*Putting Science On Its Posterior*

Plaintiffs’ and Defendants’ counsel both want the scientific and legal standard to be framed as a very high posterior probability of the truth of a claim. Plaintiffs want the scientific posterior probability to be high because they want to push the legal system in the direction of allowing weak or specious claims that are not supported by sufficient scientific evidence to support a causal conclusion. By asserting that the scientific posterior probability for a causal claim is high, and that the legal and scientific standards are different, they seek to empower courts and juries to support judgments of causality that are deemed inconclusive, speculative, or worse, by scientists themselves.

Defendants want the scientific posterior probability to be high, and claim that the legal standard should be at least as high as the scientific standard.

Both Plaintiffs and Defendants thus find common cause in committing the transposition fallacy by transmuting the coefficient of confidence, typically 95%, into a minimally necessary posterior probability for scientific causal judgments. “One wanders to the left, another to the right ; both are equally in error, but are seduced by different delusions.”[1]

In the *Goodman v. Viljoen [2]* case, both sides, plaintiffs and defendants, embraced the claim that science requires a high posterior probability, and that the p-value provided evidence of the posterior probability of the causal claim at issue. The error came mostly from the parties’ clinical expert witnesses and from the lawyers themselves; the parties’ statistical expert witnesses appeared to try to avoid the transposition fallacy. Clearly, no text would support the conflation of confidence with certainty. No scientific text, treatise, or authority was cited for the notion that scientific “proof” required 95% certainty. This notion was simply an opinion of testifying witnesses.

The principal evidence that antenatal corticosteroid (ACS) therapy can prevent cerebral palsy (CP) came from a Cochrane review and meta-analysis[3] of clinical trials. The review examined a wide range of outcomes, only one of which was CP. The trials were apparently not designed to assess CP risk, and they varied significantly in case definition, diagnostic criteria, and length of follow up for case ascertainment. Of the five included studies, four ascertained CP at follow up from two to six years, and the length of follow up was unknown in the fifth study.

Data were sparse in the Cochrane review, as expected for a relatively rare outcome. The five studies encompassed 904 children, with 490 in the treatment group, and 414 in the control group. There was a total of 48 CP cases, with 20 in the treatment, and 28 in the control, groups. Blinding was apparently not maintained over the extended reporting period.

Professor Andrew Willan, plaintiffs’ testifying expert witness on statistics, sponsored a Bayesian statistical analysis, with which he concluded that there was between a 91 and 97% probability that there was an increased risk of CP from not providing ACS in pre-term labor (or, a decreased risk of CP from administering ACS).[4] Willan’s posterior probabilities was for any increased risk, based upon the Cochrane data. Willan’s calculations were not provided in his testimony, and no information about his prior probability, was given. The data came from clinical trials, but the nature of the observations and the analyses made these trials little more than observational studies conducted within the context of clinical trials designed to look at other outcomes. The Bayesian analysis did not account for the uncertainty in the case definitions, variations in internal validity and follow up, and biases in the clinical trials. Willan’s posterior probabilities thus described a maximal probability for general causation, which surely needed to be discounted for validity and bias issues.

There was a further issue of external validity. The Goodman twins developed CP from having sustained periventricular leukomalacia (PVL), which is one among several mechanistic pathways by which CP can develop in pre-term infants. The Cochrane data did not address PVL, and the included trials were silent as to whether any of the CP cases involved PVL mechanisms. There was no basis for assuming that ACS reduced risk of CP from all mechanisms equally, or even at all.[5] The Willan posterior probabilities did not address the external validity issues as they pertained to the *Goodman* case itself.

Although Dr. Viljoen abandoned the challenge to the Bayesian analysis at trial, his statistical expert witness, Dr. Robert Platt went further to opine that he agreed with Willan’s calculations. To agree with his calculations, and the posterior probabilities that came out of those calculations, Platt had to have agreed with the analyses themselves. This agreement seems ill considered given that elsewhere in his testimony, Platt appears to advance important criticisms of the Cochrane data in the form of validity and bias issues.

Certainly, Platt’s concession about the correctness of Willan’s calculations greatly undermined Dr. Viljoen’s position with the trial and appellate court. Dr. Viljoen maintained those criticisms throughout the trial, and on appeal. *See, e.g*., Defendant (Appellant) Factum, 2012 CCLTFactum 20936, at ¶14(a)

(“(a) antenatal corticosteroids have never been shown to reduce the incidence or effect of PVL”); *id*. at ¶14(d)(“at best, even taking the Bayesian approach at face value, the use of antenatal corticosteroids showed only a 40% reduction in the incidence of cerebral palsy, but not PVL”).

How might have things gone better for Dr. Vijoen? For one thing, Platt’s concession about the correctness of Willan’s calculations had to be explained and qualified as conceding only the posterior probability on the doubtful and unproven assumptions made by Willan. Willan’s posterior, as big as it was, represented only an idealized maximal posterior probability, which in reality had to be deeply discounted by important uncertainties, biases, and validity concerns. The inconclusiveness of the data were “provable” on either a frequentist or a Bayesian analysis.

[1] Horace, in Wood, *Dictionary of Quotations* 182 (1893).

[2] *Goodman v. Viljoen*, 2011 ONSC 821 (CanLII), *aff’d*, 2012 ONCA 896 (CanLII), *leave appeal den’d*, Supreme Court of Canada No. 35230 (July 11, 2013).

[3] Devender Roberts & Stuart R Dalziel “Antenatal corticosteroids for accelerating fetal lung maturation for women at risk of preterm birth,” *Cochrane Database of Systematic *Reviews, at 8, Issue 3. Art. No. CD004454 (2006)

[4] *Notes of Testimony of Andrew Willan at 34* (April 9, 2010) (concluding that ACS reduces risk of CP, with a probability of 91 to 97 percent, depending upon whether random effects or fixed effect models are used).

[5] *See, e.g*., Olivier Baud, Laurence Laurence Foix l’Hélias, *et al.,* “Antenatal Glucocorticoid- Treatment and Cystic Periventricular Leukomalacia in Very Premature Infants,” 341 *New Engl. J. Med.* 1190, 1194 (1999) (“Our results suggest that exposure to betamethasone but not dexamethasone is associated with a decreased risk of cystic periventricular leukomalacia.”).