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 (c) 2019 Statistical Society of Canada