  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.

Methodology