Broadbent on the Relative Risk > 2 Argument

Alex Broadbent, of the University of Johannesburg, Department of Philosophy, has published a paper that contributes to the debate over whether a relative risk (RR) greater than (>) two is irrelevant, helpful, necessary, or sufficient in inferring that an exposure more likely than not caused an individual claimant’s disease. Alex Broadbent, “Epidemiological Evidence in Proof of Specific Causation,” 17 Legal Theory 237 (2011) [cited as Broadbent].  I am indebted to his having called his paper to my attention. Professor Broadbent’s essay is clearly written, which is helpful in assessing the current use of the RR > 2 argument in judicial decisions.

General vs. Specific Causation

Broadbent carefully distinguishes between general and specific causation.  By focusing exclusively upon specific causation (and assuming that general causation is accepted), he avoids the frequent confusion over when RR > 2 might play a role in legal decisions. Broadbent also “sanitizes” his portrayal of RR by asking us to assume that “the RR is not due to anything other than the exposure.” Id. at 241. This is a BIG assumption and a tall order for observational epidemiologic evidence.  The study or studies that establishes the RR we are reasoning from must be free of bias and confounding. Id.  Broadbent does not mention, however, the statistical stability of the RR, which virtually always will be based upon a sample, and thus subject to the play of random error.  He sidesteps the need for statistical significance in comparing two proportions, but the most charitable interpretation of his paper requires us to assume further that the hypothetical RR from which we are reasoning is sufficiently statistically stable that random error, along with bias and confounding, can be also ruled out as likely explanations for the RR > 1.

Broadbent sets out to show that RR > 2 may, in certain circumstances, suffices to show specific causation, but he argues that RR > 2 is never logically necessary, and must never be required to support a claim of specific causation.  Broadbent at 237.  On the same page in which he states that epidemiologic evidence of increased risk is a “last resort,” Broadbent contradicts himself by stating RR > 2 evidence “must never be required,” and then, in an apparent about face, he argues:

“that far from being epistemically irrelevant, to achieve correct and just outcomes it is in fact mandatory to take (high-quality) epidemiological evidence into account in deciding specific causation. Failing to consider such evidence when it is available leads to error and injustice. The conclusion is that in certain circumstances epidemiological evidence of RR > 2 is not necessary to prove specific causation but that it is sufficient.”

Id. at 237 (emphasis added). I am not sure how epidemiologic evidence can be mandatory but never logically necessary, and something that we should never require.

Presumably, Broadbent is using “to prove” in its legal and colloquial sense, and not as a mathematician.  Let us also give Broadbent his assumptions of “high quality” epidemiologic studies, with established general causation, and ask why, and explore when and whether, RR > 2 is not necessary to show specific causation.

The Probability of Causation vs. The Fact of Causation

Broadbent notes that he is arguing against what he perceives to be Professor Haack’s rejection of probabilistic inference, which would suggest that epidemiologic evidence is “never sufficient to establish specific causation.” Id. at 239 & n.3 (citing Susan Haack, “Risky Business: Statistical Proof of Individual Causation,” in Causación y Atribucion de Responsabilidad (J. Beltran ed., forthcoming)). He correctly points out that sometimes the probabilistic inference is the only probative inference available to support specific causation.  His point, however, does not resolve the dispute; it suffices only to show that whether we allow the probabilistic inference may be outcome determinative in many lawsuits.  Broadbent characterizes Haack’s position as one of two “serious mistakes in judicial and academic literature on this topic.”  Broadbent at 239.  The other alleged mistake is the claim that RR > 2 is needed to show specific causation:

“What follows, I conclude, is that epidemiological evidence is relevant to the proof of specific causation. Epidemiological evidence says that a particular exposure causes a particular harm within a certain population. Importantly, it quantifies: it says how often the exposure causes the harm. However, its methods are limited: they measure only the net effect of the exposure, leaving open the possibility that the exposure is causing more harm than the epidemiological evidence suggests—but ruling out the possibility that it causes less. Accordingly I suggest that epidemiological evidence can be used to estimate a lower bound on the probability of causation but that no epidemiological measure can be required. Thus a relative risk (RR, defined in Section II) of greater than 2 can be used to prove causation when there is no other evidence; but RR < 2 does not disprove causation. Given high-quality epidemiological evidence, RR > 2 is sufficient for proof of specific causation when no other evidence is available but not necessary when other evidence is available.”

