Introduction to Categorical Data Analysis

222 LOGLINEAR MODELS FOR CONTINGENCY TABLES
Table 7.12. Equivalent Loglinear and Logistic Models for
a Three-Way Table With Binary Response Variable Y
Loglinear Symbol Logistic Model Logistic Symbol
(Y, XZ) α (—)
(XY, XZ) α + β X i (X)
(YZ, XZ) α + β Z k (Z)
(XY, YZ, XZ) α + β X i + βZ k (X + Z)
(XYZ) α + β X i + βZ k
+ βXZ
ik
(X*Z)
categories, relevant loglinear models correspond to baseline-category logit models
(Section 6.1). In such cases it is more sensible to fit logistic models directly, rather
than loglinear models. Indeed, one can see by comparing equations (7.8) and (7.9)
how much simpler the logistic structure is. The loglinear approach is better suited
for cases with more than one response variable, as in studying association patterns
for the drug use example in Section 7.1.6. In summary, loglinear models are most
natural when at least two variables are response variables and we want to study their
association structure. Otherwise, logistic models are more relevant.
Selecting a loglinear model becomes more difficult as the number of variables
increases, because of the increase in possible associations and interactions. One
exploratory approach first fits the model having only single-factor terms, the model
having only two-factor and single-factor terms, the model having only three-factor
and lower terms, and so forth, as Section 7.2.6 showed. Fitting such models often
reveals a restricted range of good-fitting models.
When certain marginal totals are fixed by the sampling design or by the response–
explanatory distinction, the model should contain the term for that margin. This
is because the ML fit forces the corresponding fitted totals to be identical to those
marginal totals. To illustrate, suppose one treats the counts {ng+ℓ+} in Table 7.9 as
fixed at each combination of levels of G = gender and L = location. Then a loglinear
model should contain the GL two-factor term, because this ensures that {ˆμg+ℓ+ =
ng+ℓ+}. That is, the model should be at least as complex as model (GL,S,I).If
20,629 women had accidents in urban locations, then the fitted counts have 20,629
women in urban locations.
Related to this point, the modeling process should concentrate on terms linking
response variables and terms linking explanatory variables to response variables.
Allowing a general interaction term among the explanatory variables has the effect
of fixing totals at combinations of their levels. If G and L are both explanatory
variables, models assuming conditional independence between G and L are not of
interest.
For Table 7.9, I is a response variable, and S might be treated either as a response
or explanatory variable. If it is explanatory, we treat the {ng+ℓs} totals as fixed and fit
logistic models for the I response. If S is also a response, we consider the {ng+ℓ+}
totals as fixed and consider loglinear models that are at least as complex as (GL,S,I).