A Likelihood Ratio Test for Gene-Environment Interaction Based on the Trend Effect of Genotype under an Additive Risk Model Using the Gene-Environment Independence Assumption

Several statistical methods have been proposed for testing gene-environment (G-E) interactions under additive risk models using data from genome-wide association studies. However, these approaches have strong assumptions from underlying genetic models, such as dominant or recessive effects that are known to be less robust when the true genetic model is unknown. We aimed to develop a robust trend test employing a likelihood ratio test for detecting G-E interaction under an additive risk model, while incorporating the G-E independence assumption to increase power. We used a constrained likelihood to impose 2 sets of constraints for: 1) the linear trend effect of genotype and 2) the additive joint effects of gene and environment. To incorporate the G-E independence assumption, a retrospective likelihood was used versus a standard prospective likelihood. Numerical investigation suggests that the proposed tests are more powerful than tests assuming dominant, recessive, or general models under various parameter settings and under both likelihoods. Incorporation of the independence assumption enhances efficiency by 2.5-fold. We applied the proposed methods to examine the gene-smoking interaction for lung cancer and gene-Apolipoprotein E $\varepsilon$4 interaction for Alzheimer disease, which identified 2 interactions between apolipoprotein E $\varepsilon$4 and loci membrane-spanning 4-domains subfamily A (MS4A) and bridging integrator 1 (BIN1) genes at genome-wide significance that were replicated using independent data.