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

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.