## Generalized marginal homogeneity model and its relation to marginal equimoments for square contingency tables with ordered categories.(English)Zbl 1284.62337

Summary: For square contingency tables with ordered categories, Tomizawa gave theorems that the marginal homogeneity (MH) model is equivalent to certain two or three models holding simultaneously. This paper proposes a generalized MH model, which describes a structure of the odds that an observation will fall in row category $$i$$ or below and column category $$i+1$$ or above, instead of in column category $$i$$ or below and row category $$i+1$$ or above. In addition, this paper gives the theorems that the MH model is equivalent to the generalized MH model and some models holding simultaneously whose each indicates: (1) the equality of $$m$$-order moment of row and column variables, (2) the equality of skewness of them and (3) the equality of kurtosis of them. When the MH model fits the data poorly, these may be useful for seeing the reason for the poor fit; for instance, the poor fit of the MH model is caused by the poor fit of the equality of row and column means rather than the generalized MH model. Examples are given.

### MSC:

 62H17 Contingency tables
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### References:

 [1] Agresti A (1984) Analysis of ordinal categorical data. Wiley, New York · Zbl 0647.62052 [2] Bishop YMM, Fienberg SE, Holland PW (1975) Discrete multivariate analysis: theory and practice. MIT Press, Cambridge [3] Bowker AH (1948) A test for symmetry in contingency tables. J Am Stat Assoc 43: 572–574 · Zbl 0032.17500 [4] Caussinus H (1965) Contribution à l’analyse statistique des tableaux de corrélation. Ann Fac Sci Univ Toulouse 29: 77–182 · Zbl 0168.39904 [5] Darroch JN, Ratcliff D (1972) Generalized iterative scaling for log-linear models. Ann Math Stat 43: 1470–1480 · Zbl 0251.62020 [6] Glass DV (1954) Social mobility in Britain. Routledge & Kegan Paul, London [7] Goodman LA (1979a) Multiplicative models for square contingency tables with ordered categories. Biometrika 66: 413–418 [8] Goodman LA (1979b) Simple models for the analysis of association in cross-classifications having ordered categories. J Am Stat Assoc 74: 537–552 [9] McCullagh P (1978) A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika 65: 413–418 · Zbl 0402.62032 [10] Miyamoto N, Niibe K, Tomizawa S (2005) Decompositions of marginal homogeneity model using cumulative logistic models for square contingency tables with ordered categories. Aust J Stat 34: 361–373 [11] Stuart A (1955) A test for homogeneity of the marginal distributions in a two-way classification. Biometrika 42: 412–416 · Zbl 0066.12502 [12] Tomizawa S (1993a) A decomposition of the marginal homogeneity model for square contingency tables with ordered categories. Calcutta Stat Assoc Bull 43: 123–125 · Zbl 0806.62046 [13] Tomizawa S (1993b) Diagonals-parameter symmetry model for cumulative probabilities in square contingency tables with ordered categories. Biometrics 49: 883–887 · Zbl 0800.62296 [14] Tomizawa S (1995) A generalization of the marginal homogeneity model for square contingency tables with ordered categories. J Educ Behav Stat 20: 349–360 · Zbl 02314625 [15] Tomizawa S (1998) A decomposition of the marginal homogeneity model into three models for square contingency tables with ordered categories. Sankhyā Ser B 60: 293–300 · Zbl 0973.62048 [16] Tomizawa S, Tahata K (2007) The analysis of symmetry and asymmetry: orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables. J Soc Française Stat 148: 3–36 [17] Tomizawa S, Seo T, Yamamoto H (1998) Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories. J Appl Stat 25: 387–398 · Zbl 0936.62066 [18] Yamamoto H, Iwashita T, Tomizawa S (2007) Decomposition of symmetry into ordinal quasi-symmetry and marginal equimoment for multi-way tables. Aust J Stat 36: 291–306
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