Thursday, July 27, 2017

Overvaluing P-values

Seventy-two "big names in statistics want to shake up [the] much maligned P-value." 1/ These academic researchers give the following one-sentence summary of their proposal to change the way scientific articles are written:
We propose to change the default P-value threshold for statistical significance for claims of new discoveries from 0.05 to 0.005. 2/
The President’s Council of Advisors on Science and Technology (PCAST) effectively invoked the current "default P-value" of 0.05 as a rule for admitting scientific evidence in court. 3/ In light of this new (but not novel) call for reducing the conventional p-value, one might think that PCAST was being too generous toward forensic-science identifications. After all, if 5% is ten times the value that should be used to declare differences “statistically significant” in scientific research, then it seems way too large as a limit for what PCAST called “scientific reliability” for courtroom use. But that conclusion would rest on a misunderstanding of the objective and nature of the proposal to change the nomenclature for p-values.

The motivation for moving from 0.05 to 0.005 is “growing concern over the credibility of claims of new discoveries based on ‘statistically significant’ findings.” The authors argue that regarding 0.05 as proof of a real difference (a true positive) is “a leading cause of non-reproducibility” of published discoveries in scientific fields that could easily be corrected by referring to findings with p-values between 0.005 and 0.05 as “suggestive” rather than “significant.” The problem with the verbal tag of “statistically significant” (p < 0.05) is that, in comparison to the state of scientific research ninety years ago when Sir Ronald Fisher floated the 0.05 level, “[a] much larger pool of scientists are now asking a much larger number of questions, possibly with much lower prior odds of success,” resulting in too many apparent discoveries that cannot be replicated in later experiments.

Not only is the group of 72 addressing the perils of the p-value in a different context, but their proposal is not intended as a bright-line rule for deciding what to publish. They explain:
We emphasize that this proposal is about standards of evidence, not standards for policy action nor standards for publication. Results that do not reach the threshold for statistical significance (whatever it is) can still be important and merit publication in leading journals if they address important research questions with rigorous methods. This proposal should not be used to reject publications of novel findings with 0.005 < P < 0.05 properly labeled as suggestive evidence.
So too, “[r]esults that do not reach the threshold for statistical significance (whatever it is) can still be important” in litigation, and the desire to shake things up in the research community does not reveal much about appropriate standards for admissibility in court.

However, PCAST is on firm ground in emphasizing the need to present forensic-science findings without overstating their probative value. The 72 researchers focus on probative value when they discuss a “more direct measure of the strength of evidence.” They suggest that a “two-sided P-value of 0.05 [often] corresponds to Bayes factors ... that range from about 2.5 to 3.4.” Such evidence, they note, is weak. In contrast, they defend the "two-sided P-value of 0.005" in part on the ground that it "corresponds to Bayes factors between approximately 14 and 26." As such, it "represents ‘substantial’ to ‘strong’ evidence according to conventional Bayes factor classifications."

Forensic scientists who advocate describing the strength of evidence rather than only false-positive rates are more demanding. They usually consider Bayes factors between 10 and 100 to constitute "moderate" rather than “strong” evidence. 4/

  1. Dalmeet Singh Chawla, Big Names in Statistics Want To Shake Up Much-maligned P value, Nature News, July 27, 2017,
  2. Daniel J. Benjamin et al., Redefine Statistical Significance (2017), PsyArXiv. July 22. (forthcoming in Nature Human Behavior).
  3. For discussion, see The Source and Soundness of PCAST's 5% Rule, Forensic Sci., Stat. & L., July 23, 2017,
  4. E.g., R. Marquis et al., Discussion on How to Implement a Verbal Scale in a Forensic Laboratory: Benefits, Pitfalls and Suggestions to Avoid Misunderstandings, 56(5) Sci. & Just. 364-70 (2016), doi: 10.1016/j.scijus.2016.05.009, preprint available at (Appendix A).

