Semiparametric tests for sufficient cause interaction

Summary.  A sufficient cause interaction between two exposures signals the presence of individuals for whom the outcome would occur only under certain values of the two exposures. When the outcome is dichotomous and all exposures are categorical, then, under certain no confounding assumptions, empirical conditions for sufficient cause interactions can be constructed on the basis of the sign of linear contrasts of conditional outcome probabilities between differently exposed subgroups, given confounders. It is argued that logistic regression models are unsatisfactory for evaluating such contrasts, and that Bernoulli regression models with linear link are prone to misspecification. We therefore develop semiparametric tests for sufficient cause interactions under models which postulate probability contrasts in terms of a finite dimensional parameter, but which are otherwise unspecified. Estimation is often not feasible in these models because it would require non‐parametric estimation of auxiliary conditional expectations given high dimensional variables. We therefore develop ‘multiply robust tests’ under a union model which assumes that at least one of several working submodels holds. In the special case of a randomized experiment or a family‐based genetic study in which the joint exposure distribution is known by design or Mendelian inheritance, the procedure leads to asymptotically distribution‐free tests of the null hypothesis of no sufficient cause interaction.