s) Motion to Exclude Plaintiffs Expert Testimony on the Issue of Causation, and for Summary Judgment, and Memorandum of Law in Support, Case No. 1:14-CV-01748, MDL 2545, Document #: 1753, 2017 WL 1104501 (N.D. Ill. Feb. 20, 2017) [AbbVie Brief].

5 AbbVie Brief at 3; *see also id*. at 7-8 (“Depending upon the expert, even the basic tests of statistical significance are simply ignored, dismissed as misleading… .”) AbbVie’s definitions of statistical significance occasionally wandered off track and into the transposition fallacy, but generally its point was understandable.

6 AbbVie Brief at 63 n.16 (emphasis in original).

7 AbbVie Brief at 13 (emphasis in original).

8 AbbVie Brief at 13-14 (emphasis in original).

9 The defense brief further emphasized statistical significance almost as though it were a sufficient basis for inferring causality from observational studies: “Regardless of this debate, courts have routinely found the traditional epidemiological method—including bedrock principles of significance testing—to be the most reliable and accepted way to establish general causation. *See, e.g., In re Zoloft*, 26 F. Supp. 3d 449, 455; *see also Rosen v. Ciba-Geigy Corp*., 78 F.3d 316, 319 (7th Cir. 1996) (“The law lags science; it does not lead it.”). AbbVie Brief at 63-64 & n.16. The defense’s language about “including bedrock principles of significance testing” absolves it of having totally ignored other necessary considerations, but still the defense might have advantageously pointed out at the other needed considerations for causal inference at the same time.

10 Plaintiffs’ Steering Committee’ Memorandum of Law in Opposition to Motion of AbbVie Defendants to Exclude Plaintiffs’ Expert Testimony on the Issue of Causation, and for Summary Judgment at p.34, Case No. 1:14-CV-01748, MDL 2545, Document No. 1753 (N.D. Ill. Mar. 23, 2017) [Opp. Brief].

11 *Id*. at 35 (appending the ASA Statement and the commentary of more than two dozen interested commentators).

12 *Id*. at 38 (quoting from the ASA Statement at 131).

13 *Id*. at 38 (quoting from the ASA Statement at 132).

14 *Id*. at 38 (quoting from the ASA Statement at 132).

15 *Id*. at 38 (quoting from the ASA Statement at 132).

16 *Id*. at 38

17 *In re Testosterone Replacement Therapy Prods. Liab. Litig*., MDL No. 2545, C.M.O. No. 46, 2017 WL 1833173 (N.D. Ill. May 8, 2017) [*In re TRT*]

18* In re TRT *at *4.

19 *In re TRT *at *4.

20 *Id*.

21 *Id*. at *4.

22 ASA Statement at 131-32.