Sunday, July 23, 2017

The Source and Soundness of PCAST's 5% Rule

The President’s Council of Advisors on Science and Technology (PCAST) Report on comparative pattern matching in forensic science has a deceptively simple rule for the admissibility of evidence of a match between a questioned and a known sample: if examiners would declare that the two samples have the same source as often as one time in 20 when analyzing pairs of samples actually that come from different samples, then the comparisons are “scientifically unreliable.” The report gives no explanation of how it arrived at this rule beyond the following enigmatic paragraph: 1/
False positive rate (abbreviated FPR) is defined as the probability that the method declares a match between two samples that are from different sources (again in an appropriate population), that is, FPR = P(M|H0). For example, a value FPR = 0.01 would indicate that two samples from different sources will be (mistakenly) called as a match 1 percent of the time. Methods with a high FPR are scientifically unreliable for making important judgments in court about the source of a sample. To be considered reliable, the FPR should certainly be less than 5 percent and it may be appropriate that it be considerably lower, depending on the intended application. 2/
Five percent has a crisp, authoritative ring to it, but why is 5% “certainly” the maximum tolerable FPR for courtroom use of the test? And what “intended applications” would demand a lower FPR? Is the underlying thought that greater “scientific reliability” is required as the gravity of the case increases—from a civil case, to a misdemeanor, to a major crime, on up to a capital case?

Statistical Practice as the Basis for the 5% Rule

Inasmuch as the paragraph is found in an appendix entitled "statistical issues," we should expect statistical concepts and practice to help answer such questions. And in fact, 5% is a common number in statistics. In many applications, statistical hypothesis tests try to keep the risk of a false rejection of the “null hypothesis” H0—a false-positive conclusion—below 5%. Researchers and journal editors in many fields prize results that can be said to be “statistically significant,” usually at the 0.05 level or better. The expression p < 0.05 is therefore a common accoutrement of experimental or observational results indicating an association between variables. Likewise, the Food and Drug Administration demands clinical trials to show that a new drug is effective for its intended use (“validity,” if you will), with “the typical ‘cap’ on the type I [false positive] error rate ... set at 5% .”3/ In the forensic pattern-matching context, the null hypothesis H0 in the PCAST paragraph would be that a questioned and a known sample are not associated with the same source.

Thus, to the extent PCAST was thinking of the 5% FPR as the significance level required to reject H0, its emphasis on 5% is well grounded in statistical practice. Using certain standard levels of significance, particularly 5%, can be traced to the 1920s. The eminent British statistician Sir R. A. Fisher wrote:
It is convenient to draw the line at about the level at which we can say: ‘Either there is somethng in the treatment, or a coincidence has occurred such as does not occur more than once in twenty trials.’ ... If one in twenty does not seem high enough odds, we may, if we prefer it, draw the line at one in fifty (the 2 per cent point), or one in a hundred (the 1 per cent point). Personally, the writer prefers to set a low standard of significance at the 5 per cent point, and ignore entirely all results which fail to reach that level. A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance. 4/
For FPRs larger than 5%, the reports of criminalists do not meet (Fisher’s) criterion for establishing a “scientific fact.” Their conclusions of positive association for such error-prone procedures are not, in PCAST’s words, “scientifically reliable.”

Having equated PCAST’s unexplained choice of 5% with a common implementation of statistical hypothesis testing, we also can see why the report suggested that a “considerably lower” number might be required for scientific “reliability.” A 5% FPR lets in examiner conclusions that might be wrong about one time in twenty when defendants are innocent and there is no true association between the questioned item and the known one. False positives tend to increase the rate of false convictions, whereas false negatives tend to would increase the rate of false acquittals. The norm that false convictions are worse than false acquittals counsels caution in relying on an examiner’s conclusion to convict a defendant. And if false convictions in the most serious of cases are worse still, we can see why the PCAST report stated that “the FPR should certainly be less than 5 percent and it may be appropriate that it be considerably lower, depending on the intended application.” Five percent may be good enough for an editor to publish an interesting paper purporting to have discovered something new in social psychology, but this scientific convention does not mean that 5% is good enough for a criminal conviction, let alone one that would lead to an execution.

