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 française de statistique & Revue de statistique appliquée, Volume 148 (2007) no. 3, pp. 3-36.

For the analysis of square contingency tables, Caussinus (1965) proposed the quasi-symmetry model and gave the theorem that the symmetry model holds if and only if both the quasi-symmetry and the marginal homogeneity models hold. Bishop, Fienberg and Holland (1975, p.307) pointed out that the similar theorem holds for three-way tables. Bhapkar and Darroch (1990) gave the similar theorem for general multi-way tables. The purpose of this paper is (1) to review some topics on various symmetry models, which include the models, the decompositions of models, and the measures of departure from models, on various symmetry and asymmetry, and (2) to show that for multi-way tables, the likelihood ratio statistic for testing goodness-of-fit of the complete symmetry model is asymptotically equivalent to the sum of those for testing the quasi-symmetry model with some order and the marginal symmetry model with the corresponding order.

Pour l'analyse des tableaux carrés, Caussinus (1965) a proposé le modèle de quasi-symétrie et montré qu'un tableau est symétrique si et seulement s'il satisfait à la fois quasi-symétrie et égalité des distributions marginales. Bishop, Fienberg et Holland (1975, p. 307) ont noté qu'un théorème semblable valait pour les tableaux à trois dimensions, tandis que Bhapkar et Darroch l'ont donné pour des tableaux de dimension quelconque. Le but de cet article est (1) de passer en revue les questions de symétrie, les modèles eux-mêmes, leur décomposition et les mesures d'écart au modèle pour divers concepts de symétrie et asymétrie, (2) de montrer que, pour les tableaux multiples, la statistique du rapport de vraisemblance pour tester la symétrie est asymptotiquement équivalente à la somme des statistiques analogues testant respectivement la quasi-symétrie d'un certain ordre et l'égalité des marges pour l'ordre correspondant.

Keywords: association model, decomposition, independence, likelihood ratio statistic, marginal homogeneity, marginal symmetry, measure, model, orthogonality, quasi-symmetry, separability, square contingency table, symmetry
@article{JSFS_2007__148_3_3_0,
author = {Tomizawa, Sadao and Tahata, Kouji},
title = {The analysis of symmetry and asymmetry : orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables},
journal = {Journal de la Soci\'et\'e fran\c{c}aise de statistique & Revue de statistique appliqu\'ee},
pages = {3--36},
publisher = {Soci\'et\'e fran\c{c}aise de statistique},
volume = {148},
number = {3},
year = {2007},
language = {en},
url = {http://www.numdam.org/item/JSFS_2007__148_3_3_0/}
}
TY  - JOUR
AU  - Tomizawa, Sadao
AU  - Tahata, Kouji
TI  - The analysis of symmetry and asymmetry : orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables
JO  - Journal de la Société française de statistique & Revue de statistique appliquée
PY  - 2007
DA  - 2007///
SP  - 3
EP  - 36
VL  - 148
IS  - 3
PB  - Société française de statistique
UR  - http://www.numdam.org/item/JSFS_2007__148_3_3_0/
LA  - en
ID  - JSFS_2007__148_3_3_0
ER  - 
%0 Journal Article
%A Tahata, Kouji
%T The analysis of symmetry and asymmetry : orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables
%J Journal de la Société française de statistique & Revue de statistique appliquée
%D 2007
%P 3-36
%V 148
%N 3
%I Société française de statistique
%G en
%F JSFS_2007__148_3_3_0
Tomizawa, Sadao; Tahata, Kouji. 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 française de statistique & Revue de statistique appliquée, Volume 148 (2007) no. 3, pp. 3-36. http://www.numdam.org/item/JSFS_2007__148_3_3_0/

[1] Agresti A. (1983a). A survey of strategies for modeling cross-classifications having ordinal variables. Journal of the American Statistical Association, 78, 184-198. | MR | Zbl

[2] Agresti A. (1983b). Testing marginal homogeneity for ordinal categorical variables. Biometrics, 39, 505-510.

[3] Agresti A. (1983c). A simple diagonals-parameter symmetry and quasi-symmetry model. Statistics and Probability Letters, 1, 313-316. | MR | Zbl

[4] Agresti A. (1984). Analysis of Ordinal Categorical Data. Wiley, New York. | MR | Zbl

[5] Agresti A. (1988). A model for agreement between ratings on an ordinal scale. Biometrics, 44, 539-548. | Zbl

[6] Agresti A. (1989). An agreement model with kappa as parameter. Statistics and Probability Letters, 7, 271-273.

