Orthogonality of Decompositions of Symmetry into Extended Symmetry and Marginal Equimoment for Multi-way Tables with Ordered Categories

Abstract

Yamamoto, Iwashita, and Tomizawa (2007) gave the decompositions of the symmetry (S) model into the extended S and two or three marginal equimoment models for multi-way tables with ordered categories. However the goodness-of-fit test statistic for the S model is not asymptotically equivalent to the sum of those for the decomposed three or four models. This paper gives, for multi-way tables, the modification of their decompositions such that the test statistic for the S model is asymptotically equivalent to the sum of those for the extended S model and one modified marginal equimoment model, which is the constraints combined the marginal means equality and variances equality or combined the marginal means equality, variances equality and correlations equality. Examples are given.