So we can see that PCAST’s 5% figure did not come from thin air. Indeed, some statisticians and psychologists think that it is too weak a standard—that the general rule in science ought to be p < 0.005. 5/ Nevertheless, the general use of the arguably lenient 5% significance level does not establish that the 5% rule is legally compelled. The law incorporates the intensified concern for false positives into the burden of persuasion for the evidence as a whole. The jury is instructed to acquit unless it has no reasonable doubt that a defendant in a criminal case is guilty; in contrast, in a civil case, the plaintiff can prevail on a mere preponderance of the evidence. But these burdens do not apply to individual items of evidence. The standard for admitting scientific—and other—evidence does not change with how much is at stake in the particular case. After all, the probative value of scientific evidence is no different in a criminal case than in a civil one. Although the PCAST report insists that its statements about science are merely designed to inform courts about scientific standards, if “scientific reliability” depends on the “importance” of the “judgments in court” and varies according to “the intended application,” then PCAST's "scientific reliability" turns out to be based on what is considered socially or legally “appropriate.”

Beyond the FPR

In sum, it is (or would have been) fair for PCAST to point out that it is uncommon for results at higher significance levels than 0.05 to be credited in the scientific literature. But a more deeply analytical report would have noted that there is uneasiness in the statistical community with the hypothesis testing framework and particularly with over-reliance on the p < 0.05 rule. (Today's mail includes an invitation to attend a "Symposium on Statistical Inference: A World Beyond p < 0.05" sponsored by the American Statistical Association.)

Only part of the world beyond p < 0.05 comes from the fact that the FPR is not the only quantity that determines “scientific reliability.” Superficially, the false-positive error probability might look like the appropriate statistic for considering the probative value of a positive finding, but that cannot be right. Scientific evidence, like all circumstantial evidence, has probative value to the extent it changes the probability of a material fact. That there is much more to probative value than the FPR therefore is easily seen through the lens of Bayes’ rule. As the PCAST report notes, in this context, Bayes' theorem prescribes how probability or odds change with the introduction of evidence. The odds after learning of the examiner’s finding are the odds without that information multiplied by the Bayes factor: posterior odds = prior odds × BF.

The Bayes factor thus indicates the strength of the evidence. Stronger evidence has a larger BF and hence a greater impact on the prior odds than weaker evidence. The Bayes factor is a simple ratio. The FPR appears as the denominator, and the sensitivity—or true positive rate—forms the numerator. In symbols, BF = sensitivity / FPR.

The report acknowledges that sensitivity matters (for some purposes at least). Earlier, the report states that “[i]t is necessary to have appropriate empirical measurements of a method’s false positive rate and the method’s sensitivity. [I]t is necessary to know these two measures to assess the probative value of a method.” 6/  Because it takes both operating characteristics to express the probative value of the test, PCAST cannot sensibly dismiss a test as having so little probative value as to be considered “scientifically reliable” on the basis of only one number. Realizing this prompts the next question for devising a rule in the spirit of PCAST's—namely, what is the sensitivity that, together with an FPR of 5%, would define the threshold for “scientific reliability”?

One might imagine that PCAST would consider any false-negative rate in excess of 5% as too high. 7/ If so, it follows that the scientists are saying that, in their view of what is important or what is the dominant convention in various domains, subjective pattern matching must shift the prior odds by a factor of at least .95/.05 = 19 to be considered “scientifically reliable.” On the other hand, if the scientists on PCAST think it is appropriate for a false-negative probability to be ten times the maximum acceptable false-positive probability, then their minimum for “reliability” would become a FNR of 50% and a FPR of 5%, for a Bayes’ factor of only ten.

What Does the Law Require?

Whether the cutoff comes from the FPR alone or the more complete Bayes factor, the very notion of a sharp cutoff is questionable. The purpose of a forensic-science test for identity is to provide evidence that will assist judges or jurors. Forensic scientists who present results and reasonable estimates of the likelihoods or conditional error probabilities associated with their conclusions are staying within the bounds of what is scientifically known.

