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

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.