On the copula correlation ratio and its generalization

Abstract

The correlation ratio has been used to measure how much the behavior of one variable can be predicted by the other variable. In this paper, we derive a new expression of the correlation ratio based on copulas. We represent the copula correlation ratio in terms of Spearman’s rho of the $\ast$-product of two copulas. Our expression provides a new way to obtain the copula correlation ratio, which is especially useful when a copula is closed under the $\ast$-product operation. Moreover, we propose a Kendall’s tau copula correlation ratio that has not been considered in the literature. We apply the new expressions to investigate the theoretical properties of the copula correlation ratios, including difference and discontinuity. For multivariate copulas, we propose to define the copula correlation ratio matrices, and show their invariance property.

Introduction

In analyzing a multivariate distribution, studying dependence structure between random variables is essential. Many measures of association have been proposed by various authors to describe association structure between two random variables such as Pearson’s correlation coefficient, Spearman’s rho, and Kendall’s tau. However, many of them do not describe the asymmetric relationship between two random variables. For two random variables $X$ and $Y$, the association of $X$ on $Y$ can be different from the association of $Y$ on $X$.

To deal with this problem, Dabrowska [7] investigated a regression-based measure of association through the correlation ratio. The fundamental idea of regression association is to measure how much the behavior of one variable can be predicted by the other variable. If $Y$ is predictable from $X$, we say that there is regression association of $Y$ on $X$. The analysis of a bivariate distribution focuses on the contrast between two regressions: ($Y$ on $X$) and ($X$ on $Y$). It is possible that $Y$ can be perfectly predicted by $X$, whereas $X$ is ‘almost independent’ from $Y$ (Siburg and Stoimenov [38]).

Sungur [40] proposed a copula-based correlation ratio, that is, the correlation ratio after eliminating the influence of the marginal distributions. Dette et al. [10] introduced a general concept of regression dependence and proposed a copula-based nonparametric measure of regression dependence. Based on the work of Dette et al. [10], an almost opposite case of regression dependence was discussed by Siburg and Stoimenov [38]. Recently, Chatterjee [6] investigated the same measure as in Dette et al. [10] without the aid of copulas. For real applications of copula-based regression association and prediction, we refer to Chang and Joe [5], Emura et al. [19], Kim and Kim [26], Kim et al. [24], and Kim et al. [25].

In this paper, we derive a new expression of the copula-based correlation ratio that was defined by Sungur [40]. By utilizing the $\ast$-product operator introduced by Darsow et al. [8], we show that the copula correlation ratio is equal to Spearman’s rho of the $\ast$-product of two copulas. Our expression provides a new way to obtain the copula correlation ratio, which is especially useful when a copula is closed under the $\ast$-product operation. Moreover, our new expression also suggests a natural generalization of the copula correlation ratio by allowing Spearman’s rho to be replaced by any other measure of association. Based on our new expression, we investigate the theoretical properties of the copula correlation ratios, including difference and discontinuity. For multivariate copulas, we propose to define the copula correlation ratio matrices, and show their invariance property.

This paper is organized as follows: Section 2 reviews some basic properties of copulas and the correlation ratio. Section 3 provides a new expression for the copula correlation ratio and proposes its generalization. Section 4 studies some theoretical properties of the copula correlation ratios. Section 5 concludes the paper. Additional results for the paper including estimation, simulation, and data analysis are provided in the Supplementary Material.