Canonical analysis of contingency tables by maximum likelihood

Canonical analysis has often been employed instead of log-linear models to analyze the relationship of two polytomous random variables; however, until the last few years, analysis has been informal. In this article, models are examined that place nontrivial restrictions on the values of the canonical parameters so that a parsimonious description of association is obtained. Maximum likelihood is used to obtain parameter estimates for these restricted models. Approximate confidence intervals are derived for parameters, and chi-squared tests are used to check adequacy of models. The resulting models may be used to determine the appropriateness of latent-class analysis or to determine whether a set of canonical scores has specified patterns. Results are illustrated through analysis of two tables previously analyzed in the statistical literature. Comparisons are made with alternate methods of analysis based on a log-linear parameterization of cell probabilities. It is shown that canonical analysis, which uses interpretations based on regression and correlation, is an alternative to log-linear parameterizations interpreted in terms of cross-product ratios.