Multiply Robust Inference for Statistical Interactions

A primary focus of an increasing number of scientific studies is to determine whether two exposures interact in the effect that they produce on an outcome of interest. Interaction is commonly assessed by fitting regression models in which the linear predictor includes the product between those exposures. When the man interest lies in the interaction, this approach is not entirely satisfactory, because it is prone to (possibly severe) bias when the main exposure effects or the associations between outcome and extraneous factors are misspecified. In this article we consider conditional mean models with identity or log link that postulate the statistical interaction in terms of a finite-dimensional parameter but are otherwise unspecified. We show that estimation of the interaction parameter often is not feasible in this model, because it requires nonparametric estimation of auxiliary conditional expectations given high-dimensional variables. We thus consider multiply robust estimation under a union model that assumes that at least one of several working submodels holds. Our approach is novel in that it makes use of information on the joint distribution of the exposures conditional on the extraneous factors in making inferences about the interaction parameter of interest. In the special case of a randomized trial or a family-based genetic study in which the joint exposure distribution is known by design or by Mendelian inheritance, the resulting multiply robust procedure leads to asymptotically distribution-free tests of the null hypothesis of no interaction on an additive scale. We illustrate the methods through simulation and analysis of a randomized follow-up study.