Multiple-Robust Estimation of an Odds Ratio Interaction

An important scienti…c goal of genetic epidemiology studies is increasingly to determine whether an interaction between a genetic factor and an environmental factor or two genetic factors is present in the e¤ect that they produce on the risk of a disease outcome. In such studies, interaction is commonly assessed by …tting a logistic regression model to case-control or cohort data, in which the linear predictors includes on the log-odds scale, the product between the two factors of interest. Unfortunately, inferences on an interaction using standard logistic regression methods are prone to bias due to model mis-speci…cation of main exposure e¤ects or of the association model between extraneous factors and the outcome. In this paper, an alternative semiparametric logistic regression model is considered, which postulates the statistical interaction in terms of a …nite-dimensional parameter, but which is otherwise unspeci…ed. We show that estimation is generally not feasible in this model because of the curse of dimensionality associated with the required estimation of auxiliary conditional densities given high-dimensional covariates. We thus consider ‘multiply robust estimation’and propose a more general model which assumes at least