Models for the Analysis of Association in Multivariate Contingency Tables

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Abstract
The present work describes a family of models that can be used for the analysis of association in multiway cross-classifications. The models are complementary to the usual (i.e., the so-called hierarchical) log-linear models. An advantage of this family of models is that it is frequently possible to give a more parsimonious description of the association than is possible with the usual log-linear models approach. The models will be especially useful when some or all of the variables are ordinal. The models are derived from a reparameterization of the saturated model based on singular value decompositions of interaction terms in the log-linear parameterization. Multivariate association models given in Goodman (1979, 1986), Clogg (1982a,b), Agresti and Kezouh (1983), Gilula and Haberman (1988), and Becker and Clogg (1989) are obtained as special cases. The utility of the models developed in this article is demonstrated with the analysis of two examples. The first example, a cross-classification of Danish youths by education, father's income, and father's social rank, illustrates how association models with multidimensional scores can be used to provide a straightforward description of the association in a seemingly complex table; a graphical display of estimated score parameters facilitates the analysis. The usefulness of association models for modeling three-factor interaction is demonstrated in the second example, a cross-classification of married couples in the 1974 General Social Survey by sex of respondent, husband's highest degree, and wife's highest degree.

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