Visualization for Departures from Symmetry with the Power-Divergence-Type Measure in Square Contingency Tables.

Abstract

When the row and column variables consist of the same category in a two-way contingency table, it is called a square contingency table. Since square contingency tables have an association structure due to the concentration of observed values near the main diagonal, a primary objective is to examine symmetric relationships and transitions between variables. Various models and measures have been proposed to analyze these structures to understand the changes between two variables' behavior at two-time points or cohorts. This is necessary for a detailed investigation of individual categories and their interrelationships, such as shifts in brand preferences. We propose a novel approach to correspondence analysis (CA) for evaluating departures from symmetry in square contingency tables with nominal categories, using a modified divergence statistic. This approach ensures that well-known divergence statistics can also be visualized and regardless of the divergence statistics used, the CA plot consists of two principal axes with equal contribution rates. Notably, the scaling of the departures from symmetry provided by the modified divergence statistic is independent of sample size, allowing for meaningful comparisons and unification of results across different tables. Confidence regions are also constructed to enhance the accuracy of the CA plot.