Orthogonal decomposition of symmetry model using the ordinal quasi-symmetry model based on \(f\)-divergence for square contingency tables. (English) Zbl 1328.62362

Summary: For square contingency tables, H. Caussinus [Ann. Fac. Sci. Univ. Toulouse, IV. Sér. 29, 77–183 (1966; Zbl 0168.39904)] considered the quasi-symmetry (QS) model. M. Kateri and A. Agresti [Stat. Probab. Lett. 77, No. 6, 598–603 (2007; Zbl 1116.62067)] considered the ordinal quasi-symmetry (OQS[\(f\)]) model based on \(f\)-divergence. The present paper gives a decomposition of the symmetry (\(S\)) model into the OQS[\(f\)] and marginal mean equality models. It also shows that the test statistic for goodness-of-fit of the \(S\) model is asymptotically equivalent to the sum of those for the decomposed models.


62H17 Contingency tables
Full Text: DOI


[1] Agresti, A., Analysis of ordinal categorical data, (2010), Wiley Hoboken, New Jersey · Zbl 1263.62007
[2] Aitchison, J., Large-sample restricted parametric tests, J. R. Stat. Soc. Ser. B, 24, 234-250, (1962) · Zbl 0113.13604
[3] Bishop, Y. M.M.; Fienberg, S. E.; Holland, P. W., Discrete multivariate analysis: theory and practice, (1975), MIT Press Cambridge · Zbl 0332.62039
[4] Bowker, A. H., A test for symmetry in contingency tables, J. Amer. Statist. Assoc., 43, 572-574, (1948) · Zbl 0032.17500
[5] Caussinus, H., Contribution à l’analyse statistique des tableaux de corrélation, Ann. Fac. Sci. Univ. Toulouse, 29, 77-182, (1965) · Zbl 0168.39904
[6] Csiszár, I.; Shields, P., (Information Theory and Statistics: A Tutorial, Foundations and Trends in Communications and Information Theory, vol. 1, (2004)), 417-528
[7] Darroch, J. N.; Silvey, S. D., On testing more than one hypothesis, Ann. Math. Statist., 34, 555-567, (1963) · Zbl 0115.14003
[8] Kateri, M.; Agresti, A., A class of ordinal quasi-symmetry models for square contingency tables, Statist. Probab. Lett., 77, 598-603, (2007) · Zbl 1116.62067
[9] Kateri, M.; Papaioannou, T., Asymmetry models for contingency tables, J. Amer. Statist. Assoc., 92, 1124-1131, (1997) · Zbl 0889.62050
[10] Read, C. B., Partitioning chi-square in contingency tables: a teaching approach, Comm. Statist. Theory Methods, 6, 553-562, (1977) · Zbl 0365.62043
[11] Stuart, A., A test for homogeneity of the marginal distributions in a two-way classification, Biometrika, 42, 412-416, (1955) · Zbl 0066.12502
[12] Tahata, K.; Yamamoto, H.; Tomizawa, S., Orthogonality of decompositions of symmetry into extended symmetry and marginal equimoment for multi-way tables with ordered categories, Austral. J. Statist., 37, 185-194, (2008)
[13] Tallis, G. M., The maximum likelihood estimation of correlation from contingency tables, Biometrics, 18, 342-353, (1962) · Zbl 0107.14005
[14] Tomizawa, S., Three kinds of decompositions for the conditional symmetry model in a square contingency table, J. Japan Statist. Soc., 14, 35-42, (1984) · Zbl 0556.62031
[15] Tomizawa, S., Decomposing the marginal homogeneity model into two models for square contingency tables with ordered categories, Calcutta Statist. Assoc. Bull., 41, 161-164, (1991) · Zbl 0850.62444
[16] Tomizawa, S.; Tahata, K., The analysis of symmetry and asymmetry: orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables, J. Soc. Fr. Statist., 148, 3-36, (2007)
[17] Yamamoto, H.; Iwashita, T.; Tomizawa, S., Decomposition of symmetry into ordinal quasi-symmetry and marginal equimoment for multi-way tables, Austral. J. Statist., 36, 291-306, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.