Double symmetry model and its orthogonal decomposition for multi-way tables

Abstract

For two-way contingency tables, Tomizawa (1985b) gave a theorem that the double symmetry $(DS)$ model holds if and only if both the quasi $DS$ and the marginal $DS$ models hold. The present paper proposes, for multiway tables, the $DS$, some quasi $DS$ and marginal $DS$ models, and extends Tomizawa℉s theorem into multi-way tables. It also shows that for multi-way tables the likelihood ratio statistic for testing goodness of fit of the $DS$ model is asymptotically equivalent to the sum of those for testing the quasi $DS$ model with some order and the marginal $DS$ with the corresponding order. An example is given.