Introduction to Categorical Data Analysis

212 LOGLINEAR MODELS FOR CONTINGENCY TABLES
Table 7.6 shows software output, using constraints for which parameters at the second
level of any variable equal 0. Thus, ˆλ AC
22 = ˆλ AC
12 = ˆλ AC
21 = 0, and the estimated
) = exp(2.05) = 7.8.
conditional AC odds ratio is exp(ˆλ AC
11
7.2 INFERENCE FOR LOGLINEAR MODELS
Table 7.5 shows that estimates of conditional and marginal odds ratios are highly
dependent on the model. This highlights the importance of good model selection.
An estimate from this table is informative only if its model fits well. This section
shows how to check goodness of fit, conduct inference, and extend loglinear models
to higher dimensions.
7.2.1 Chi-Squared Goodness-of-Fit Tests
Consider the null hypothesis that a given loglinear model holds.As usual, large-sample
chi-squared statistics assess goodness of fit by comparing the cell fitted values to the
observed counts. In the three-way case, the likelihood-ratio and Pearson statistics are
G 2 = 2 � � �
nij k
nij k log , X
ˆμij k
2 = � (nij k −ˆμij k) 2
ˆμij k
The G 2 statistic is the deviance for the model (recall Section 3.4.3). The degrees of
freedom equal the number of cell counts minus the number of model parameters. The
df value decreases as the model becomes more complex. The saturated model has
df = 0.
For the student drug survey (Table 7.3), Table 7.7 presents goodness-of-fit tests
for several models. For a given df , larger G 2 or X 2 values have smaller P -values
Table 7.7. Goodness-of-Fit Tests for Loglinear Models Relating Alcohol (A),
Cigarette (C), and Marijuana (M) Use
Model G 2 X 2 df P -value ∗
(A, C, M) 1286.0 1411.4 4