Re: “Invited Commentary: How Big Is That Interaction (In My Community)—And in Which Direction?”

In their invited commentary on an article by Turner et al. (1), Panagiotou and Wacholder (2) discuss the estimation of population attributable risks (PARs), or population attributable fractions as they call them, when multiple risk factors are involved. In particular, they consider the difference between the PAR associated with the combined effects of 2 factors and the sum of the effects for each factor separately. In referring to much earlier work of mine on this topic (3), they state that I showed that “for a rare outcome” (2, p. 1154) this difference would be zero under an additive model of disease risks. In fact, in my earlier paper (3), I established quite general conditions under which these 2 methods of calculating PARs would be equivalent, and, contrary to Panagiotou and Wacholder's claim, no rare outcome assumption is required. In particular, my equation results show that the combined versus summative formulations of 2 PARs will be equal if the disease risks are additive or if no one is exposed to both factors (see equation 9 in my paper (3)). The only sense in which rarity of outcomes might be invoked in practice is if the additive model becomes empirically untenable—for instance, if it were to predict risks greater than 100% for doubly exposed persons, based on the risks associated with single exposures. Note that my paper also considered an important related issue—specifically, identification of the conditions under which a marginal estimate of PAR for a risk factor of interest would be valid in the presence of other risk factors. (If the conditions are satisfied, then a marginal approach to PAR estimation might be adopted, with only more limited data then being required.) These conditions were that either the exposures are independently distributed in the population or the other factors do not increase risk for persons unexposed to the factor of interest. When there are only 2 risk factors, one of these conditions must necessarily be true; however, in the situation of more than 2 risk factors, analogous conditions yield validity of a marginal PAR—but then the conditions need not necessarily be true. Again, no rare outcome assumption is required for any of these results. Conflict of interest: none declared.

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