Introduction to Categorical Data Analysis

208 LOGLINEAR MODELS FOR CONTINGENCY TABLES
Model (7.2) has a single constant parameter (λ), (I − 1) nonredundant λX i parameters,
(J − 1) nonredundant λY j parameters, and (I − 1)(J − 1) nonredundant λXY
ij
parameters. The total number of parameters equals 1 + (I − 1) + (J − 1) + (I − 1)
(J − 1) = IJ. The model has as many parameters as observed cell counts. It is
the saturated loglinear model, having the maximum possible number of parameters.
Because of this, it is the most general model for two-way tables. It describes perfectly
any set of expected frequencies. It gives a perfect fit to the data. The estimated odds
ratios just reported are the same as the sample odds ratios. In practice, unsaturated
models are preferred, because their fit smooths the sample data and has simpler
interpretations.
When a model has two-factor terms, be cautious in interpreting the single-factor
terms. By analogy with two-way ANOVA, when there is two-factor interaction, it can
be misleading to report main effects. The estimates of the main effect terms depend
on the coding scheme used for the higher-order effects, and the interpretation also
depends on that scheme. Normally, we restrict our attention to the highest-order terms
for a variable.
7.1.4 Loglinear Models for Three-Way Tables
With three-way contingency tables, loglinear models can represent various independence
and association patterns. Two-factor association terms describe the conditional
odds ratios between variables.
For cell expected frequencies {μij k}, consider loglinear model
log μij k = λ + λ X i + λY j + λZ k
+ λXZ
ik
+ λYZ
jk
(7.4)
Since it contains an XZ term (λXZ ik ), it permits association between X and Z, controlling
for Y . This model also permits a YZ association, controlling for X. It does not contain
an XY association term. This loglinear model specifies conditional independence
between X and Y , controlling for Z.
We symbolize this model by (XZ, YZ). The symbol lists the highest-order terms
in the model for each variable. This model is an important one. It holds, for instance,
if an association between two variables (X and Y ) disappears when we control for a
third variable (Z).
Models that delete additional association terms are too simple to fit most data
sets well. For instance, the model that contains only single-factor terms, denoted by
(X, Y, Z), is called the mutual independence model. It treats each pair of variables as
independent, both conditionally and marginally. When variables are chosen wisely
for a study, this model is rarely appropriate.
A model that permits all three pairs of variables to have conditional associations is
log μij k = λ + λ X i + λY j + λZ k
+ λXY
ij
+ λXZ
ik
+ λYZ
jk
(7.5)