Introduction to Categorical Data Analysis

224 LOGLINEAR MODELS FOR CONTINGENCY TABLES
The loglinear model (WX, XY, YZ) has independence graph
W ——– X ——– Y ——– Z
Here, W and Z are separated by X, byY , and by X and Y . So, W and Z are
independent given X alone or given Y alone or given both X and Y . Also, W and
Y are independent, given X alone or given X and Z, and X and Z are independent,
given Y alone or given Y and W.
7.4.2 Collapsibility Conditions for Three-Way Tables
Sometimes researchers collapse multiway contingency tables to make them simpler
to describe and analyze. However, Section 2.7.5 showed that marginal associations
may differ from conditional associations. For example, if X and Y are conditionally
independent, given Z, they are not necessarily marginally independent. Under the
following collapsibility conditions, a model’s odds ratios are identical in partial tables
as in the marginal table:
For three-way tables, XY marginal and conditional odds ratios are identical if either Z and
X are conditionally independent or if Z and Y are conditionally independent.
The conditions state that the variable treated as the control (Z) is conditionally
independent of X or Y , or both. These conditions correspond to loglinear models
(XY, YZ) and (XY, XZ). That is, the XY association is identical in the partial tables
and the marginal table for models with independence graphs
X ——– Y ——– Z and Y ——– X ——– Z
or even simpler models.
For Table 7.3 from Section 7.1.6 with A = alcohol use, C = cigarette use, and
M = marijuana use, the model (AM, CM) of AC conditional independence has
independence graph
A ——– M ——– C
Consider the AM association, identifying C with Z in the collapsibility conditions. In
this model, since C is conditionally independent of A, the AM conditional odds ratios
are the same as the AM marginal odds ratio collapsed over C. In fact, from Table 7.5,
both the fitted marginal and conditional AM odds ratios equal 61.9. Similarly, the
CM association is collapsible. The AC association is not, however. The collapsibility
conditions are not satisfied, because M is conditionally dependent with both A and
C in model (AM, CM). Thus, A and C may be marginally dependent, even though
they are conditionally independent in this model. In fact, from Table 7.5, the model’s
fitted AC marginal odds ratio equals 2.7, not 1.0.