Consider a hypothetical pattern-matching test for identity for which FPR = 10% and sensitivity = 70% as shown by extensive experiments, each of which demonstrates an ability to distinguish sources from nonsources with accuracy above what would be expected by chance (p < 0.05). According to the PCAST report, this test would be inadmissible for want of “scientific reliability” or “foundational validity” because the FPR of 10% is too high. But if this were a test for a disease, would we really want a diagnosing physician to ignore the positive test result just because the FPR is greater than 5%? The positive finding from the lab would raise the prior odds from, say, 1 to 2, to 7 to 2 (corresponding to an increase in probability from 33% to 78%). Like the physician trying to reach the best possible diagnosis, the judge or jury trying to reach the best possible reconstruction of the events could benefit from knowing that an examiner, who can perform at the empirically established level of accuracy, has found a positive association.

The logic behind a high hurdle for scientific evidence is that “it is likely to be shrouded with an aura of near infallibility, akin to the ancient oracle of Delphi.” 8/ As one federal judge (an advisor to PCAST) wrote in excluding the testimony of a handwriting expert:
[I]t is the Court's role to ensure that a given discipline does not falsely lay claim to the mantle of science, cloaking itself with the aura of unassailability that the imprimatur of ‘science’ confers and thereby distorting the truth-finding process. There have been too many pseudo-scientific disciplines that have since been exposed as profoundly flawed, unreliable, or baseless for any Court to take this role lightly. 9/
Under this rationale, a court should be able to admit the positive test result if the jury is informed of and can appreciate the limitations of the finding. A result that is ten time more probable when the samples have the reported source than when they have different sources is not unreliable “junk science.” Of course, it may not be the product of a particularly scientific (or even a very standardized) procedure, and that must be made clear to the factfinder. When the criminalists employing the highly subjective procedure truly have specialized knowledge—as evidenced by rigorous and repeated tests of their ability to arrive at correct answers—their findings can be presented along with their known error rates without creating “an aura of near infallibility.” If this view of what judges and juries can understand is correct, then a blanket rule against all expert evidence that has known error rates in excess of 5% is unsound.

This criticism of PCAST's 5% rule does not reject the main theme of the report—that when a forensic identification procedure relies on a vaguely defined judgmental process (such as "sufficient similarities and explicable dissimilarities in the light of the examiner's training and experience"), well-founded estimates of the ability of examiners to make the correct judgments are vital to admitting source attributions in court. Of course, Daubert v. Merrell Pharmaceuticals 9/ did not make any single factor, including a "known or potential rate of error," absolutely necessary for admitting all types of scientific evidence. But the Daubert Court painted with an amazingly broad brush. The considerations that will be most important can vary from one type of evidence to another.  When it comes to source attributions from entirely subjective assessments of the similarities and differences in feature sets, there is a cogent argument that the only acceptable way to validate the psychological process is to study how often examiners reach the right conclusions when confronted with same-source and different-source samples.

  1. Thanks to Ken Melson for calling to my attention to this paragraph.
  2. PCAST Report at 161-52.
  3. Russell Katz, FDA: Evidentiary Standards for Drug Development and Approval, 1(3) NeuroRx 307–316, (2004), doi: 10.1602/neurorx.1.3.307.
  4. R.A. Fisher, The Arrangement of Field Experiments, 33 J. Ministry Agric. Gr. Brit. 504 (1926), as quoted in L. Savage, On Rereading R.A. Fisher, 4 Annals of Statistics 471 (1976).
  5. Kelly Servick, It Will Be Much Harder To Call New Findings ‘Significant’ If This Team Gets Its Way, Jul. 25, 2017, 2:30 PM, Science, DOI: 10.1126/science.aan7154.
  6. PCAST Report at 50 (emphasis added).
  7. However, the report made no mention of the fact that the false-negative rate was higher than that in at least one of the two experiments on latent print identification of which it approved.
  8. United States v. Alexander, 526 F.2d 161, 168 (8th Cir. 1975).
  9. Almeciga v. Center for Investigative Reporting, Inc., 185 F. Supp. 3d 401, 415 (S.D.N.Y. 2016) (Rakoff, J.).
  10. 509 U.S. 579 (1993).

