The analysis of symmetry and asymmetry : orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables

Tomizawa, Sadao; Tahata, Kouji

Journal de la Société française de statistique & Revue de statistique appliquée, Tome 148 (2007) no. 3, pp. 3-36.

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.

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.

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