Cumulative ordinal quasi-symmetry model and its separation for square contingency tables with ordered categories

This study deals with square contingency tables, which are two-way contingency tables in which the row and column variables consist of the same classification. When the categorical variables are grouped into ordered categories from quantitative variables by cut points, a score that approximates the distance between the midpoints of the quantitative scale categories may be used to reflect the characteristics of the data. As an asymmetry model based on scores, this study proposes the cumulative ordinal quasi-symmetry model and ordinal marginal homogeneity model based on cumulative probabilities. The asymmetric parameter in the proposed models would be useful for making inferences such as that a row variable is stochastically less than a column variable or vice versa. This study also gives a separation of the cumulative ordinal quasi-symmetry model, that is, the cumulative ordinal quasi-symmetry model holds if and only if both the cumulative quasi-symmetry and ordinal marginal homogeneity models hold. The usefulness of the proposed models and the proposed separation is demonstrated through real data analysis.