Some of this seems reasonable enough.  Contrary to the claims of authors such as Haack and Wright, Broadbent maintains that some RR evidence is relevant and indeed probative of specific causation.  In a tobacco lung cancer, with a plaintiff who has smoked three packs a day, for 50 years (and RR > 50), we can confidently attribute the lung cancer to smoking, and rest assured that background cosmic radiation did not likely play a substantial role. The RR quantifies the strength of the association, and it does lead us to a measure of “attributable risk” (AR), also known as the attributable fraction (AF):

AR = 1 – 1/RR.

So far, so good.

Among the perplexing statements above, however, Broadbent suggests that:

1. The methods of epidemiologic evidence measure only the net effect of the exposure.  Epidemiologic evidence (presumably the RR or other risk ratio) provides a lower bound on the probability of causation.  I take up this suggestion in discussing Broadbent’s distinction between the “excess fraction,” and the “etiologic fraction,” below.

2. A RR > 2 “can be used to prove causation when there is no other evidence; but RR < 2 does not disprove causation.” (My emphasis.) When an author is usually clear about his qualifications, and his language generally, it is distressing for him to start comparing apples to oranges.  Note that RR > 2 suffices “when there is no other evidence,” but the parallel statement about RR < 2 is not similarly qualified, and the statement about RR < 2 is framed in terms of disproof of causation. Even if the RR < 2 did not “disprove” specific causation, when there was no other evidence, it would not prove causation.  And if there is no other evidence, judgment for the defense must result. Broadbent fails to provide us a persuasive scenario in which a RR ≤ 2, with no other evidence, would support an inference of specific causation.

Etiological Fraction vs. Excess Fraction — Occam’s Disposable Razor

Broadbent warns that the expression “attributable risk” (AR or “attributable fraction,” AF) is potentially misleading.  The numerical calculation identifies the excess number of cases, above “expected” per base rate, and proceeds from there.  The AR thus identifies the “excess fraction,” and not the “etiological fraction,” which is the fraction of all cases in which exposure makes a contribution. Broadbent tells us that:

“Granted a sound causal inference, we can infer that all the excess cases are caused by the exposure. But we cannot infer that the remaining cases are not caused by the exposure. The etiologic fraction—the cases in which the exposure makes a causal contribution—could be larger. Roughly speaking, this is because, in the absence of substantive biological assumptions, it is possible that the exposure could contribute to cases that would have occurred12 even without the exposure.13 For example, it might be that smoking is a cause of lung cancer even among some of those who would have developed it anyway. The fact that a person would have developed lung cancer anyway does not offer automatic protection against the carcinogenic effects of cigarette smoke (a point we return to in Section IV).”

Id. at 241. In large measure here, Broadbent has adopted (and acknowledged) his borrowings from Professor Sander Greenland.  Id. at 242 n.11. The argument  still fails.  What Broadbent has interposed is a “theoretical possibility” that the exposure in question may contribute to those cases that would have occurred anyway.  Note that raising theoretical possibilities here now alters the hypothetical; Broadbent is no longer working from a hypothetical that we have a RR and no other evidence.  Even more important, we are left guessing what it means to say that an exposure causes some cases that would have occurred anyway.  If we accept the postulated new evidence at face value, we can say confidently that the exposure is not the “but for” cause of the case at issue.  Without sufficient evidence of “but for” causation, plaintiff will lose. Furthermore, we are being told to add a new fact to the hypothetical, namely that the non-excess cases are causally over-determined.  If this is the only additional new fact being added, a court might invoke the rule in Summers v. Tice, but even so, the defense will be entitled to a directed verdict if the RR < 2. (If the RR = 2, I suppose, the new fact, and the change in the controlling rule, might alter the result.)

