Research ArticleOpen AccessGoogle Scholar indexed
Decompositions of Symmetry Using Generalized Linear Diagonals-Parameter Symmetry Model and Orthogonality of Test Statistic for Square Contingency Tables
Department of Medical Innovation, Osaka University Hospital, Osaka, Japan
Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba, Japan
Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba, Japan
- 1 Department of Medical Innovation, Osaka University Hospital, Osaka, Japan
- 2 Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba, Japan
- 3 Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba, Japan
Open Journal of Statistics·Volume 03 (2013)·Pages 9–13·Published 28 December 2013·DOI10.4236/ojs.2013.36A002
Copy link · social · email
Abstract
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.
Keywordsumulative ProbabilityGlobal SymmetryLinear Diagonals-Parameter SymmetryMean EqualityOrdinal CategoryOrthogonal Test Statistic
- 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