  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. The separability of the proposed model and marginal
moment equality model is a significant development in the analysis of multi-way
contingency tables. 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 set data, paving the way for new discoveries and insights.
