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**The analysis of symmetry and asymmetry: orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables.**
*(English.
French summary)*
Zbl 1441.62153

Summary: For the analysis of square contingency tables, H. Caussinus [Ann. Fac. Sci. Univ. Toulouse, IV. Sér. 29, 77–183 (1966; Zbl 0168.39904)] 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. Y. M. Bishop, S. E. Fienberg and P. W. Holland [Discrete multivariate analysis: Theory and practice. Cambridge, MA: MIT Press (1975), p. 307] pointed out that the similar theorem holds for three-way tables. V. P. Bhapkar and J. N. Darroch [J. Multivariate Anal. 34, No. 2, 173–184 (1990; Zbl 0735.62057)] 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.

### MSC:

62H17 | Contingency tables |

### Keywords:

association model; decomposition; independence; likelihood ratio statistic; marginal homogeneity; marginal symmetry; measure; model; orthogonality; quasi-symmetry; separability; square contingency table; symmetry
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\textit{S. Tomizawa} and \textit{K. Tahata}, J. Soc. Fr. Stat. \& Rev. Stat. Appl. 148, No. 3, 3--36 (2007; Zbl 1441.62153)

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