[7] Agresti A. (1995). Logit models and related quasi-symmetric log-linear models for comparing responses to similar items in a survey. Sociological Methods and Research, 24, 68-95.

[8] Agresti A. (2002a). Categorical Data Analysis, 2nd edition. Wiley, New York. | MR | Zbl

[9] Agresti A. (2002b). Links between binary and multi-category logit item response models and quasi-symmetric loglinear models. Annales de la Faculté des Sciences de Toulouse, 11, 443-454. | Numdam | MR | Zbl

[10] Agresti A. and Lang J. B. (1993). Quasi-symmetric latent class models, with application to rater agreement. Biometrics, 49, 131-139.

[11] Agresti A. and Natarajan R. (2001). Modeling clustered ordered categorical data: A survey. International Statistical Review, 69, 345-371.

[12] Aitchison J. (1962). Large-sample restricted parametric tests. Journal of the Royal Statistical Society, Series B, 24, 234-250. | MR | Zbl

[13] Andersen E. B. (1994). The Statistical Analysis of Categorical Data, 3rd edition. Springer, Berlin. | Zbl

[14] Bartolucci F., Forcina A. and Dardanoni V. (2001). Positive quadrant dependence and marginal modeling in two-way tables with ordered margins. Journal of the American Statistical Association, 96, 1497-1505. | MR | Zbl

[15] Becker M. P. (1990). Quasisymmetric models for the analysis of square contingency tables. Journal of the Royal Statistical Society, Series B, 52, 369-378. | MR

[16] Bhapkar V. P. (1966). A note on the equivalence of two test criteria for hypotheses in categorical data. Journal of the American Statistical Association, 61, 228-235. | MR | Zbl

[17] Bhapkar V. P. (1979). On tests of marginal symmetry and quasi-symmetry in two and three-dimensional contingency tables. Biometrics, 35, 417-426. | MR | Zbl

[18] Bhapkar V. P. and Darroch J. N. (1990). Marginal symmetry and quasi symmetry of general order. Journal of Multivariate Analysis, 34, 173-184. | MR | Zbl

[19] Bishop Y. M. M., Fienberg S. E. and Holland P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. The MIT Press, Cambridge, Massachusetts. | MR | Zbl

[20] Bowker A. H. (1948). A test for symmetry in contingency tables. Journal of the American Statistical Association, 43, 572-574. | Zbl

[21] Bradley R. A. and Terry M. E. (1952). Rank analysis of incomplete block designs I. The method of paired comparisons. Biometrika, 39, 324-345. | MR | Zbl

[22] Caussinus H. (1965). Contribution à l'analyse statistique des tableaux de corrélation. Annales de la Faculté des Sciences de l'Université de Toulouse, 29, 77-182. | Numdam | MR | Zbl

[23] Caussinus H. (2002). Some concluding observations. Annales de la Faculté des Sciences de Toulouse, 11, 587-591. | Numdam | MR

[24] Caussinus H. and Thélot C. (1976). Note complémentaire sur l'analyse statistique des migrations. Annales de l'INSEE, 22-23, 135-146.

[25] Chuang C., Gheva D. and Odoroff C. (1985). Methods for diagnosing multiplicative-interaction models for two-way contingency tables. Communications in Statistics-Theory and Methods, 14, 2057-2080. | MR | Zbl

[26] Clogg C. C. and Shihadeh E. S. (1994). Statistical Models for Ordinal Variables. Sage Publications, California. | MR

[27] Cohen J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20, 37-46.

[28] Constantine A. G. and Gower J. C. (1978). Graphical representation of asymmetric matrices. Applied Statistics, 27, 297-304. | MR | Zbl

[29] Cressie N. A. C. and Read T. R. C. (1984). Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, Series B, 46, 440-464. | MR | Zbl

[30] Darroch J. N. and Mccloud P. I. (1986). Category distinguishability and observer agreement. Australian Journal of Statistics, 28, 371-388. | MR | Zbl

[31] Darroch J. N. and Silvey S. D. (1963). On testing more than one hypothesis. Annals of Mathematical Statistics, 34, 555-567. | MR | Zbl

[32] Dossou-Gbété S. and Grorud A. (2002). Biplots for matched two-way tables. Annales de la Faculté des Sciences de Toulouse, 11, 469-483. | Numdam | MR | Zbl

