Visualization for departures from symmetry with the power-divergence-type measure in two-way contingency tables

TL;DR

This paper introduces a new correspondence analysis method using power-divergence measures to visualize and compare asymmetries in square contingency tables, with confidence regions for improved accuracy.

Contribution

It proposes a novel CA approach that visualizes departures from symmetry using power-divergence measures, ensuring equal contribution axes and sample size independence.

Findings

The method visualizes various divergence-based departures from symmetry.

Axes in the CA plot have equal contribution rates.

Sample size independence allows for comparison across multiple tables.

Abstract

When the row and column variables consist of the same category in a two-way contingency table, it is specifically called a square contingency table. Since it is clear that the square contingency tables have an association structure, a primary objective is to examine symmetric relationships and transitions between variables. While various models and measures have been proposed to analyze these structures understanding changes between two variables in behavior at two-time points or cohorts, it is also necessary to require a detailed investigation of individual categories and their interrelationships, such as shifts in brand preferences. This paper proposes a novel approach to correspondence analysis (CA) for evaluating departures from symmetry in square contingency tables with nominal categories, using a power-divergence-type measure. The approach ensures that well-known divergences can also be visualized and, regardless of the divergence used, the CA plot consists of two principal axes with equal contribution rates. Additionally, the scaling is independent of sample size, making it well-suited for comparing departures from symmetry across multiple contingency tables. Confidence regions are also constructed to enhance the accuracy of the CA plot.