For square (i.e., two-way) contingency tables with ordered categories, the decomposition of the symmetry model holds into the conditional symmetry and global symmetry models (Read [8]). For three-way contingency tables, Tomizawa [11] considered the conditional symmetry model. For three-way tables with ordered categories, the present paper proposes the global symmetry model and gives the decomposition of the symmetry model into the conditional symmetry and global symmetry models. The present paper also shows that for three-way tables the likelihood ratio statistic for testing goodness-of-fit of symmetry is equal to the sum of those for testing conditional symmetry and global symmetry. The meteorological data are analyzed.

conditional symmetry, global symmetry, likelihood ratio statistic, orthogonality, symmetry.