Loglinear modeling with mixed numerical and categorical predictor variables through an Extended Stereotype Model

Loglinear analysis is most useful when we have two or more categorical response variables. Loglinear analysis, however, requires categorical predictor variables, such that the data can be represented in a contingency table. Researchers often have a mix of categorical and numerical predictors. We present a new statistical methodology for the analysis of multiple categorical response variables with a mix of numeric and categorical predictor variables. Therefore, the stereotype model, a reduced rank regression model for multinomial outcome variables, is extended with a design matrix for the profile scores and one for the dependencies among the responses. An MM algorithm is presented for estimation of the model parameters. Three examples are presented. The first shows that our method is equivalent to loglinear analysis when we only have categorical variables. With the second example, we show the differences between marginal logit models and our extended stereotype model, which is a conditional model. The third example is more extensive, and shows how to analyze a data set, how to select a model, and how to interpret the final model.