A measure of departure from symmetry via the Fisher-Rao distance for contingency tables

TL;DR

This paper introduces a new geometric measure based on Fisher-Rao distance and cosine similarity to quantify asymmetry in square contingency tables, offering a natural, robust, and sensitive way to detect departures from symmetry.

Contribution

The paper proposes a novel asymmetry measure using Fisher-Rao distance and cosine similarity, providing a more natural and robust quantification compared to existing methods.

Findings

The proposed measure is geometrically simple and natural.

It is less affected by extreme asymmetry in a few cells.

It more sensitively detects departures from symmetry in simulations.

Abstract

A measure of asymmetry is a quantification method that allows for the comparison of categorical evaluations before and after treatment effects or among different target populations, irrespective of sample size. We focus on square contingency tables that summarize survey results between two time points or cohorts, represented by the same categorical variables. We propose a measure to evaluate the degree of departure from a symmetry model using cosine similarity. This proposal is based on the Fisher-Rao distance, allowing asymmetry to be interpreted as a geodesic distance between two distributions. Various measures of asymmetry have been proposed, but visualizing the relationship of these quantification methods on a two-dimensional plane demonstrates that the proposed measure provides the geometrically simplest and most natural quantification. Moreover, the visualized figure indicates that the proposed method for measuring departures from symmetry is less affected by very few cells with extreme asymmetry. A simulation study shows that for square contingency tables with an underlying asymmetry model, our method can directly extract and quantify only the asymmetric structure of the model, and can more sensitively detect departures from symmetry than divergence-type measures.