Introduction to Categorical Data Analysis

CHAPTER 7
Loglinear Models for Contingency
Tables
Section 3.3.1 introduced loglinear models as generalized linear models (GLMs) for
count data. One use of them is modeling cell counts in contingency tables. The models
specify how the size of a cell count depends on the levels of the categorical variables
for that cell. They help to describe association patterns among a set of categorical
response variables.
Section 7.1 introduces loglinear models. Section 7.2 discusses statistical inference
for model parameters and model checking. When one variable is a binary response
variable, logistic models for that response are equivalent to certain loglinear models.
Section 7.3 presents the connection. We shall see that loglinear models are mainly
of use when at least two variables in a contingency table are response variables.
Section 7.4 introduces graphical representations that portray a model’s association
patterns and indicate when conditional odds ratios are identical to marginal odds ratios.
The loglinear models of Sections 7.1–7.4 treat all variables as nominal. Section 7.5
presents a loglinear model that describes association between ordinal variables.
7.1 LOGLINEAR MODELS FOR TWO-WAY AND
THREE-WAY TABLES
Consider an I × J contingency table that cross-classifies n subjects. When the
responses are statistically independent, the joint cell probabilities {πij } are determined
by the row and column marginal totals,
πij = πi+π+j , i = 1,...,I, j = 1,...,J
An Introduction to Categorical Data Analysis, Second Edition. By Alan Agresti
Copyright © 2007 John Wiley & Sons, Inc.
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