Introduction to Categorical Data Analysis

CHAPTER 6
Multicategory Logit Models
Logistic regression is used to model binary response variables. Generalizations of it
model categorical responses with more than two categories. We will now study models
for nominal response variables in Section 6.1 and for ordinal response variables in
Sections 6.2 and 6.3. As in ordinary logistic regression, explanatory variables can be
categorical and/or quantitative.
At each setting of the explanatory variables, the multicategory models assume that
the counts in the categories of Y have a multinomial distribution. This generalization
of the binomial distribution applies when the number of categories exceeds two (see
Section 1.2.2).
6.1 LOGIT MODELS FOR NOMINAL RESPONSES
Let J denote the number of categories for Y . Let {π1,...,πJ } denote the response
probabilities, satisfying �
j πj = 1. With n independent observations, the probability
distribution for the number of outcomes of the J types is the multinomial. It specifies
the probability for each possible way the n observations can fall in the J categories.
Here, we will not need to calculate such probabilities.
Multicategory logit models simultaneously use all pairs of categories by specifying
the odds of outcome in one category instead of another. For models of this section,
the order of listing the categories is irrelevant, because the model treats the response
scale as nominal (unordered categories).
6.1.1 Baseline-Category Logits
Logit models for nominal response variables pair each category with a baseline
category. When the last category (J ) is the baseline, the baseline-category logits
An Introduction to Categorical Data Analysis, Second Edition. By Alan Agresti
Copyright © 2007 John Wiley & Sons, Inc.
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