Modeling sign concordance of quantile regression residuals with multiple outcomes

TL;DR

This paper introduces a two-step method for modeling the sign concordance of residuals in multivariate quantile regression, enabling analysis of correlation structures without needing a multivariate quantile definition.

Contribution

The paper proposes a simple, effective approach combining separate quantile regressions with multinomial modeling of residual signs for multivariate responses.

Findings

Method captures correlation structure in multivariate responses.

Applicable to datasets with multiple outcomes.

Provides insights into covariate effects on residual sign dependence.

Abstract

Quantile regression permits describing how quantiles of a scalar response variable depend on a set of predictors. Because a unique definition 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 first step, quantile regression is applied to each response 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 definition of quantiles, and can be used to capture important features of a multivariate response and assess the effects of covariates on the correlation structure. We apply the proposed method to analyze two different datasets.