# Generalized Exponential Symmetry Model and Orthogonal Decomposition of Symmetry for Square Tables

Hiroyuki Kurakami*, Kouji Yamamoto, Sadao Tomizawa
Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda City, Chiba, 278-8510, Japan.

© 2011 Kurakami et al.;

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: https://creativecommons.org/licenses/by/4.0/legalcode. This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda City, Chiba, 278-8510, Japan. Tel: +81 (0) 4-7124-1501; Fax: +81 (0) 4-7123-9764; E-mail: h-kurakami@mti.biglobe.ne.jp

## Abstract

For the analysis of square contingency tables with ordered categories, some models that the log odds for two symmetric cell probabilities is a linear function of the row and column values have been considered. This paper proposes a generalization of these models. This paper also proposes the model that the weighted sum of the probability that an observation will fall in one of the cells in upper right triangle of the table is equal to the weighted sum of the probability that it falls in one of the cells in lower left triangle of the table. In addition, this paper gives the theorem that the symmetry model is equivalent to both the proposed models holding simultaneously. Moreover, this paper shows that the likelihood ratio statistic for testing goodness-of-fit of the symmetry model is asymptotically equivalent to the sum of those for testing the proposed models. Examples are given.

Keywords: Generalized exponential symmetry, generalized weighted global symmetry, likelihood ratio statistic, orthogonality, square contingency table, symmetry..