title: Stochastic counterfactuals and stochastic sufficient causes.
abstract: Most work in causal inference concerns deterministic counterfactuals; the literature on stochastic counterfactuals is small. In the stochastic counterfactual setting, the outcome for each individual under each possible set of exposures follows a probability distribution so that for any given exposure combination, outcomes vary not only between individuals but also probabilistically for each particular individual. The deterministic sufficient cause framework supplements the deterministic counterfactual framework by allowing for the representation of counterfactual outcomes in terms of sufficient causes or causal mechanisms. In the deterministic sufficient cause framework it is possible to test for the joint presence of two causes in the same causal mechanism, referred to as a sufficient cause interaction. In this paper, these ideas are extended to the setting of stochastic counterfactuals and stochastic sufficient causes. Formal definitions are given for a stochastic sufficient cause framework. It is shown that the empirical conditions that suffice to conclude the presence of a sufficient cause interaction in the deterministic sufficient cause framework suffice also to conclude the presence of a sufficient cause interaction in the stochastic sufficient cause framework. Two examples from the genetics literature, in which there is evidence that sufficient cause interactions are present, are discussed in light of the results in this paper.
authors.affiliation: Departments of Epidemiology and Biostatistics, Harvard School of Public Health.
authors.forename: Tyler J
authors.initials: TJ
authors.lastname: Vanderweele
authors.affiliation: Departments of Epidemiology and Biostatistics, Harvard School of Public Health.
authors.forename: James M
authors.initials: JM
authors.lastname: Robins
keywords: Causal inference; Interaction; Stochastic counterfactual; Sufficient cause; Synergism