[33] Erosheva E. A., Fienberg S. E. and Junker B. W. (2002). Alternative statistical models and representations for large sparse multi-dimensional contingency tables. Annales de la Faculté des Sciences de Toulouse, 11, 485-505. | Numdam | MR | Zbl

[34] Everitt B. S. (1992). The Analysis of Contingency Tables, 2nd edition. Chapman and Hall, London. | MR | Zbl

[35] De Falguerolles A. and Van Der Heijden, P. G. M. (2002). Reduced rank quasi-symmetry and quasi-skew symmetry: a generalized bi-linear model approach. Annales de la Faculté des Sciences de Toulouse, 11, 507-524. | Numdam | MR | Zbl

[36] Gilula Z. and Haberman S. J. (1988). The analysis of multivariate contingency tables by restricted canonical and restricted association models. Journal of the American Statistical Association, 83, 760-771. | MR | Zbl

[37] Goodman L. A. (1972). Some multiplicative models for the analysis of cross-classified data. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 1, 649-696. | MR | Zbl

[38] Goodman L. A. (1979a). Multiplicative models for square contingency tables with ordered categories. Biometrika, 66, 413-418.

[39] Goodman L. A. (1979b). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74 537-552. | MR

[40] Goodman L. A. (1981a). Association models and canonical correlation in the analysis of cross-classifications having ordered categories. Journal of the American Statistical Association, 76, 320-334. | MR

[41] Goodman L. A. (1981b). Association models and the bivariate normal for contingency tables with ordered categories. Biometrika, 68, 347-355. | MR | Zbl

[42] Goodman L. A. (1985). The analysis of cross-classified data having ordered and/or unordered categories: association models, correlation models, and asymmetry models for contingency tables with or without missing entries. Annals of Statistics, 13, 10-69. | MR | Zbl

[43] Goodman L. A. (1986). Some useful extensions of the usual correspondence analysis approach and the usual log-linear models approach in the analysis of contingency tables. International Statistical Review, 54, 243-309. | MR | Zbl

[44] Goodman L. A. (2002). Contributions to the statistical analysis of contingency tables: Notes on quasi-symmetry, quasi-independence, log-linear models, log-bilinear models, and correspondence analysis models. Annales de la Faculté des Sciences de Toulouse, 11, 525-540. | Numdam | MR | Zbl

[45] Gower J. C. (1977). The analysis of asymmetry and orthogonality. In Recent Developments in Statistics, J. R. Barra et al. ed., North-Holland, Amsterdam, 109-123. | MR | Zbl

[46] Greenacre M. (2000). Correspondence analysis of square asymmetric matrices. Applied Statistics, 49, 297-310. | MR | Zbl

[47] Grizzle J. E., Starmer C. F. and Koch G. G. (1969). Analysis of categorical data by linear models. Biometrics, 25, 489-504. | MR | Zbl

[48] Haber M. (1985). Maximum likelihood methods for linear and log-linear models in categorical data. Computational Statistics and Data Analysis, 3, 1-10. | MR | Zbl

[49] Haberman S. J. (1979). Analysis of Qualitative Data, Volume 2. Academic Press, New York.

[50] Ireland C. T., Ku H. H. and Kullback S. (1969). Symmetry and marginal homogeneity of an $r×r$ contingency table. Journal of the American Statistical Association, 64, 1323-1341. | MR

[51] Lang J. B. (1996). On the partitioning of goodness-of-fit statistics for multivariate categorical response models. Journal of the American Statistical Association, 91, 1017-1023. | MR | Zbl

[52] Lang J. B. and Agresti A. (1994). Simultaneously modeling joint and marginal distributions of multivariate categorical responses. Journal of the American Statistical Association, 89, 625-632. | Zbl

[53] Lovison G. (2000). Generalized symmetry models for hypercubic concordance tables. International Statistical Review, 68, 323-338. | Zbl

[54] Mccullagh P. (1977). A logistic model for paired comparisons with ordered categories. Biometrika, 64, 449-453. | MR | Zbl

[55] Mccullagh P. (1978). A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika, 65, 413-418. | Zbl

[56] Mccullagh P. (1982). Some applications of quasisymmetry. Biometrika, 69, 303-308. | MR | Zbl

[57] Mccullagh P. (2002). Quasi-symmetry and representation theory. Annales de la Faculté des Sciences de Toulouse, 11, 541-561. | Numdam | MR | Zbl

[58] Miyamoto N., Niibe K. and Tomizawa S. (2005). Decompositions of marginal homogeneity model using cumulative logistic models for square contingency tables with ordered categories. Austrian Journal of Statistics, 34, 361-373.

