Semiparametrically efficient estimation in quantile regression of secondary analysis

Summary Analysing secondary outcomes is a common practice for case–control studies. Traditional secondary analysis employs either completely parametric models or conditional mean regression models to link the secondary outcome to covariates. In many situations, quantile regression models complement mean‐based analyses and provide alternative new insights on the associations of interest. For example, biomedical outcomes are often highly asymmetric, and median regression is more useful in describing the ‘central’ behaviour than mean regressions. There are also cases where the research interest is to study the high or low quantiles of a population, as they are more likely to be at risk. We approach the secondary quantile regression problem from a semiparametric perspective, allowing the covariate distribution to be completely unspecified. We derive a class of consistent semiparametric estimators and identify the efficient member. The asymptotic properties of the resulting estimators are established. Simulation results and a real data analysis are provided to demonstrate the superior performance of our approach with a comparison with the only existing approach so far in the literature.