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The CATMOD Procedure 
Over the course of one school year, third graders from three different schools are exposed to three different styles of mathematics instruction: a selfpaced computerlearning style, a team approach, and a traditional class approach. The students are asked which style they prefer and their responses, classified by the type of program they are in (a regular school day versus a regular day supplemented with an afternoon school program) are displayed in Table 22.3. The data set is from Stokes, Davis, and Koch (1995).
Table 22.3: School Program DataLearning Style Preference  
School  Program  Self  Team  Class 
1  Regular  10  17  26 
1  Afternoon  5  12  50 
2  Regular  21  17  26 
2  Afternoon  16  12  36 
3  Regular  15  15  16 
3  Afternoon  12  12  20 
The levels of the response variable (self, team, and class) have no essential ordering, hence a logistic regression is performed on the generalized logits. The model to be fit is
The following statements create the data set school and request the analysis. Generalized logits are the default response functions, and maximum likelihood estimation is the default method for analyzing generalized logits, so only the WEIGHT and MODEL statements are required. The option ORDER=DATA means that the response variable levels are ordered as they exist in the data set: self, team, and class; thus the logits are formed by comparing self to class and by comparing team to class. The results of this analysis are shown in Figure 22.8 and Figure 22.9.
data school; length Program $ 9; input School Program $ Style $ Count @@; datalines; 1 regular self 10 1 regular team 17 1 regular class 26 1 afternoon self 5 1 afternoon team 12 1 afternoon class 50 2 regular self 21 2 regular team 17 2 regular class 26 2 afternoon self 16 2 afternoon team 12 2 afternoon class 36 3 regular self 15 3 regular team 15 3 regular class 16 3 afternoon self 12 3 afternoon team 12 3 afternoon class 20 ; proc catmod order=data; weight Count; model Style=School Program School*Program; run;
A summary of the data set is displayed in Figure 22.8; the variable levels that form the three responses and six populations are listed in the "Response Profiles" and "Population Profiles" table, respectively. A table containing the iteration history is also produced, but it is not displayed here.

The analysis of variance table is displayed in Figure 22.9. Since this is a saturated model, there are no degrees of freedom remaining for a likelihood ratio test, and missing values are displayed in the table. The interaction effect is clearly nonsignificant, so a main effects model is fit.
Since PROC CATMOD is an interactive procedure, you can analyze the main effects model by simply submitting the new MODEL statement as follows.
model Style=School Program; run;

You can check the population and response profiles (not shown) to confirm that they are the same as those in Figure 22.8. The analysis of variance table is shown in Figure 22.10. The likelihood ratio chisquare statistic is 1.78 with a pvalue of 0.7766, indicating a good fit; the Wald chisquare tests for the school and program effects are also significant. Since School has three levels, two parameters are estimated for each of the two logits they modeled, for a total of four degrees of freedom. Since Program has two levels, one parameter is estimated for each of the two logits, for a total of two degrees of freedom.

The parameter estimates and tests for individual parameters are displayed in Figure 22.11. The ordering of the parameters corresponds to the order of the population and response variables as shown in the profile tables (see Figure 22.8), with the levels of the response variables varying most rapidly. So, for the first response function, which is the logit that compares self to class, Parameter 1 is the intercept, Parameter 3 is the parameter for the differential effect for School=1, Parameter 5 is the parameter for the differential effect for School=2, and Parameter 7 is the parameter for the differential effect for Program=regular. The even parameters are interpreted similarly for the second logit, which compares team to class.
The Program variable (Parameters 7 and 8) has nearly the same effect on both logits, while School=1 (Parameters 3 and 4) has the largest effect of the schools.
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