For square contingency tables with ordered categories, the present paper gives several theorems that the symmetry model holds if and only if the generalized linear diagonals-parameter symmetry model for cell probabilities and for cumulative probabilities and the mean nonequality model of row and column variables hold. It also shows the orthogonality of statistic for testing goodness-of-fit of the symmetry model. An example is given.
Cite this paper
K. Yamamoto, M. Ohama and S. Tomizawa, "Decompositions of Symmetry Using Generalized Linear Diagonals-Parameter Symmetry Model and Orthogonality of Test Statistic for Square Contingency Tables," Open Journal of Statistics
, Vol. 3 No. 6, 2013, pp. 9-13. doi: 10.4236/ojs.2013.36A002
 A. H. Bowker, “A Test for Symmetry in Contingency Tables,” Journal of the American Statistical Association, Vol. 43, No. 244, 1948, pp. 572-574.http://dx.doi.org/10.1080/01621459.1948.10483284
 A. Agresti, “A Simple Diagonals-Parameter Symmetry and Quasi-Symmetry Model,” Statistics and Probability Letters, Vol. 1, No. 6, 1983, pp. 313-316.http://dx.doi.org/10.1016/0167-7152(83)90051-2
 K. Yamamoto and S. Tomizawa, “Statistical Analysis of Case-Control Data of Endometrial Cancer Based on New Asymmetry Models,” Journal of Biometrics and Biostatistics, Vol. 3, No. 5, 2012, pp. 1-4.http://dx.doi.org/10.4172/2155-6180.1000147
 N. Miyamoto, W. Ohtsuka and S. Tomizawa, “Linear Diagonals-Parameter Symmetry and Quasi-Symmetry Models for Cumulative Probabilities in Square Contingency Tables with Ordered Categories,” Biometrical Journal, Vol. 46, No. 6, 2004, pp. 664-674.http://dx.doi.org/10.1002/bimj.200410066
 H. Yamamoto, T. Iwashita and S. Tomizawa, “Decomposition of Symmetry into Ordinal Quasi-Symmetry and Marginal Equimoment for Multi-way Tables,” Austrian Journal of Statistics, Vol. 36, No. 4, 2007, pp. 291-306.
 K. Yamamoto and S. Tomizawa, “Analysis of Unaided Vision Data Using New Decomposition of Symmetry,” American Medical Journal, Vol. 3, No. 1, 2012, pp. 3742. http://dx.doi.org/10.3844/amjsp.2012.37.42
 J. B. Lang and A. Agresti, “Simultaneously Modeling Joint and Marginal Distributions of Multivariate Categorical Responses,” Journal of the American Statistical Association, Vol. 89, No. 426, 1994, pp. 625-632.http://dx.doi.org/10.1080/01621459.1994.10476787
 J. B. Lang, “On the Partitioning of Goodness-of-Fit Statistics for Multivariate Categorical Response Models,” Journal of the American Statistical Association, Vol. 91, No. 435, 1996, pp. 1017-1023.http://dx.doi.org/10.1080/01621459.1996.10476972
 J. Aitchison, “Large-Sample Restricted Parametric Tests,” Journal of the Royal Statistical Society: Series B, Vol. 24, No. 1, 1962, pp. 234-250.
 C. B. Read, “Partitioning Chi-Square in Contingency Tables: A Teaching Approach,” Communications in Statistics-Theory and Methods, Vol. 6, No. 6, 1977, pp. 553562. http://dx.doi.org/10.1080/03610927708827513
 J. N. Darroch and S. D. Silvey, “On Testing More than One Hypothesis,” Annals of Mathematical Statistics, Vol. 34, No. 2, 1963, pp. 555-567.http://dx.doi.org/10.1214/aoms/1177704168
 S. Tomizawa and K. Tahata, “The Analysis of Symmetry and Asymmetry: Orthogonality of Decomposition of Symmetry into Quasi-Symmetry and Marginal Symmetry for Multi-Way Tables,” Journal de la Société Francaise de Statistique, Vol. 148, No. 3, 2007, pp. 3-36.
 K. Tahata, H. Yamamoto and S. Tomizawa, “Orthogonality of Decompositions of Symmetry into Extended Symmetry and Marginal Equimoment for Multi-Way Tables with Ordered Categories,” Austrian Journal of Statistics, Vol. 37, No. 2, 2008, pp. 185-194.
 K. Tahata and S. Tomizawa, “Orthogonal Decomposition of Point-Symmetry for Multiway Tables,” Advances in Statistical Analysis, Vol. 92, No. 3, 2008, pp. 255-269.http://dx.doi.org/10.1007/s10182-008-0070-5
 M. Haber, “Maximum Likelihood Methods for Linear and Log-Linear Models in Categorical Data,” Computational Statistics and Data Analysis, Vol. 3, No. 1, 1985, pp. 110. http://dx.doi.org/10.1016/0167-9473(85)90053-2
 K. Tahata and S. Tomizawa, “Double Linear DiagonalsParameter Symmetry and Decomposition of Double Symmetry for Square Tables,” Statistical Methods and Applications, Vol. 19, No. 3, 2010, pp. 307-318.http://dx.doi.org/10.1007/s10260-009-0127-y
 C. R. Rao, “Linear Statistical Inference and its Applications,” 2nd Edition, John Wiley, New York, 1973.http://dx.doi.org/10.1002/9780470316436
 A. Agresti, “Analysis of Ordinal Categorical Data,” 2nd Edition, John Wiley, Hoboken, 2010.http://dx.doi.org/10.1002/9780470594001