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.

翻译：本文针对有序类别的匹配集数据，探讨多维列联表建模中面临的挑战。尽管完全对称模型与边际齐性模型已较为完善，但其对数据的拟合效果未必总能令人满意。为解决此问题，我们提出一种广义序次拟对称模型，该模型在完全对称模型无法捕捉数据潜在结构时展现出更高的灵活性。我们研究了这一新模型的性质，并给出了信息论视角的解读，阐明其与序次拟对称模型的关联。此外，我们重新审视了Agresti的研究成果，提出了完全对称模型的一个新的充分必要条件，并证明所提模型与边际矩相等模型为可分离假设。该模型与边际矩相等模型的可分性是多维列联表分析的重要进展，使得研究者能够更精确地检验数据中的对称结构，从而更深入地理解潜在模式。这一强大框架为研究者探索匹配集数据中有序变量关系的复杂性提供了必要工具，为新的发现与洞见开辟了道路。