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, Volume 148 (2007) no. 3, p. 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 caise de statistique},
     publisher = {Soci\'et\'e fran\c caise de statistique},
     volume = {148},
     number = {3},
     year = {2007},
     pages = {3-36},
     language = {en},
     url = {http://www.numdam.org/item/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, 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 696864 | Zbl 0508.62047

[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 721443 | Zbl 0528.62050

[4] Agresti A. (1984). Analysis of Ordinal Categorical Data. Wiley, New York. | MR 747468 | Zbl 0647.62052

[5] Agresti A. (1988). A model for agreement between ratings on an ordinal scale. Biometrics, 44, 539-548. | Zbl 0707.62227

[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 1044993 | Zbl 0716.62001

[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 2032351 | Zbl 1044.62117

[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. | Zbl pre02124689

[12] Aitchison J. (1962). Large-sample restricted parametric tests. Journal of the Royal Statistical Society, Series B, 24, 234-250. | MR 140164 | Zbl 0113.13604

[13] Andersen E. B. (1994). The Statistical Analysis of Categorical Data, 3rd edition. Springer, Berlin. | Zbl 0805.62002

[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 1946593 | Zbl 1073.62542

[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 1064423

[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 193708 | Zbl 0147.18402

[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 535778 | Zbl 0419.62045

[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 1073104 | Zbl 0735.62057

[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 381130 | Zbl 0332.62039

[20] Bowker A. H. (1948). A test for symmetry in contingency tables. Journal of the American Statistical Association, 43, 572-574. | Zbl 0032.17500

[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 70925 | Zbl 0047.12903

[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 242341 | Zbl 0168.39904

[23] Caussinus H. (2002). Some concluding observations. Annales de la Faculté des Sciences de Toulouse, 11, 587-591. | Numdam | MR 1508502

[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 812358 | Zbl 0591.62052

[26] Clogg C. C. and Shihadeh E. S. (1994). Statistical Models for Ordinal Variables. Sage Publications, California. | MR 1304491

[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 534997 | Zbl 0436.62049

[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 790631 | Zbl 0571.62017

[30] Darroch J. N. and Mccloud P. I. (1986). Category distinguishability and observer agreement. Australian Journal of Statistics, 28, 371-388. | MR 876747 | Zbl 0609.62140

[31] Darroch J. N. and Silvey S. D. (1963). On testing more than one hypothesis. Annals of Mathematical Statistics, 34, 555-567. | MR 148173 | Zbl 0115.14003

[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 2032353 | Zbl 1042.62056

[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 2032354 | Zbl 1042.62057

[34] Everitt B. S. (1992). The Analysis of Contingency Tables, 2nd edition. Chapman and Hall, London. | MR 1214789 | Zbl 0803.62047

[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 2032355 | Zbl 1044.62081

[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 963804 | Zbl 0662.62063

[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 400611 | Zbl 0248.62032

[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 548257

[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 624334

[41] Goodman L. A. (1981b). Association models and the bivariate normal for contingency tables with ordered categories. Biometrika, 68, 347-355. | MR 626393 | Zbl 0477.62038

[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 773152 | Zbl 0613.62070

[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 963345 | Zbl 0611.62060

[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 2032356 | Zbl 1122.62316

[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 480599 | Zbl 0366.62058

[46] Greenacre M. (2000). Correspondence analysis of square asymmetric matrices. Applied Statistics, 49, 297-310. | MR 1824542 | Zbl 0959.62056

[47] Grizzle J. E., Starmer C. F. and Koch G. G. (1969). Analysis of categorical data by linear models. Biometrics, 25, 489-504. | MR 381144 | Zbl 1149.62317

[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 812132 | Zbl 0586.62084

[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 251877

[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 1424604 | Zbl 0882.62051

[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 0799.62063

[53] Lovison G. (2000). Generalized symmetry models for hypercubic concordance tables. International Statistical Review, 68, 323-338. | Zbl 1107.62335

[54] Mccullagh P. (1977). A logistic model for paired comparisons with ordered categories. Biometrika, 64, 449-453. | MR 478469 | Zbl 0374.62069

[55] 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

[56] Mccullagh P. (1982). Some applications of quasisymmetry. Biometrika, 69, 303-308. | MR 671967 | Zbl 0497.62051

[57] Mccullagh P. (2002). Quasi-symmetry and representation theory. Annales de la Faculté des Sciences de Toulouse, 11, 541-561. | Numdam | MR 2032357 | Zbl 1042.62058

[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 2108610

[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 2224352 | Zbl 1106.62063

[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 675883 | Zbl 0511.62113

[62] Plackett P. L. (1981). The Analysis of Categorical Data, 2nd edition. Charles Griffin, London. | Zbl 0479.62046

[63] Rao C. R. (1973). Linear Statistical Inference and Its Applications, 2nd edition. Wiley, New York. | MR 346957 | Zbl 0256.62002

[64] Read C. B. (1977). Partitioning chi-square in contingency table: A teaching approach. Communications in Statistics-Theory and Methods, 6, 553-562. | MR 440767 | Zbl 0365.62043

[65] Read T. R. C. and Cressie N. A. C. (1988). Goodness-of-Fit Statistics for Discrete Multivariate Data. Springer, New York. | MR 955054 | Zbl 0663.62065

[66] Stuart A. (1953). The estimation and comparison of strengths of association in contingency tables. Biometrika, 40, 105-110. | MR 56240 | Zbl 0050.36405

[67] Stuart A. (1955). A test for homogeneity of the marginal distributions in a two-way classification. Biometrika, 42, 412-416. | MR 72413 | Zbl 0066.12502

[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 2365938 | Zbl pre05217612

[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 2059211

[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 2266419 | Zbl 1134.62341

[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 765026 | Zbl 0556.62031

[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 872764 | Zbl 0579.62037

[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 886358 | Zbl 0655.62062

[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 1039416 | Zbl 0707.62117

[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 1192083 | Zbl 0781.62087

[78] Tomizawa S. (1992b). Multiplicative models with further restrictions on the usual symmetry model. Communications in Statistics-Theory and Methods, 21, 693-710. | MR 1173718 | Zbl 0800.62297

[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 1243499 | Zbl 0800.62296

[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 1234154 | Zbl 0772.62032

[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 1282878 | Zbl 0823.62054

[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 1721527 | Zbl 0973.62048

[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 1891657

[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 2037926 | Zbl 1121.62549

[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 2061383 | Zbl 1121.62499

[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 1859123 | Zbl 0991.62040

[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 2355059 | Zbl 1121.62058

[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 2145474 | Zbl 1115.62331

[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 2352654

[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 1718064 | Zbl 1069.62532

[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 1745525 | Zbl 1132.62330

[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 1647284 | Zbl 0936.62066

[97] Upton G. J. G. (1978). The Analysis of Cross-tabulated Data. Wiley, New York. | MR 513227

[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 1001477 | Zbl 0707.62114

[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 0335.62030

[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 668608

[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.