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For square contingency tables, Caussinus[ 1 ] demonstrated that the symmetry model holds if and only if both the quasi-symmetry and marginal homogeneity models hold. Bishop, Fienberg, and Holland[ 2 , p. 307] and Bhapkar and Darroch[ 3 ] provided similar theorems for multi-way tables. The present article proposes a moment symmetry model and unique decompositions of the symmetry model. For two-way tables, the second-order moment symmetry model decomposes the symmetry model into the second-order moment symmetry and marginal homogeneity models, while the [Formula: see text]th-order moment symmetry model decomposes the [Formula: see text]th-order marginal symmetry model using the [Formula: see text]th-order marginal symmetry model for multi-way tables. AMS 2000 subject classification: 62H17

Summary For square contingency tables with ordered categories, Iki, Tahata and Tomizawa (2012) considered a measure to represent the degree of departure from marginal homogeneity. However, the maximum value of this measure cannot distinguish two kinds of marginal inhomogeneity. The present paper proposes a measure which can distinguish two kinds of marginal inhomogeneity. In particular, the proposed measure is useful for representing the degree of departure from marginal homogeneity when the marginal cumulative logistic model holds.

For square contingency tables with ordered categories, Iki, Ishihara and Tomizawa (2013) considered the t-distribution type symmetry model and Iki, Okada and Tomizawa (2018) extended this model. These models are appropriate for a square contingency table if it is reasonable to assume an underlying bivariate t-distribution having any degrees of freedom. This study proposes three kinds of parsimonious models for these models. Additionally, this paper provides the decompositions of the parsimonious symmetry model using the proposed model. Some simulation studies based on bivariate t-distribution show the performances of the proposed models.

The purpose of this paper is to propose a new model of asymmetry for square contingency tables with ordered categories. The new model may be appropriate for a square contingency table if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances having any degrees of freedom. As the degrees of freedom becomes larger, the proposed model approaches the extended linear diagonals-parameter symmetry model, which may be appropriate for a square table if it is reasonable to assume an underlying bivariate normal distribution. The simulation study based on bivariate t-distribution is given. An example is given.

For the analysis of two-way contingency tables with ordered categories, Yamamoto, Tahata, Suzuki, and Tomizawa (2011) considered a measure to represent the degree of departure from marginal point-symmetry. The maximum value of the measure cannot distinguish two kinds of marginal complete asymmetry with respect to the midpoint. A measure is proposed which can distinguish two kinds of marginal asymmetry with respect to the midpoint. It also gives large-sample confidence interval for the proposed measure.

For the analysis of square contingency tables with the same row and column ordinal classifications, this article proposes new models which indicate the structures of symmetry with respect to the anti-diagonal of the table. Also, this article gives a simple decomposition in 3 ′ 3 contingency table using the proposed models. The proposed models are applied to grip strength data.

For square contingency tables, we propose a quasi-symmetry model with an exponential form along subdiagonal and give the theorem that Tomizawa’s (1992) diagonal exponent symmetry model holds if and only if the proposed model and marginal means equality model hold with the orthogonality of test statistics.

For the analysis of square contingency tables with the same row and column ordinal classifications, McCullagh (1978) considered the palindromic symmetry (PS) model, which has a multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (> i) or above. The present paper proposes a modified PS model which indicates that (1) the symmetric odds for cumulative probabilities with distance 1 from main diagonal of the table are constant and (2) there is the structure of quasi-symmetry for them with distance k (≥ 2). Also the present paper gives the decomposition of the symmetry model using the proposed model. Examples are given.

For square contingency tables with ordered categories, the present paper proposes an asymmetry model with m-additional parameters, which indicates (1) the generalized marginal homogeneity and (2) the structure of quasi-symmetry for cumulative probabilities. The proposed model includes a modified palindromic symmetry model by Iki, Oda and Tomizawa [7]. Also the present paper gives the decomposition of the symmetry model using the proposed model. Examples are given.

For square contingency tables with ordered categories, this paper proposes a measure to represent the degree of departure from the marginal homogeneity model. It is expressed as the weighted sum of the power-divergence or Patil–Taillie diversity index, and is a function of marginal log odds ratios. The measure represents the degree of departure from the equality of the log odds that the row variable is i or below instead of i+1 or above and the log odds that the column variable is i or below instead of i+1 or above for every i. The measure is also extended to multi-way tables. Examples are given.

For two-way contingency tables with ordered categories, Tomizawa (1992) considered the parsimonious Linear-by-Linear association model. This model can be described in terms of fewer parameters than the Linear-by-Linear association model (Agresti, 1983). The purpose of this paper is (i) to define the parsimonious independence model, (ii) to show the parsimonious independence model holds if and only if the parsimonious Linear-by-Linear association model holds and the each one of various correlation coefficients is equal to zero, and (iii) show the statistic for testing the parsimonious independence model is asymptotically equivalent to the sum of test statistics for the decomposed models.

Problem statement: For square contingency tables with ordered categor ies, if the Marginal Homogeneity (MH) model holds, then the probability that X selected at random from the row marginal distribution is less than Y selected independently at random from the column marginal distribution equals the probability that X is greater than Y. Ho wever, the converse does not hold. We are intereste d in the condition in order that the converse holds. Approach: This study gave a decomposition theorem that the MH model holds if and only if both the mar ginal cumulative logistic model and the model of equality of mean ridits for the row and column marg inal distributions hold. Examples are given. Results: For the data of cross-classification of ewes accor ding to number of lambs born in consecutive years and cross-classification of unaided distance vision of women in British, the decomposition of th e MH model is applied and the detailed analysis is gi ven. Conclusion: When the MH model fits the data poorly, this decomposition is useful for seeing whi ch of decomposed two models influences stronger.