A simple approximation to the bias of gene-environment interactions in case-control studies with silent disease

One of the most important research areas in case-control Genome-Wide Association Studies is to determine how the effect of a genotype varies across the environment or to measure the gene-environment interaction (G × E). We consider the scenario when some of the "healthy" controls actually have the disease and when the frequency of these latent cases varies by the environmental variable of interest. In this scenario, performing logistic regression with the clinically diagnosed disease status as an outcome variable and will result in biased estimates of G × E interaction. Here, we derive a general theoretical approximation to the bias in the estimates of the G × E interaction and show, through extensive simulation, that this approximation is accurate in finite samples. Moreover, we apply this approximation to evaluate the bias in the effect estimates of the genetic variants related to mitochondrial proteins a large-scale prostate cancer study.