Generalized R-squared for Detecting Dependence

TL;DR

This paper introduces G-squared, a new dependence measure that generalizes R-squared to effectively detect nonlinear and heteroscedastic relationships between variables, outperforming existing methods.

Contribution

The paper proposes G-squared, a novel dependence measure that extends R-squared to nonlinear and heteroscedastic contexts, with consistent estimators and superior power in tests.

Findings

G-squared effectively detects nonlinear dependence.

G-squared outperforms existing dependence measures in simulations.

Two consistent estimators of G-squared are proposed.

Abstract

Detecting dependence between two random variables is a fundamental problem. Although the Pearson correlation is effective for capturing linear dependency, it can be entirely powerless for detecting nonlinear and/or heteroscedastic patterns. We introduce a new measure, G-squared, to test whether two univariate random variables are independent and to measure the strength of their relationship. The G-squared is almost identical to the square of the Pearson correlation coefficient, R-squared, for linear relationships with constant error variance, and has the intuitive meaning of the piecewise R-squared between the variables. It is particularly effective in handling nonlinearity and heteroscedastic errors. We propose two estimators of G-squared and show their consistency. Simulations demonstrate that G-squared estimates are among the most powerful test statistics compared with several state-of-the-art methods.