Wednesday, July 5, 2017

Multiple Hypothesis Testing in Karlo v. Pittsburgh Glass Works

The following posting is adapted from a draft of an annual update to the legal treatise The New Wigmore on Evidence: Expert Evidence. I am not sure of the implications of the calculations in note 23 and the fact that the age-based groups are overlapping. Advice is welcome.

The Age Discrimination in Employment Act of 1967 (ADEA) 1/ covers individuals who are at least forty years old. The federal circuit courts are split as to whether a disparate-impact claim is viable when it is limited to a subgroup of employees such as those aged fifty and older. In Karlo v. Pittsburgh Glass Works, 2/ the Third Circuit held that statistical proof of disparate impact on such a subgroup can support a claim for recovery. The court countered the employer’s argument that “plaintiffs will be able to ‘gerrymander’ arbitrary age groups in order to manufacture a statistically significant effect” 3/ by promising that “the Federal Rules of Evidence and Daubert jurisprudence [are] a sufficient safeguard against the menace of unscientific methods and manipulative statistics.” 4/ In Daubert v. Merrell Dow Pharmaceuticals, the Supreme Court famously reminded trial judges applying the Federal Rules of Evidence that they are gatekeepers responsible for ensuring that scientific evidence presented at trials is based on sound science. By the end of the Karlo opinion, however, the court appeals held that the Senior District Judge Terrence F. McVerry had been too vigorous a gatekeeper when he found inadmissible a statistical analysis of reductions in force offered by laid-off older workers.

The basic problem was that plaintiffs claimed to have observed statistically significant disparities in various overlapping age groups without correcting for the fact that by performing a series of hypothesis tests, they had more than one opportunity to discover something "significant." By way of analogy, if you flip a coin five times and observe five heads, you might begin to suspect that the coin is not fair. The probability of five heads in a row with a fair coin is p = (1/2)5 = 1/32 = 0.03. We can say that the five heads in the sample are "statistically significant" proof (at the conventional 0.05 level) that the coin is unfair.

But suppose you get to repeat the experiment five times. Now the probability of at least one sample of 5 flips with 5 heads is about five times larger. It is 1 - (1 - 1/32)5 = 0.146785, to be exact. This outcome is not so far out line with what is expected of a fair coin. It would be seen about 15% of the time for a fair coin. This is weak evidence that the coin is unfair; certainly, it is not as compelling as the 3% p-value. So the extra testing, with the opportunity to select any one or more of the five samples as proof of unfairness, has reduced the weight of the statistical evidence of unfairness. The effect of the opportunity to search for significance is sometimes known as "selection bias" or, of late, "p-hacking."

In Karlo, Dr. Michael Campion—a distinguished professor of management at Purdue University with degrees in industrial and organizational psychology—compared proportions of Pittsburgh Glass workers older than 40, 45, 50, 55, and 60 who were laid off to the proportion of younger workers who were laid off. He found that the disparities in three of the five categories were statistically significant at the 0.05 level. 5/ The disparity for the 40-and-older range, he said, fell “just short,” being “ significant at the 13% level.” Dr. Campion maintained that “[t]hese results suggest that there is evidence of disparate impact.” 6/ He also misconstrued the 0.05 level as “a 95% probability that the difference in termination rates of the subgroups is [] due to chance alone.” 7/ The district court expressed doubt as to whether Dr. Campion was a qualified statistical expert 8/ and excluded the testimony under Daubert as inadequate “data snooping.” 9/