[59] Miyamoto N., Ohtsuka W. and Tomizawa S. (2004). Linear diagonals-parameter symmetry and quasi-symmetry models for cumulative probabilities in square contingency tables with ordered categories. Biometrical Journal, 46, 664-674. | MR

[60] Miyamoto N., Tahata K., Ebie H. and Tomizawa S. (2006). Marginal inhomogeneity models for square contingency tables with nominal categories. Journal of Applied Statistics, 33, 203-215. | MR | Zbl

[61] Patil G. P. and Taillie C. (1982). Diversity as a concept and its measurement. Journal of the American Statistical Association, 77, 548-561. | MR | Zbl

[62] Plackett P. L. (1981). The Analysis of Categorical Data, 2nd edition. Charles Griffin, London. | Zbl

[63] Rao C. R. (1973). Linear Statistical Inference and Its Applications, 2nd edition. Wiley, New York. | MR | Zbl

[64] Read C. B. (1977). Partitioning chi-square in contingency table: A teaching approach. Communications in Statistics-Theory and Methods, 6, 553-562. | MR | Zbl

[65] Read T. R. C. and Cressie N. A. C. (1988). Goodness-of-Fit Statistics for Discrete Multivariate Data. Springer, New York. | MR | Zbl

[66] Stuart A. (1953). The estimation and comparison of strengths of association in contingency tables. Biometrika, 40, 105-110. | MR | Zbl

[67] Stuart A. (1955). A test for homogeneity of the marginal distributions in a two-way classification. Biometrika, 42, 412-416. | MR | Zbl

[68] Tahata K., Katakura S. and Tomizawa S. (2007). Decompositions of marginal homogeneity model using cumulative logistic models for multi-way contingency tables. Revstat, 5, 163-176. | MR | Zbl

[69] Tahata K., Miyamoto N. and Tomizawa S. (2004). Measure of departure from quasi-symmetry and Bradley-Terry models for square contingency tables with nominal categories. Journal of the Korean Statistical Society, 33, 129-147. | MR

[70] Tahata K. and Tomizawa S. (2006). Decompositions for extended double symmetry model in square contingency tables with ordered categories. Journal of the Japan Statistical Society, 36, 91-106. | MR | Zbl

[71] Tanner M. A. and Young M. A. (1985). Modeling agreement among raters. Journal of the American Statistical Association, 80, 175-180.

[72] Tomizawa S. (1984). Three kinds of decompositions for the conditional symmetry model in a square contingency table. Journal of the Japan Statistical Society, 14, 35-42. | MR | Zbl

[73] Tomizawa S. (1985a). Analysis of data in square contingency tables with ordered categories using the conditional symmetry model and its decomposed models. Environmental Health Perspectives, 63, 235-239.

[74] Tomizawa S. (1985b). The decompositions for point-symmetry models in two-way contingency tables. Biometrical Journal, 27, 895-905. | MR | Zbl

[75] Tomizawa S. (1987). Diagonal weighted marginal homogeneity models and decompositions for linear diagonals-parameter symmetry model. Communications in Statistics-Theory and Methods, 16, 477-488. | MR | Zbl

[76] Tomizawa S. (1989). Decompositions for conditional symmetry model into palindromic symmetry and modified marginal homogeneity models. Australian Journal of Statistics, 31, 287-296. | MR | Zbl

[77] Tomizawa S. (1992a). A decomposition of conditional symmetry model and separability of its test statistic for square contingency tables. Sankhyā, Series B, 54, 36-41. | MR | Zbl

[78] Tomizawa S. (1992b). Multiplicative models with further restrictions on the usual symmetry model. Communications in Statistics-Theory and Methods, 21, 693-710. | MR | Zbl

[79] Tomizawa S. (1992c). An agreement model having structure of symmetry plus main-diagonal equiprobability. Journal of the Korean Statistical Society, 21, 179-185.

