Some Models for the Multiway Contingency Table with a One-to-One Correspondence among Categories

Goodman (1985) discusses a class of models for the R X R contingency table with a one-to-one correspondence between the row and column variables. The most restrictive model in this class combines the features of symmetry and independence, and the least restrictive is the model of quasi symmetry or symmetric association. In this paper, I extend the class of models discussed by Goodman to the R X R X K (K 2 2) contingency table and to the R X R X R contingency table. Several of the models I discuss have been proposed by previous investigators, but many have not been previously considered. I show how to parameterize these models in new and informative ways, and I develop the relationship between the model parameters and various measures of dependence that are especially useful in tables of the type considered here. To illustrate the types of inferences and interpretations yielded by these models, I present two empirical examples.