Genome-wide association studies (GWAS) often measure gene-environment interactions (G × E). We consider the problem of accurately estimating a G × E in a case-control GWAS when a subset of the controls have silent, or undiagnosed, disease and the frequency of the silent disease varies by the environmental variable. We show that using case-control status without accounting for misdiagnosis can lead to biased estimates of the G × E. We further propose a pseudolikelihood approach to remove the bias and accurately estimate how the relationship between the genetic variant and the true disease status varies by the environmental variable. We demonstrate our method in extensive simulations and apply our method to a GWAS of prostate cancer.

Case-control genome-wide association studies (CC-GWAS) might provide valuable clues to the underlying pathophysiologic mechanisms of complex diseases, such as neurodegenerative disease and cancer. A commonly overlooked complication is that multiple distinct disease states might present with the same set of symptoms and hence share a clinical diagnosis. These disease states can only be distinguished based on a biomarker evaluation that might not be feasible in the whole set of cases in the large number of samples that are typically needed for CC-GWAS. Instead, the biomarkers are measured on a subset of cases. Or an external reliability study estimates the frequencies of the disease states of interest within the clinically diagnosed set of cases. These frequencies often vary by the genetic and/or nongenetic variables. We derive a simple approximation that relates the genetic effect estimates obtained in a traditional logistic regression model with the clinical diagnosis as the outcome variable to the genetic effect estimates in the relationship to the true disease state of interest. We performed simulation studies to assess the accuracy of the approximation that we have derived. We next applied the derived approximation to the analysis of the genetic basis of the innate immune system of Alzheimer's disease.

Case-control Genome-Wide Association Studies (GWAS) provide a rich resource for studying the genetic architecture of complex diseases. A key is to elucidate how the genetic effects vary by the environment, what is traditionally defined by Gene-Environment interactions (GxE). The overlooked complication is that multiple, distinct pathophysiologic mechanisms may lead to the same clinical diagnosis and often these mechanisms have distinct genetic bases. In this paper, we first show that using the clinically diagnosed status can lead to severely biased estimates of GxE interactions in situations when the frequency of the pathologic diagnosis of interest, as compared to other diagnoses, depends on the environment. We then propose a pseudo-likelihood solution to correct the bias. Finally, we demonstrate our method in extensive simulations and in a GWAS of Alzheimer's disease.

Abstract: Large‐scale genome‐wide analyses scans on massive numbers of various cases and controls are archived in the genetic databases that are publically available, for example, the Database of Genotypes and Phenotypes (https://www.ncbi.nlm.nih.gov/gap/). These databases offer unprecscendented opportunity to study the genetic effects. Yet, the set of nongenetic variables in these databases is often brief. From the statistical literature, we know that omitting a continuous variable from a logistic regression model can result in biased estimates of odds ratios (OR), even when the omitted and the included variables are independent. We are interested in assessing what information is needed to recover the bias in the OR estimate of genotype due to omitting a continuous variable in settings when the actual values of the omitted variable are not available. We derive two estimating procedures that can recover the degree of bias based on a conditional density of the omitted variable given the disease status and the genotype or the known distribution of the omitted variable and frequency of the disease in the population. Importantly, our derivations show that omitting a continuous variable can result in either under‐ or over‐estimation of the genetic effects. We performed extensive simulation studies to examine bias, variability, false‐positive rate, and power in the model that omits a continuous variable. We show the application to two genome‐wide studies of Alzheimer's disease.

Genetic studies provide valuable information to assess if the effect of genetic variants varies by the nongenetic "environmental" variables, what is traditionally defined to be gene-environment interaction (GxE). A common complication is that multiple disease states present with the same set of symptoms, and hence share the clinical diagnosis. Because (a) disease states might have distinct genetic bases; and (b) frequencies of the disease states within the clinical diagnosis vary by the environmental variables, analyses of association with the clinical diagnosis as an outcome variable might result in false positive or false negative findings. We develop estimates for this setting to be able to assess GxE in a case-only study and we compare the case-control and case-only estimates. We report extensive simulation studies that evaluate empirical properties of the estimates and show the application to a study of Alzheimer's 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.