New perspective on the benefits of the gene–environment independence in case–control studies

We study the benefit of exploiting the gene–environment independence (GEI) assumption for inferring the joint effect of genotype and environmental exposure on disease risk in a case–control study. By transforming the problem into a constrained maximum likelihood estimation problem we derive the asymptotic distribution of the maximum likelihood estimator (MLE) under the GEI assumption (MLE‐GEI) in a closed form. Our approach uncovers a transparent explanation of the efficiency gained by exploiting the GEI assumption in more general settings, thus bridging an important gap in the existing literature. Moreover, we propose an easy‐to‐implement numerical algorithm for estimating the model parameters in practice. Finally, we conduct simulation studies to compare the proposed method with the traditional prospective logistic regression method and the case‐only estimator. The Canadian Journal of Statistics 47: 473–486; 2019 © 2019 Statistical Society of Canada