A family of divergence-based correlation measures for contingency tables under bivariate normality

TL;DR

This paper introduces a new family of correlation measures for contingency tables under bivariate normality, offering improved accuracy and computational efficiency over existing methods.

Contribution

The authors develop divergence-based correlation measures that approximate latent correlation in contingency tables, with closed-form expressions and enhanced stability.

Findings

Proposed measures better approximate true latent correlation than traditional divergence measures.

The new measures are significantly faster to compute than polychoric correlation.

They effectively distinguish weak and moderate associations where other measures fail.

Abstract

We propose a family of association measures for two-way contingency tables whose latent distribution can be assumed to be bivariate normal. When this assumption holds, the power-divergence measuring departure from independence can be approximated in closed form as a function of the latent correlation coefficient. By inverting this relationship, we obtain a family of measures $\rho_{(\lambda)}$, indexed by a scalar parameter $-1 \leq \lambda \leq 1$, that directly approximates the latent correlation. Special cases include the informational measure of correlation proposed by Linfoot (1957) at $\lambda = 0$ and Pearson's contingency coefficient $C$ at $\lambda = 1$. Additionally, we derive asymptotic distributions via the delta method and construct two families of confidence intervals. Simulation studies confirm that the proposed measures approximate the true latent correlation more faithfully than conventional divergence-based measures, and that they successfully distinguish between weak and moderate associations where existing measures tend to give indistinguishable values. Compared with the polychoric correlation coefficient, the proposed measures are computed several thousand times faster and remain numerically stable even when the latent correlation is close to one.