Modeling sign concordance of quantile regression residuals with multiple outcomes

Quantile regression permits describing how quantiles of a scalar response vari- able depend on a set of predictors. Because a unique de nition of multivariate quantiles is lacking, extending quantile regression to multivariate responses is somewhat complicated. In this paper, we describe a simple approach based on a two-step procedure: in the  rst step, quantile regression is applied to each re- sponse separately; in the second step, the joint distribution of the signs of the residuals is modeled through multinomial regression. The described approach does not require a multidimensional de nition of quantiles, and can be used to capture important features of a multivariate response and assess the e ects of co- variates on the correlation structure. We apply the proposed method to analyze two di erent datasets.