Moment Symmetry Models and Decompositions of Symmetry for Multi-way Contingency Tables

For square contingency tables, Caussinus[ 1 ] demonstrated that the symmetry model holds if and only if both the quasi-symmetry and marginal homogeneity models hold. Bishop, Fienberg, and Holland[ 2 , p. 307] and Bhapkar and Darroch[ 3 ] provided similar theorems for multi-way tables. The present article proposes a moment symmetry model and unique decompositions of the symmetry model. For two-way tables, the second-order moment symmetry model decomposes the symmetry model into the second-order moment symmetry and marginal homogeneity models, while the [Formula: see text]th-order moment symmetry model decomposes the [Formula: see text]th-order marginal symmetry model using the [Formula: see text]th-order marginal symmetry model for multi-way tables. AMS 2000 subject classification: 62H17