Apparently, Judge McVerry was more impressed with the report of Defendant’s expert, James Rosenberger — a statistics professor at Pennsylvania State University and a fellow of the American Statistical Association and the American Association for the Advancement of Science. The report advocated adjusting the significance level to account for the five groupings of over-40 workers. The Chief Judge of the Third Circuit, D. Brooks Smith (also an adjunct professor at Penn State), described the recommended correction as follows:
The Bonferroni procedure adjusts for that risk [of a false positive] by dividing the “critical” significance level by the number of comparisons tested. In this case, PGW's rebuttal expert, Dr. James L. Rosenberger, argues that the critical significance level should be p < 0.01, rather than the typical p < 0.05, because Dr. Campion tested five age groups (0.05 / 5 = 0.01). Once the Bonferroni adjustment is applied, Dr. Campion's results are not statistically significant. Thus, Dr. Rosenberger argues that Dr. Campion cannot reject the null hypothesis and report evidence of disparate impact. 10/
Another way to apply the Bonferroni correction is to change the p-value. That is, when M independent comparisons have been conducted, the Bonferroni correction is either to set “the critical significance level . . . at 0.05/M” (as Professor Rosenberger recommended) or “to inflate all the calculated P values by a factor of M before considering against the conventional critical P value (for example, 0.05).” 11/

The Court of Appeals was not so sure that this conservative adjustment was essential to the admissibility of the p-values or assertions of statistical significance. It held that the district court erred in excluding the subgroup analysis and granting summary judgment. It remanded “for further Daubert proceedings regarding plaintiffs' statistical evidence.” 12/ Further proceedings were said to be necessary partly because the district court had applied “an incorrectly rigorous standard for reliability.” 13/ The lower court had set “a higher bar than what Rule 702 demands” 14/ because “it applied a bright-line exclusionary rule” for all studies with multiple comparisons that have no Bonferroni correction. 15/

But the district court did not clearly articulate such a rule. It wrote that “Dr. Campion does not apply any of the generally accepted statistical procedures (i.e., the Bonferroni procedure) to correct his results for the likelihood of a false indication of significance.” 16/ The sentence is grammatically defective (and hence confusing). On the one hand, it refers to "generally accepted statistical procedures." On the other hand, the parenthetical phrase suggests that only one "procedure" exists. Had the district court written “e.g.” instead of “i.e.,” it would have been clear that it was not promulgating a dubious rule that only the Bonferroni adjustment to p-values or significance levels would satisfy Daubert. To borrow from Mark Twain, "the difference between the almost right word and the right word is really a large matter—'tis the difference between the lightning-bug and the lightning." 17/

Understanding the district court to be demanding a Bonferroni correction in all cases of multiple testing, the court of appeals essentially directed it to reconsider its exclusionary ruling in light of the fact that other procedures could be superior. Indeed, there are many adjustment methods in common use, of which Bonferroni’s is merely the simplest. 18/ However, plaintiff’s expert apparently had no other method to offer, which makes it hard to see why the possibility of some alternative adjustment, suggested by neither expert in the case, made the district court's decision to exclude Dr. Campion's proposed testimony an abuse of discretion.

A rule insisting on a suitable response to the multiple-comparison problem does not seem “incorrectly rigorous.” To the contrary, statisticians usually agree that “the proper use of P values requires that they be ... appropriately adjusted for multiple testing when present.” 19/ It is widely understood that when multiple comparisons are made, reported p-values will exaggerate the significance of the test statistic. 20/ The court of appeal’s statement that “[i]n certain cases, failure to perform a statistical adjustment may simply diminish the weight of an expert's finding.” 21/ is therefore slightly misleading. In virtually all cases, multiple comparisons degrade the meaning of a p-value. Unless the statistical tests are all perfectly correlated, multiple comparisons always make the true probability of the disparity (or a larger one) under the model of pure chance greater than the nominal value. 22/

Even so, whether the fact that an unadjusted p-value exaggerates the weight of evidence invariably makes unadjusted p-values or reports of significance inadmissible under Daubert is a more delicate question. If no reasonable adjustment can be devised for the type of analysis used and no better analysis can be done, then the nominal p-values might be presented along with a cautionary statement about selection bias. In addition, in extreme cases, the adjustment will be small and the degree of exaggeration will not be so formidable as to render the unadjusted p-value inadmissible. For instance, if the nominal p-value were 0.001, the fact that the corrected figure is 0.005 would not be a fatal flaw. The disparity would be highly statistically significant even with the correction. But that was not the situation in Karlo. In this case, statistical significance was not apparent. It was undisputed that as soon as one considered the number of tests performed, not a single subgroup difference was significant at the 0.05 level. 23/

