Abstract: Analyzing the tail quantiles of a response distribution is sometimes more important than analyzing the mean in biomarker studies. Inferences in a quantile regression are complicated when there exist a large number of candidate markers, together with some prespecified controlled covariates. In this study, we develop a new and simple testing procedure to detect the effects of biomarkers in a high-dimensional quantile regression in the presence of protected covariates. The test is based on the maximum-score-type statistic obtained from a conditional marginal regression. We establish the asymptotic properties of the proposed test statistic under both null and alternative hypotheses and propose an alternative multiplier bootstrap method, with theoretical justifications. We use numerical studies to show that the proposed method provides adequate controls of the family-wise error rate with competitive power, and that it can also be used as a stopping rule in a forward regression. The proposed method is applied to a motivating genome-wide association study to detect single nucleotide polymorphisms associated with low glomerular filtration rates in type 1 diabetes patients.

Key words and phrases: Conditional marginal regression, extreme value distribution, high dimensional, maximal score statistic, multiplier bootstrap.