In Chapter 5 we presented generalized linear models (GLMs) for binary response variables that assume a binomial random component. GLMs for multicategory response variables assume a multinomial random component. In this chapter we present generalizations of logistic regression for multinomial response variables. Separate models are available for nominal response variables and for ordinal response variables.
In Section 6.1 we present a model for nominal response variables. It uses a separate binary logistic equation for each pair of response categories. An important type of application analyzes effects of explanatory variables on a person's choice from a discrete set of options, such as a choice of product brand to buy. In Section 6.2 we present a model for ordinal response variables. It applies the logit or some other link simultaneously to all the cumulative response probabilities, such as to model whether the importance of religion to a person is below or above some point on a scale (unimportant, slightly important, moderately important, very important). A parsimonious version of the model uses the same effect parameters for each logit. Section 6.3 presents examples and discusses model selection for multicategory responses.
We denote the number of response categories by c. For subject i, let πij denote the probability of response in category j, with ∑cj = 1πij = 1. The category choice is the result of a single multinomial trial. Let