Genetic effect estimates in case-control studies when a continuous variable is omitted from the model

Large-scale genome-wide analyses scans provide massive volumes of genetic variants on large number of cases and controls that can be used to estimate the genetic effects. Yet, the sets of non-genetic variables available in publicly available databases are often brief. It is known that omitting a continuous variable from a logistic regression model can result in biased estimates of odds ratios (OR) (e.g., Gail et al (1984), Neuhaus et al (1993), Hauck et al (1991), Zeger et al (1988)). We are interested to assess 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 or knowing the distribution of the omitted variable. 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 Alzheimers disease.