Robust estimation for homoscedastic regression in the secondary analysis of case-control data.

Primary analysis of case-control studies focuses on the relationship between disease and a set of covariates of interest (, ). A secondary application of the case-control study, which is often invoked in modern genetic epidemiologic association studies, is to investigate the interrelationship between the covariates themselves. The task is complicated owing to the case-control sampling, where the regression of on is different from what it is in the population. Previous work has assumed a parametric distribution for given and derived semiparametric efficient estimation and inference without any distributional assumptions about . We take up the issue of estimation of a regression function when given follows a homoscedastic regression model, but otherwise the distribution of is unspecified. The semiparametric efficient approaches can be used to construct semiparametric efficient estimates, but they suffer from a lack of robustness to the assumed model for given . We take an entirely different approach. We show how to estimate the regression parameters consistently even if the assumed model for given is incorrect, and thus the estimates are model robust. For this we make the assumption that the disease rate is known or well estimated. The assumption can be dropped when the disease is rare, which is typically so for most case-control studies, and the estimation algorithm simplifies. Simulations and empirical examples are used to illustrate the approach.