Exploiting Gene‐Environment Independence for Analysis of Case–Control Studies: An Empirical Bayes‐Type Shrinkage Estimator to Trade‐Off between Bias and Efficiency

Summary Standard prospective logistic regression analysis of case–control data often leads to very imprecise estimates of gene‐environment interactions due to small numbers of cases or controls in cells of crossing genotype and exposure. In contrast, under the assumption of gene‐environment independence, modern “retrospective” methods, including the “case‐only” approach, can estimate the interaction parameters much more precisely, but they can be seriously biased when the underlying assumption of gene‐environment independence is violated. In this article, we propose a novel empirical Bayes‐type shrinkage estimator to analyze case–control data that can relax the gene‐environment independence assumption in a data‐adaptive fashion. In the special case, involving a binary gene and a binary exposure, the method leads to an estimator of the interaction log odds ratio parameter in a simple closed form that corresponds to an weighted average of the standard case‐only and case–control estimators. We also describe a general approach for deriving the new shrinkage estimator and its variance within the retrospective maximum‐likelihood framework developed by Chatterjee and Carroll (2005, Biometrika92, 399–418). Both simulated and real data examples suggest that the proposed estimator strikes a balance between bias and efficiency depending on the true nature of the gene‐environment association and the sample size for a given study.