Consequently, the rejection of the district court’s conclusion that the particular statistical analysis in the expert’s report was unsound seems harsh. It should be within the trial court’s discretion to prevent an expert from testifying to the statistical significance of disparities (or their p-values) unless the expert avoids multiple comparisons that would seriously degrade the claims of significance or modifies those claims to reflect the negative impact of the repeated tests on the strength of the statistical evidence. 24/ The logic of Daubert does not allow an expert to dismiss the problem of selection bias on the theory -- advanced by plaintiffs in Karlo -- that “adjusting the required significance level [is only] required [when the analyst performs] ‘a huge number of analyses of all possibilities to try to find something significant.'’’ 25/ The threat to the correct interpretation of a significance probability does not necessarily disappear when the number of comparisons is moderate rather than “huge.” Given the lack of highly significant results here (even nominally), it is not statistically acceptable to ignore the threat. 26/ Although the Third Circuit was correct to observe that not all statistical imperfections render studies invalid within the meaning of Daubert, the reasoning offered in support of the claim of significant disparities in Karlo was not statistically acceptable. 27/

l. 29 U.S.C. §§ 621–634.
2. 849 F.3d 61 (3d Cir. 2017).
3. Id. at 76.
4. Id.
5. He testified that he did not compute a z-score (a way to analyze the difference between two proportions when the sample sizes are large) for the 60-and-over group “because ‘[t]here are only 14 terminations, which means the statistical power to detect a significant effect is very low.’” Karlo, 849 F.2d at 82 n.15.
6. Karlo v. Pittsburgh Glass Works, LLC, 2015 WL 4232600, at *11, No. 2:10–cv–1283 (W.D. Penn. July 13, 2015), vacated, 849 F.3d 61 (3d Cir. 2017).
7. Id. at *11 n.13. "A P value measures a sample's compatibility with a hypothesis, not the truth of the hypothesis." Naomi Altman & Martin Krzywinski, Points of Significance: Interpreting P values, 14 Nature Methods 213, 213 (2017).
8. Id. at *12.
9. Id. at *13.
10. 849 F.3d at 82 (notes omitted).
11. Pak C. Sham & Shaun M. Purcell, Statistical Power and Significance Testing in Large-scale Genetic Studies, 15 Nature Reviews Genetics 335 (2014) (Box 3).
12. Id. at 80 (note omitted).
13. Id. at 82.
14. Id at 83.
15. Id. (internal quotation marks and ellipsis deleted).
16. Karlo, 2015 WL 4232600, at *1.
17. George Bainton, The Art of Authorship 87–88 (1890.
18. Martin Krzywinski & Naomi Altman, Points of Significance: Comparing Samples — Part II, 11 Nature Methods 355, 355 (2014)
19. Naomi Altman & Martin Krzywinski, Points of Significance: Interpreting P values, 14 Nature Methods 213, 214 (2017)
20. Krzywinski & Altman, supra note 18
21. Id. at 83 (emphasis added).
22. Because each age group included some of the same older workers, the tests here were not completely independent. But neither were they completely dependent.
23. However, that three out of five groups exhibited significant associations between age and terminations is surprising under the null hypothesis that those variables are uncorrelated. If each test were independent, then the probability of a significant result in each group would be 0.05. The probability of one or more significant results in five tests would be 0.226; that of two or more would be 0.0226; of three or more, 0.00116.
24. Joseph Gastwirth, Case Comment: An Expert's Report Criticizing Plaintiff's Failure to Account for Multiple Comparisons Is Deemed Admissible in EEOC v. Autozone, 7 Law, Probability & Risk 61, 62 (2008).
25. Karlo, 849 F.3d at 82.
26. Dr. Campion also believed that “his method [was] analogous to ‘cross-validating the relationship between age and termination at different cut-offs,’ or ‘replication with different samples.’” Id. at 83. Although the court of appeals seemed to take these assertions at face value, cross-validation involves applying the same statistical model to different data sets (or distinct subsets of one larger data set). For instance, a equation that predicts law school grades as a function of such variables as undergraduate grades and LSAT test scores might be derived from one data set, then checked to ensure that it performs well in an independent data set. Findings in one large data set of statistically significant associations between particular genetic loci and a disease could be checked to see if the associations were present in an independent data set. No such validation or replication was performed in this case.
