On relationships between Chatterjee's and Spearman's correlation coefficients

TL;DR

This paper explores the probabilistic relationship between Chatterjee's and Spearman's correlation coefficients, revealing their joint distribution under independence and dependence, and proposes a new combined independence test.

Contribution

It establishes the joint asymptotic normality and independence of the two correlations under independence, and introduces a novel test combining both measures for better sensitivity.

Findings

Under independence, correlations are asymptotically jointly normal and independent.

The two correlations can differ significantly under dependence.

The new combined test shows good sensitivity to various correlation patterns.

Abstract

In his seminal work, Chatterjee (2021) introduced a novel correlation measure which is distribution-free, asymptotically normal, and consistent against all alternatives. In this paper, we study the probabilistic relationships between Chatterjee's correlation and the widely used Spearman's correlation. We show that, under independence, the two sample-based correlations are asymptotically joint normal and asymptotically independent. Under dependence, the magnitudes of two correlations can be substantially different. We establish some extremal cases featuring large differences between these two correlations. Motivated by these findings, a new independence test is proposed by combining Chatterjee's and Spearman's correlations into a maximal strength measure of variable association. Our simulation study and real data application show the good sensitivity of the new test to different correlation patterns.