Abstract
For multiway contingency tables, Wall and Lienert (Biom. J. 18:259–264, 1976) considered the point-symmetry model. For square contingency tables, Tomizawa (Biom. J. 27:895–905, 1985) gave a theorem that the point-symmetry model holds if and only if both the quasi point-symmetry and the marginal point-symmetry models hold. This paper proposes some quasi point-symmetry models and marginal point-symmetry models for multiway tables, and extends Tomizawa’s (Biom. J. 27:895–905, 1985) theorem into multiway tables. We also show that for multiway tables the likelihood ratio statistic for testing goodness of fit of the point-symmetry model is asymptotically equivalent to the sum of those for testing the quasi point-symmetry model with some order and the marginal point-symmetry model with the corresponding order. An example is given.
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References
Agresti, A.: Categorical Data Analysis, 2nd edn. Wiley, New York (2002)
Aitchison, J.: Large-sample restricted parametric tests. J. R. Stat. Soc. Ser. B 24, 234–250 (1962)
Bhapkar, V.P., Darroch, J.N.: Marginal symmetry and quasi symmetry of general order. J. Multivar. Anal. 34, 173–184 (1990)
Bishop, Y.M.M., Fienberg, S.E., Holland, P.W.: Discrete Multivariate Analysis: Theory and Practice. MIT Press, Cambridge (1975)
Bowker, A.H.: A test for symmetry in contingency tables. J. Am. Stat. Assoc. 43, 572–574 (1948)
Caussinus, H.: Contribution à l’analyse statistique des tableaux de corrélation. Ann. Fac. Sci. Univ. Toulouse 29, 77–182 (1965)
Darroch, J.N., Ratcliff, D.: Generalized iterative scaling for log-linear models. Ann. Math. Stat. 43, 1470–1480 (1972)
Darroch, J.N., Silvey, S.D.: On testing more than one hypothesis. Ann. Math. Stat. 34, 555–567 (1963)
Darroch, J.N., Speed, T.P.: Additive and multiplicative models and interactions. Ann. Stat. 11, 724–738 (1983)
Haber, M.: Maximum likelihood methods for linear and log-linear models in categorical data. Comput. Stat. Data Anal. 3, 1–10 (1985)
Lang, J.B.: On the partitioning of goodness-of-fit statistics for multivariate categorical response models. J. Am. Stat. Assoc. 91, 1017–1023 (1996)
Lang, J.B., Agresti, A.: Simultaneously modeling joint and marginal distributions of multivariate categorical responses. J. Am. Stat. Assoc. 89, 625–632 (1994)
Rao, C.R.: Linear Statistical Inference and Its Applications, 2nd edn. Wiley, New York (1973)
Read, C.B.: Partitioning chi-square in contingency tables: a teaching approach. Commun. Stat. Theory Methods 6, 553–562 (1977)
Stuart, A.: A test for homogeneity of the marginal distributions in a two-way classification. Biometrika 42, 412–416 (1955)
Tomizawa, S.: The decompositions for point symmetry models in two-way contingency tables. Biom. J. 27, 895–905 (1985)
Tomizawa, S.: A decomposition of conditional symmetry model and separability of its test statistic for square contingency tables. Sankhyā Ser. B 54, 36–41 (1992)
Tomizawa, S.: Orthogonal decomposition of point-symmetry model for two-way contingency tables. J. Stat. Plan. Inference 36, 91–100 (1993)
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. Stat. 148, 3–36 (2007)
Wall, K.D., Lienert, G.A.: A test for point-symmetry in J-dimensional contingency-cubes. Biom. J. 18, 259–264 (1976)
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Tahata, K., Tomizawa, S. Orthogonal decomposition of point-symmetry for multiway tables. Adv Stat Anal 92, 255–269 (2008). https://doi.org/10.1007/s10182-008-0070-5
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DOI: https://doi.org/10.1007/s10182-008-0070-5