Exposures that Cause Some and Prevent Some Cases of Disease

Broadbent raises yet another hypothetical possibility, which adds to, and materially alters,  his original hypothetical.  If the exposure in question, causes some cases, and prevents others, then the RR ≤ 2 will not permit us to infer that a given case is less likely than not the result of the exposure.  (Broadbent might have given an example of what he had in mind, from well-established biological causal relationships; I am skeptical that he would have found one that would have satisfactorily made his argument.) The bimodal distribution of causal effects is certainly not typical of biological processes, but even if we indulge the “possibility,” we are now firmly in the realm of speculation.  This is a perfectly acceptable realm for philosophers, but in court, we want evidence.  Assuming that the claimant could present such evidence, finders of fact would still founder because the new evidence would leave them guessing whether the claimant was a person who would have gotten the disease anyway, or got it because of the exposure, or even got it in spite of the exposure.

Many commentators who urge a “probability of [specific] causation” approach equate the probability of causation (PC) with the AR.  Broadbent argues that because of the possibility that some biological model results in the etiologic fraction exceeded the excess fraction, the usual equation of PC = AR, must be represented as an equality:


While the point is logically unexceptional, Broadbent must concede that some other evidence, which supports and justifies the postulated biological model, is required to change the equality to an inequality.  If no other evidence besides the RR is available, we are left with the equality.  Broadbent tells us that the biological model “often” requires that the etiological fraction exceeds the excess fraction, but he never tells us how often, or how we would ascertain the margin of error.  Id. at 256.

Broadbent does not review any of the decided judicial cases to point out which ones involved biological models that invalidated the equality.  Doing so would be an important exercise because it might well show that even where PC ≥ AR, with a non-quantified upper bound, the plaintiff might still fail in presenting a prima facie case of specific causation.  Suppose the population RR for the exposure in question were 1.1, and we “know” (and are not merely speculating) that the etiological fraction > excess fraction.   Unless we know how much greater is the etiological fraction, such that we can recalculate the PC, then we are left agnostic about specific causation.

Broadbent treats us to several biological scenarios in which PC possibly is greater than AR.  All of these scenarios violate his starting premiss that we have a RR with no other evidence. For instance, Broadbent hypothesizes that exposure might accelerate onset of a disease.  Id. at 256. This biological model of acceleration can be established with the same epidemiologic evidence that established the RR for the population.  Epidemiologists will frequently look at time windows from onset of exposure to explore whether there is an acceleration of onset of cases in a younger age range that offsets a deficit later in the lives of the exposed population.  If there were firm evidence of such a phenomenon, then we would look to the RR within the relevant time window.  If the relevant RR ≤ 2, the biological model will have added nothing to the plaintiff’s case.

Broadbent cites Greenland for the proposition that PC > AR:

“We know of no cancer or other important chronic disease for which current biomedical knowledge allows one to exclude mechanisms that violate the assumptions needed to claim that PC = [AF].”

Id. at 259, quoting form Sander Greenland & James Robins, “Epidemiology, Justice, and the Probability of Causation,” 40 Jurimetrics J. 321, 325 (2000).  Here, not only has Broadbent postulated a mechanism that makes PC > AR, but he has shifted the burden of proof to the defense to exclude it!

The notion that the etiological fraction may exceed the excess fraction is an important caveat.  Courts and lawyers should take note.  It will not do, however, wave hands and exclaim that the RR > 2 is not a “litmus test,” and proceed to let any RR > 1, or even RR ≤ 1 support a verdict.  The biological models that may push the etiological fraction higher than the excess fraction can be tested, and quantified, with the same epidemiologic approaches that provided a risk ratio, in the first place.  Broadbent gives us an example of this sort of hand waving:

“Thus, for example, evidence that an exposure would be likely to aggravate an existing predisposition to the disease in question might suffice, along with RR between 1 and 2, to make it more likely than not that the claimant’s disease was caused by the exposure.”

Id. at 275. This is a remarkable, and unsupported claim.  The magnitude of the aggravation might still leave the RR ≤ 2.  What is needed is evidence that would allow quantification of the risk ratio in the scenario presented. Speculation will not do the trick; nor will speculation get the case to a jury, or support a verdict.


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