Models for multicategorical responses: multivariate extensions of generalized linear models

Abstract

In this chapter the concept of generalized linear models is extended to the case of a vector-valued response variable. Consider Example 2.1, where we were interested in the effect of risk factors and antibiotics on infection following birth by Caesarean section. In this example the response was binary, only distinguishing between occurrence and nonoccurrence of infection, and thereby ignores that the data originally provided information on the type of infection (type I or II) as well. It is possible, however, to use this information by introducing a response variable with three categories (no infection / infection type I / infection type II). Naturally, these categories cannot be treated as a unidimensional response. We have to introduce a (dummy) variable for each category, thus obtaining a multivariate response variable. Therefore, link and response functions for the influence term will be vector-valued functions in this chapter. The focus is on multicategorial response variables and multinomial models. Variables of this type are often called polychotomous, the possible values are called categories. Extension to other multivariate exponential family densities is possible but not considered in this text.