For square contingency tables that have nominal categories, Tomizawa considered two kinds of measure to represent the degree of departure from symmetry. This paper proposes a generalization of those measures. The proposed measure is expressed by using the average of the power divergence of Cressie and Read, or the average of the diversity index of Patil and Taillie. Special cases of the proposed measure include Tomizawa's measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.
Previously, the diagonals-parameter symmetry model based on f -divergence (denoted by DPS[f ]) was reported to be equivalent to the diagonals-parameter symmetry model regardless of the function f , but the proof was omitted.Here, we derive the DPS[f ] model and the proof of the relation between the two models. We can obtain various interpretations of the diagonalsparameter symmetry model from the result. Additionally, the necessary and sufficient conditions for symmetry and property between test statistics for goodness of fit are discussed.
The decomposition for the complete point symmetry model in a rectangular contingency table is shown. Also the respective decompositions for the local point symmetry model and the reverse local point symmetry model in a square contingency table are given. Moreover teat procedures for the decomposed models and an example are given.
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