Global quantile regression

TL;DR

This paper introduces a global quantile regression method that tests whether covariates affect the entire data distribution, extending traditional regression and distribution comparison techniques with permutation-based global envelope tests.

Contribution

It develops a novel global testing framework for covariate effects across all quantiles simultaneously, incorporating multiple test adjustment and graphical interpretation.

Findings

The proposed methods control family-wise error rate effectively.

Alternative permutation strategies perform well for extreme quantiles.

Application to real data demonstrates practical utility.

Abstract

Quantile regression is used to study effects of covariates on a particular quantile of the data distribution. Here we are interested in the question whether a covariate has any effect on the entire data distribution, i.e., on any of the quantiles. To this end, we treat all the quantiles simultaneously and consider global tests for the existence of the covariate effect in the presence of nuisance covariates. This global quantile regression can be used as the extension of linear regression or as the extension of distribution comparison in the sense of Kolmogorov-Smirnov test. The proposed method is based on pointwise coefficients, permutations and global envelope tests. The global envelope test serves as the multiple test adjustment procedure under the control of the family-wise error rate and provides the graphical interpretation which automatically shows the quantiles or the levels of categorical covariate responsible for the rejection. The Freedman-Lane permutation strategy showed liberality of the test for extreme quantiles, therefore we propose four alternatives that work well even for extreme quantiles and are suitable in different conditions. We present a simulation study to inspect the performance of these strategies, and we apply the chosen strategies to two data examples.