For the analysis of square contingency tables, it is necessary to estimate an unknown distribution with high confidence from an obtained observation. For that purpose, we need to introduce a statistical model that fits the data well and has parsimony. This study proposes asymmetry models based on cumulative probabilities for square contingency tables with the same row and column ordinal classifications. In the proposed models, the odds, for all i<j, that an observation will fall in row category i or below, and column category j or above, instead of row category j or above, and column category i or below, depend on only row category i or column category j. This is notwithstanding that the odds are constant without relying on row and column categories under the conditional symmetry (CS) model. The proposed models constantly hold when the CS model holds. However, the converse is not necessarily true. This study also shows that it is necessary to satisfy the extended marginal homogeneity model, in addition to the proposed models, to satisfy the CS model. These decomposition theorems explain why the CS model does not hold. The proposed models provide a better fit for application to a single data set of real-world occupational data for father-and-son dyads.

中文翻译：

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平方列联表中条件对称模型的累积概率和分解的奇偶对称模型

对于方形列联表的分析，有必要从获得的观察中以高置信度估计未知分布。为此，我们需要引入一个能够很好地拟合数据并具有简约性的统计模型。本研究提出了基于具有相同行列序分类的方形列联表的累积概率的不对称模型。在所提出的模型中，对于所有i < j，观测值将落在行类别i或以下，列类别j或以上，而不是行类别j或以上，列类别i或以下的几率取决于仅行类别i或列类别Ĵ。尽管在条件对称 (CS) 模型下，在不依赖行和列类别的情况下，几率是恒定的。当 CS 模型成立时，所提出的模型始终成立。然而，反过来不一定是正确的。该研究还表明，除了所提出的模型外，还需要满足扩展的边际同质性模型来满足 CS 模型。这些分解定理解释了为什么 CS 模型不成立。所提出的模型更适合应用于父子二人组真实世界职业数据的单一数据集。