Multivariate Analysis: Discrete Variables (Correspondence Models)

Correspondence models are a special class of statistical models for the association between categorical variables. The specific parametric structure of such models is based, as in the common (descriptive) correspondence analysis, on the canonical form of joint discrete distributions. Bivariate correspondence models are considered first, based on a simple canonical form involving canonical correlations. Such models are discussed in great detail, where interpretation of parameters and inferential aspects receive special attention. Joint multivariate distributions do not have a universal canonical form, and correspondence models for the multivariate case require therefore a modified methodology. Such methodology is presented next, in which discussion on multivariate correspondence models is divided into two cases. One case is where the variables can be partitioned into two natural groups of variables (response and explanatory, say); the second case is where all variables are treated on an equal footing. Correspondence models for three variables are discussed in detail as a special case from which extensions to a higher number of variables is relatively straightforward. Although the main concern here is with inference from correspondence probability models, attention is also given to the use of the parameters of such models as a descriptive tool.