27. The Karlo opinion suggested that the state of statistical knowledge or practice might be different in social science than in the broader statistical community. The court pointed to a statement (in a footnote on regression coefficients) in a treatise on statistical evidence in discrimination cases that “the Bonferroni adjustment [is] ‘good statistical practice,’ but ‘not widely or consistently adopted’ in the behavioral and social sciences.” Id. (quoting Ramona L. Paetzold & Steve L. Willborn, The Statistics of Discrimination: Using Statistical Evidence in Discrimination Cases § 6:7, at 308 n.2 (2016 Update)). The treatise writers were referring to an unreported case in which the district court found itself unable to resolve the apparent conflict between the generally recognized problem of multiple comparisons and an EEOC expert’s insistence that labor economists do not make such corrections and courts do not require them. E.E.O.C. v. Autozone, Inc., No. 00-2923, 2006 WL 2524093, at *4 (W.D. Tenn. Aug. 29, 2006). In the face of these divergent perceptions, the district judge decided not to grant summary judgment just because of this problem. Id. (“[T]he Court does not have a sufficient basis to find that ... the non-utilization [of the Bonferroni adjustment] makes [the expert's] results unreliable.”). The notion that multiple comparisons generally can be ignored in labor economics or employment discrimination cases is false, Gastwirth, supra note 23, at 62 (“In fact, combination methods and other procedures that reduce the number of individual tests used to analyse data in equal employment cases are basic statistical procedures that have been used to analyse data in discrimination cases.”), and any tendency to overlook multiple comparisons in “behavioral and social science” more generally is statistically indefensible.
That said, the outcome on appeal in Karlo might be defended as a pragmatic response to the lower court's misunderstanding of the meaning of the ADEA. The court excluded the unadjusted findings of significance for several reasons. In addition to criticizing Professor Campion's refusal to make any adjustment for his series of hypothesis tests across age groups, Judge McVerry noted that "the subgrouping analysis would only be helpful to the factfinder if this Court held that Plaintiffs could maintain an over-fifty disparate impact claim." Karlo, 2015 WL 4232600, at *13 n.16. He sided with "the majority view amongst the circuits that have considered this issue ... that a disparate impact analysis must compare employees aged 40 and over with those 39 and younger ... ." Id. (Petruska v. Reckitt Benckiser, LLC, No. CIV.A. 14–03663 CCC, 2015 WL 1421908, at *6 (D.N.J. Mar.26, 2015)). The Third Circuit decisively rejected this construction of the ADEA, pulling this rug out from under the district court. Having held that the district court erred in interpreting the ADEA, requiring the district court to re-examine the statistical showing under the ADEA, correctly understood, might seem appropriate.
Of course, ordinarily an evidentiary ruling that can be supported on several independent grounds will be upheld on appeal as long as at least one of the independent grounds is valid. Here, the ADEA argument was literally a footnote to the independent ground that the failure to adjust for multiple comparisons invalidated the expert's claim of significant disparities. Nevertheless, the independent-grounds rule normally applies after a trial. It avoids retrials when the trial judge would or could rule the same way on retrial. Because Karlo is a summary judgment case, there is less reason to sustain the evidentiary ruling. But even so, the court of appeals did not have to vacate the judgment. Instead, it could have followed the usual independent-grounds rule to affirm the summary judgment while noting that district court could reconsider its Daubert ruling in light of the court of appeals' explanation of the proper reach of the ADEA and the range of statistically valid responses to the problem of multiple hypothesis tests. As a practical matter, however, there may be little difference between having counsel address the issue in the context of a motion to reconsider and a renewed motion for summary judgment.