Modeling asymmetry in multi-way contingency tables with ordinal categories: A maximum-entropy approach with f-divergence

This study introduces a novel model that effectively captures asymmetric structures in multivariate contingency tables with ordinal categories. Leveraging the principle of maximum entropy, our approach employs f-divergence to provide a rational model under the presence of a ''prior guess.'' Inspired by the constraints used in the derivation of multivariate normal distributions, we demonstrate that the proposed model minimizes f-divergence from complete symmetry under specific constraints. The proposed model encompasses existing asymmetry models as special cases while offering remarkably high interpretability. By modifying divergence measures included in f-divergence, the model provides the flexibility to adapt to specific probabilistic structures of interest. Furthermore, we established theorems that show that a complete symmetry model can be decomposed into two or more models, each imposing less restrictive parameter constraints. We also investigated the properties of the goodness-of-fit statistics with an emphasis on the likelihood ratio and Wald test statistics. Extensive Monte Carlo simulations confirmed the nominal size, high power, and robustness of the choice of f-divergence. Finally, an application to real-world data highlights the practical utility of the proposed model for analyzing asymmetric structures in ordinal contingency tables.