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

Sadao Tomizawa; Kouji Tahata

Journal de la société française de statistique (2007)

  • Volume: 148, Issue: 3, page 3-36
  • ISSN: 1962-5197

Abstract

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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.

How to cite

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Tomizawa, Sadao, and 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 148.3 (2007): 3-36. <http://eudml.org/doc/93465>.

@article{Tomizawa2007,
abstract = {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.},
author = {Tomizawa, Sadao, Tahata, Kouji},
journal = {Journal de la société française de statistique},
keywords = {association model; decomposition; independence; likelihood ratio statistic; marginal homogeneity; marginal symmetry; measure; model; orthogonality; quasi-symmetry; separability; square contingency table; symmetry},
language = {eng},
number = {3},
pages = {3-36},
publisher = {Société française de statistique},
title = {The analysis of symmetry and asymmetry : orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables},
url = {http://eudml.org/doc/93465},
volume = {148},
year = {2007},
}

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
PY - 2007
PB - Société française de statistique
VL - 148
IS - 3
SP - 3
EP - 36
AB - 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.
LA - eng
KW - association model; decomposition; independence; likelihood ratio statistic; marginal homogeneity; marginal symmetry; measure; model; orthogonality; quasi-symmetry; separability; square contingency table; symmetry
UR - http://eudml.org/doc/93465
ER -

References

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