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

Jiawei Wei, Raymond J. Carroll, Ursula U. Müller, Ingrid Van Keilegom, Nilanjan Chatterjee

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2013 · Volume 75 · Issue 1

DOI: 10.1111/j.1467-9868.2012.01052.x

Abstract

Summary.Primary analysis of case–control studies focuses on the relationship between diseaseDand a set of covariates of interest (Y,X). 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 ofYonXis different from what it is in the population. Previous work has assumed a parametric distribution forYgivenXand derived semiparametric efficient estimation and inference without any distributional assumptions aboutX. We take up the issue of estimation of a regression function whenYgivenXfollows a homoscedastic regression model, but otherwise the distribution ofYis 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 forYgivenX. We take an entirely different approach. We show how to estimate the regression parameters consistently even if the assumed model forYgivenXis 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.

Keywords

Biased samples, Homoscedastic regression, Secondary data, Secondary phenotypes, Semiparametric inference, Two-stage samples

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