17.7 Deriving Models
The saturated model is certainly not the only possible model that would fit these data, and
we are going to search for a model that will fit nearly as well and have a simpler structure.
I am not going to show you output for all possible models, but for an example suppose that
we start with a model that tries to explain the cell frequencies on the basis of Verdict, Fault,
Moral, and the Verdict 3 Moral interaction. (I chose this model for an example almost at
random.) The generating class for this model would be F, MV, but GENLOG requires that
we specify it explicitly as “/Design Verdict Fault Moral Verdict * Moral.”The result fol-
lows in Exhibit 17.2.
Notice that the likelihood ratio chi-square for this model is 40.163 on 5 degrees of free-
dom, which is statistically significant at p 5 .000. Thus this model does not present an ad-
equate fit to the data. Notice that this conclusion is bolstered by the substantial differences
between the observed and expected cell counts. One valuable thing about hierarchical mod-
els is that they allow us to compare individual models by subtracting their corresponding
likelihood ratio chi-squares. For model F, MVin Exhibit 17.2, chi-square 5 40.163 on 5df.
The saturated model had a chi-square 5 0 on 0 df. We can ask the question “Does F, MV
represent a significantly worse fit than F 3 M 3 V?” by taking the difference between the
two values of chi-square and treating that as a chi-square on the difference in the degrees
of freedom. Here
5 40.163 – 0 5 40.163
on 5 – 0 55 df. The critical value for 5 dfis 11.07, which means that the new model fits
significantly worse than the saturated one.
Partly for completeness and partly to help in arriving at an optimal model, I have run
the syntax for each of the 17 possible models. (The eighteenth would be the model with no
predictors.) Some programs will generate all of the possible models on command, but
GENLOG will not. The results of these analyses are presented in Exhibit 17.3. The modelsx^2648 Chapter 17 Log-Linear Analysis
Cell Counts and ResidualsbVerdict Fault Moral
111
2
3
21
2
3
211
2
3
21
2
3Observed
Count
42.500
79.500
32.500
23.500
65.500
17.500
4.500
12.500
8.500
11.500
41.500
24.500%
11.7%
21.8%
8.9%
6.5%
18.0%
4.8%
1.2%
3.4%
2.3%
3.2%
11.4%
6.7%Count
42.500
79.500
32.500
23.500
65.500
17.500
4.500
12.500
8.500
11.500
41.500
24.500%
11.7%
21.8%
8.9%
6.5%
18.0%
4.8%
1.2%
3.4%
2.3%
3.2%
11.4%
6.7%Residual
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000Standardized
Residual
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000Adjusted
Residual.000
.000.000
.000
.000
.000
.000
.000
.000
.000Deviance
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000ExpectedaModel: Poisson
bDesign: Constant 1 Fault 1 Moral 1 Verdict 1 Fault*Moral 1 Verdict*Fault 1 Verdict*Moral 1 Verdict*Fault*MoralExhibit 17.1 (continued)