The analysis of multivariate contingency tables by restricted canonical and restricted association models

Restricted canonical models and restricted association models are proposed and applied to multiway contingency tables. These models have been previously applied to two-way contingency tables; however, multivariate generalization has been impeded in the past, since canonical and association models both depend on singular value decompositions that apply only to two-way arrays. In this article, this restriction to two-way arrays is overcome by division of the cross-classified variables into explanatory and response variables. The explanatory variables are treated as a single polytomous variable, and the response variables are treated as a second single polytomous variable. In this fashion, the multiway table is reduced to a two-way array to which traditional canonical and association models may be applied. Use of linear restrictions on parameters in canonical and association models is especially important in multiway tables if useful models are to be constructed. The class of models considered in this article is sufficiently broad to include models of conditional independence, homogeneity models, conditional symmetry models, log-linear models of no three-factor interaction, conditional quasisymmetry models, and models that express association in terms of preassigned scores. The proposed models can be applied to tables in which some or all variables are ordered as well as to tables with no ordered variables. To illustrate results, appropriate restricted canonical and restricted association models are applied to a three-way cross-classification of abortion attitudes (a response variable) and religion and education (explanatory variables). Insights into the table are obtained that are not readily available from the analysis of Haberman (1979, chap. 6).