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

Abstract Once the symmetry model is rejected by statistical tests, we aim to compare several contingency tables with respect to their deviations from symmetry, regardless of the sample size. In this study, we propose a measure to evaluate the degree of departure from symmetry using cosine similarity for square contingency tables that are represented by the same categories. This proposal is based on the Fisher-Rao distance, which allows asymmetry to be interpreted as the geodesic distance between two distributions. Unlike the divergence-based measures proposed in previous studies, the proposed measure provides a geometrically simple way to represent departures from symmetry, allowing intuitive interpretation through visualization on a two-dimensional plane. A simulation study demonstrates that, for square contingency tables under an asymmetry model, the proposed measure complements symmetry tests by providing an interpretable quantification of the degree of asymmetry after rejection, facilitating comparisons across tables.