A generalized ordinal quasi-symmetry model and its separability for analyzing multi-way tables

Abstract: This paper addresses the challenge of modeling multi-way contingency tables for matched set data with ordinal categories. Although the complete symmetry and marginal homogeneity models are well established, they may not always provide a satisfactory fit to the data. To address this issue, we propose a generalized ordinal quasi-symmetry model that offers increased flexibility when the complete symmetry model fails to capture the underlying structure. We investigate the properties of this new model and provide an information-theoretic interpretation, elucidating its relationship to the ordinal quasi-symmetry model. Moreover, we revisit Agresti's findings and present a new necessary and sufficient condition for the complete symmetry model, proving that the proposed model and the marginal moment equality model are separable hypotheses. We demonstrate the practical application of our model through empirical studies on medical and public opinion datasets. Comprehensive simulation studies evaluate the proposed model under various scenarios, including model's performance for multivariate normal data and asymptotic behavior. It enables researchers to examine the symmetry structure in the data with greater precision, providing a more thorough understanding of the underlying patterns. This powerful framework equips researchers with the necessary tools to explore the complexities of ordinal variable relationships in matched data sets, paving the way for new discoveries and insights.