Generalized Cram\'er's coefficient via $f$-divergence for contingency tables

TL;DR

This paper introduces a generalized measure of association in contingency tables based on $f$-divergence, broadening the scope beyond power-divergence, and provides theoretical properties, numerical experiments, and interpretative insights.

Contribution

It proves that $f$-divergence-based measures have desirable properties for association strength, enabling easy construction of new measures and deeper interpretation.

Findings

Proven desirable properties of $f$-divergence measures for contingency tables

Numerical experiments with $ heta$-divergence demonstrate practical applicability

Established relationship between proposed measures and correlation in latent variables

Abstract

Various measures in two-way contingency table analysis have been proposed to express the strength of association between row and column variables in contingency tables. Tomizawa et al. (2004) proposed more general measures, including Cram\'er's coefficient, using the power-divergence. In this paper, we propose measures using the $f$-divergence that has a wider class than the power-divergence. Unlike statistical hypothesis tests, these measures provide quantification of the association structure in contingency tables. The contribution of our study is proving that a measure applying a function that satisfies the condition of the $f$-divergence has desirable properties for measuring the strength of association in contingency tables. With this contribution, we can easily construct a new measure using a divergence that has essential properties for the analyst. For example, we conducted numerical experiments with a measure applying the $\theta$-divergence. Furthermore, we can give further interpretation of the association between the row and column variables in the contingency table, which could not be obtained with the conventional one. We also show a relationship between our proposed measures and the correlation coefficient in the bivariate normal distribution of latent variables in the contingency tables.