# Polynomial Columns-Parameter Symmetry Model and its Decomposition for Square Contingency Tables

## DOI:

https://doi.org/10.6092/issn.1973-2201/12090## Keywords:

Asymmetry, Marginal symmetry, Odds, Ordered category, Test statistic## Abstract

This study proposes a polynomial columns-parameter symmetry model for square contingency tables with the same row and column ordinal classifications. In the proposed model, the odds for all i < j that an observation will fall in row category i and column category j instead of row category j and column category i depend on only the value of column category j . The proposed model is original because many asymmetry models in square contingency tables depend on the both values of row and column category. The proposed model constantly holds when the columns-parameter symmetry model holds; but the converse does not necessarily hold. This study shows that it is necessary to satisfy the polynomial columns-marginal symmetry model, in addition to the columns-parameter symmetry model, to satisfy the proposed model. This decomposition theorem is useful for explaining why the proposed model does not hold. Moreover, this study shows the value of likelihood ratio chi-square statistic for testing the proposed model is equal to the sum of that for testing the decomposed two models.

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*Statistica*,

*81*(2), 123–134. https://doi.org/10.6092/issn.1973-2201/12090

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