Families of Generalized Quasisymmetry Models: A ϕ-Divergence Approach

The quasisymmetry (QS) model for square contingency tables is revisited, highlighting properties and features on the basis of its alternative definitions. More parsimonious QS-type models, such as the ordinal QS model for ordinal classification variables and models based on association models (AMs) with homogeneous row and column scores, are discussed. All these models are linked to the local odds ratios (LOR). QS-type models and AMs were extended in the literature for generalized odds ratios other than LOR. Furthermore, in an information-theoretic context, they are expressed as distance models from a parsimonious reference model (the complete symmetry for QS and the independence for AMs), while they satisfy closeness properties with respect to Kullback–Leibler (KL) divergence. Replacing the KL by ϕ divergence, flexible classes of QS-type models for LOR, AMs for LOR, and AMs for generalized odds ratios were generated. However, special QS-type models that are based on homogeneous AMs for LOR have not been extended to ϕ-divergence-based classes so far, or the QS-type models for generalized odds ratios. In this work, we develop these missing extensions, and discuss QS-type models and their generalizations in depth. These flexible families enrich the modeling options, leading to models of better fit and sound interpretation, as illustrated by representative examples.

Orthogonal decomposition of the sum-symmetry model for square contingency tables with ordinal categories: Use of the exponential sum-symmetry model

Summary In the existing decomposition theorem, the sum-symmetry model holds if and only if both the exponential sum-symmetry and global symmetry models hold. However, this decomposition theorem does not satisfy the asymptotic equivalence for the test statistic. To address the aforementioned gap, this study establishes a decomposition theorem in which the sum-symmetry model holds if and only if both the exponential sum-symmetry and weighted global-sum-symmetry models hold. The proposed decomposition theorem satisfies the asymptotic equivalence for the test statistic. We demonstrate the advantages of the proposed decomposition theorem by applying it to datasets comprising real data and artificial data.

A Generalized Two-Dimensional Index to Measure the Degree of Deviation from Double Symmetry in Square Contingency Tables

The double symmetry model satisfies both the symmetry and point symmetry models simultaneously. To measure the degree of deviation from the double symmetry model, a two-dimensional index that can concurrently measure the degree of deviation from symmetry and point symmetry is considered. This two-dimensional index is constructed by combining two existing indexes. Although the existing indexes are constructed using power divergence, the existing two-dimensional index that can concurrently measure both symmetries is constructed using only Kullback-Leibler information, which is a special case of power divergence. Previous studies note the importance of using several indexes of divergence to compare the degrees of deviation from a model for several square contingency tables. This study, therefore, proposes a two-dimensional index based on power divergence in order to measure deviation from double symmetry for square contingency tables. Numerical examples show the utility of the proposed two-dimensional index using two datasets.

Two-Dimensional Index of Departure from the Symmetry Model for Square Contingency Tables with Nominal Categories

In the analysis of two-way contingency tables, the degree of departure from independence is measured using measures of association between row and column variables (e.g., Yule’s coefficients of association and of colligation, Cramér’s coefficient, and Goodman and Kruskal’s coefficient). On the other hand, in the analysis of square contingency tables with the same row and column classifications, we are interested in measuring the degree of departure from symmetry rather than independence. Over past years, many studies have proposed various types of indexes based on their power divergence (or diversity index) to represent the degree of departure from symmetry. This study proposes a two-dimensional index to measure the degree of departure from symmetry in terms of the log odds of each symmetric cell with respect to the main diagonal of the table. By measuring the degree of departure from symmetry in terms of the log odds of each symmetric cell, the analysis results are easier to interpret than existing indexes. Numerical experiments show the utility of the proposed two-dimensional index. We show the usefulness of the proposed two-dimensional index by using real data.

Measure of Departure from Conditional Symmetry Based on Cumulative Probabilities for Square Contingency Tables

For the analysis of square contingency tables with ordered categories, a measure was developed to represent the degree of departure from the conditional symmetry model in which there is an asymmetric structure of the cell probabilities with respect to the main diagonal of the table. The present paper proposes a novel measure for the departure from conditional symmetry based on the cumulative probabilities from the corners of the square table. In a given example, the proposed measure is applied to Japanese occupational status data, and the interpretation of the proposed measure is illustrated as the departure from a proportional structure of social mobility.