Introduction to Categorical Data Analysis

7.2 INFERENCE FOR LOGLINEAR MODELS 215
For model (AC, AM, CM), the 95% confidence intervals are (8.0, 49.2) for the AM
conditional odds ratio and (12.5, 23.8) for the CM conditional odds ratio. The intervals
are wide, but these associations also are strong. In summary, this model reveals strong
conditional associations for each pair of drugs. There is a strong tendency for users of
one drug to be users of a second drug, and this is true both for users and for nonusers of
the third drug. Table 7.5 shows that estimated marginal associations are even stronger.
Controlling for outcome on one drug moderates the association somewhat between
the other two drugs.
The analyses in this section pertain to association structure. A different analysis
pertains to comparing marginal distributions, for instance to determine if one drug
has more usage than the others. Section 8.1 presents this type of analysis.
7.2.5 Loglinear Models for Higher Dimensions
Loglinear models are more complex for three-way tables than for two-way tables,
because of the variety of potential association patterns. Basic concepts for models
with three-way tables extend readily, however, to multiway tables.
We illustrate this for four-way tables, with variables W, X, Y , and Z. Interpretations
are simplest when the model has no three-factor terms. Such models are special
cases of (WX, WY, WZ, XY, XZ, YZ), which has homogenous associations. Each pair
of variables is conditionally associated, with the same odds ratios at each combination
of levels of the other two variables. An absence of a two-factor term implies
conditional independence for those variables. Model (WX, WY, WZ, XZ, YZ) does
not contain an XY term, so it treats X and Y as conditionally independent at each
combination of levels of W and Z.
A variety of models have three-factor terms. A model could contain WXY, WXZ,
WYZ,orXYZ terms. The XYZ term permits the association between any pair of those
three variables to vary across levels of the third variable, at each fixed level of W .
The saturated model contains all these terms plus a four-factor term.
7.2.6 Example: Automobile Accidents and Seat Belts
Table 7.9 shows results of accidents in the state of Maine for 68,694 passengers in
autos and light trucks. The table classifies passengers by gender (G), location of
accident (L), seat-belt use (S), and injury (I). The table reports the sample proportion
of passengers who were injured. For each GL combination, the proportion of injuries
was about halved for passengers wearing seat belts.
Table 7.10 displays tests of fit for several loglinear models. To investigate the
complexity of model needed, we consider model (G, I, L, S) containing only singlefactor
terms, model (GI, GL, GS, IL, IS, LS) containing also all the two-factor terms,
and model (GIL, GIS, GLS, ILS) containing also all the three-factor terms. Model
(G, I, L, S) implies mutual independence of the four variables. It fits very poorly
(G 2 = 2792.8,df = 11). Model (GI, GL, GS, IL, IS, LS) fits much better (G 2 =
23.4,df = 5) but still has lack of fit (P