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 (D) and a set of covariates of interest (Y,X). A secondary application of the case-control study, often invoked in modern genetic epidemiologic association studies, is to investigate the interrelationship between the covariates themselves. The task is complicated due to case-control sampling. Previous work has assumed a parametric distribution for Y given X and derived semiparametric efficient estimation and inference without any distributional assumptions about X. In this paper, we take up the issue of estimation of a regression function when Y given X follows a homoscedastic regression model, but otherwise the distribution of Y 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 Y given X. We take an entirely different and novel approach in the case that the disease is rare. We show how to estimate the regression parameters in the rare disease case even if the assumed model for Y given X is incorrect, and thus the estimates are model-robust. Simulations and empirical examples are used to illustrate the approach.