[80] Tomizawa S. (1993a). Diagonals-parameter symmetry model for cumulative probabilities in square contingency tables with ordered categories. Biometrics, 49, 883-887. | MR | Zbl

[81] Tomizawa S. (1993b). Orthogonal decomposition of point-symmetry model for two-way contingency tables. Journal of Statistical Planning and Inference, 36, 91-100. | MR | Zbl

[82] Tomizawa S. (1994). Two kinds of measures of departure from symmetry in square contingency tables having nominal categories. Statistica Sinica, 4, 325-334. | MR | Zbl

[83] Tomizawa S. (1995a). Measures of departure from marginal homogeneity for contingency tables with nominal categories. The Statistician, 44, 425-439.

[84] Tomizawa S. (1995b). A generalization of the marginal homogeneity model for square contingency tables with ordered categories. Journal of Educational and Behavioral Statistics, 20, 349-360.

[85] Tomizawa S. (1998). A decomposition of the marginal homogeneity model into three models for square contingency tables with ordered categories. Sankhyā, Series B, 60, 293-300. | MR | Zbl

[86] Tomizawa S. and Makii T. (2001). Generalized measures of departure from marginal homogeneity for contingency tables with nominal categories. Journal of Statistical Research, 35, 1-24. | MR

[87] Tomizawa S., Miyamoto N. and Ashihara N. (2003). Measure of departure from marginal homogeneity for square contingency tables having ordered categories. Behaviormetrika, 30, 173-193. | MR | Zbl

[88] Tomizawa S., Miyamoto N. and Funato R. (2004). Conditional difference asymmetry model for square contingency tables with nominal categories. Journal of Applied Statistics, 31, 271-277. | MR | Zbl

[89] Tomizawa S., Miyamoto N. and Hatanaka Y. (2001). Measure of asymmetry for square contingency tables having ordered categories. Australian and New Zealand Journal of Statistics, 43, 335-349. | MR | Zbl

[90] Tomizawa S., Miyamoto N., Yamamoto K. and Sugiyama A. (2007). Extensions of linear diagonals-parameter symmetry and quasi-symmetry models for cumulative probabilities in square contingency tables. Statistica Neerlandica, 61, 273-283. | MR | Zbl

[91] Tomizawa S., Miyamoto N. and Yamane S. (2005). Power-divergence-type measure of departure from diagonals-parameter symmetry for square contingency tables with ordered categories. Statistics, 39, 107-115. | MR | Zbl

[92] Tomizawa S., Miyamoto N. and Yamamoto K. (2006). Decomposition for polynomial cumulative symmetry model in square contingency tables with ordered categories. Metron, 64, 303-314. | MR

[93] Tomizawa S. and Murata M. (1992). Gauss discrepancy type measure of degree of residuals from symmetry for square contingency tables. Journal of the Korean Statistical Society, 21, 59-69.

[94] Tomizawa S. and Saitoh K. (1999a). Measure of departure from conditional symmetry for square contingency tables with ordered categories. Journal of the Japan Statistical Society, 29, 65-78. | MR | Zbl

[95] Tomizawa S. and Saitoh K. (1999b). Kullback-Leibler information type measure of departure from conditional symmetry and decomposition of measure from symmetry for contingency tables. Calcutta Statistical Association Bulletin, 49, 31-39. | MR | Zbl

[96] Tomizawa S., Seo T. and Yamamoto H. (1998). Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories. Journal of Applied Statistics, 25, 387-398. | MR | Zbl

[97] Upton G. J. G. (1978). The Analysis of Cross-tabulated Data. Wiley, New York. | MR

[98] Van Der Heijden P. G. M., De Falguerolles A. and De Leeuw J. (1989). A combined approach to contingency table analysis using correspondence analysis and log-linear analysis. Applied Statistics, 38, 249-292. | MR | Zbl

[99] Van Der Heijden P. G. M. and Mooijaart A. (1995). Some new log-bilinear models for the analysis of asymmetry in a square contingency table. Sociological Methods and Research, 24, 7-29.

[100] Wall K. D. and Lienert G. A. (1976). A test for point-symmetry in J-dimensional contingency-cubes. Biometrical Journal, 18, 259-264. | Zbl

[101] White A. A., Landis J. R. and Cooper M. M. (1982). A note on the equivalence of several marginal homogeneity test criteria for categorical data. International Statistical Review, 50, 27-34. | MR

[102] Yamamoto H. (2004). A measure of departure from symmetry for multi-way contingency tables with nominal categories. Japanese Journal of Biometrics, 25, 69-88.

[103] Yamamoto K. and Tomizawa S. (2007). Decomposition of measure for marginal homogeneity in square contingency tables with ordered categories. Austrian Journal of Statistics, 36, 105-114.