Over the course of one school year, thirdgraders 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 school day supplemented with an afternoon school program), are displayed in Table 30.3. The data set is from Stokes, Davis, and Koch (2000), and it is also analyzed in the section Nominal Response Data: Generalized Logits Model of Chapter 54: The LOGISTIC Procedure.
Table 30.3: School Program Data
Learning 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, so a logistic regression is performed on the generalized logits. The model to be fit is

where is the probability that a student in school h and program i prefers teaching style j, , and style r is the class style. There are separate sets of intercept parameters and regression parameters for each logit, and the matrix is the set of explanatory variables for the population. Thus, two logits are modeled for each school and program combination (population): the logit comparing self to class and the logit comparing team to class.
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; the logits are
formed by comparing self to class and by comparing team to class. The results of this analysis are shown in Figure 30.6 and Figure 30.7.
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 30.6; the variable levels that form the three responses and six populations are listed in the “Response Profiles” and “Population Profiles” tables, respectively.
Figure 30.6: Model Information and Profile Tables
Data Summary  

Response  Style  Response Levels  3 
Weight Variable  Count  Populations  6 
Data Set  SCHOOL  Total Frequency  338 
Frequency Missing  0  Observations  18 
Population Profiles  

Sample  School  Program  Sample Size 
1  1  regular  53 
2  1  afternoon  67 
3  2  regular  64 
4  2  afternoon  64 
5  3  regular  46 
6  3  afternoon  44 
Response Profiles  

Response  Style 
1  self 
2  team 
3  class 
The analysis of variance table is displayed in Figure 30.7. 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 maineffects model is fit.
Figure 30.7: Saturated Model: ANOVA Table
Maximum Likelihood Analysis of Variance  

Source  DF  ChiSquare  Pr > ChiSq 
Intercept  2  40.05  <.0001 
School  4  14.55  0.0057 
Program  2  10.48  0.0053 
School*Program  4  1.74  0.7827 
Likelihood Ratio  0  .  . 
Since PROC CATMOD is an interactive procedure, you can analyze the maineffects 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 30.6. The analysis of variance table is shown in Figure 30.8. 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.
Figure 30.8: MainEffects Model: ANOVA Table
Maximum Likelihood Analysis of Variance  

Source  DF  ChiSquare  Pr > ChiSq 
Intercept  2  39.88  <.0001 
School  4  14.84  0.0050 
Program  2  10.92  0.0043 
Likelihood Ratio  4  1.78  0.7766 
The parameter estimates and tests for individual parameters are displayed in Figure 30.9. The order of the parameters corresponds to the order of the population and response variables as shown in the profile tables
(see Figure 30.6), with the levels of the response variables varying most rapidly. The first response function is the logit that compares
self to class, and the corresponding parameters have Function Number=1. The second logit (Function Number=2) compares team
to class. The School
=1 parameters are the differential effects versus School
=3 for their respective logits, and the School
=2 parameters are likewise differential effects versus School
=3. The Program
parameters are the differential effects of 'regular' versus 'afternoon' for the two response functions.
Figure 30.9: Parameter Estimates
Analysis of Maximum Likelihood Estimates  

Parameter  Function Number 
Estimate  Standard Error 
Chi Square 
Pr > ChiSq  
Intercept  1  0.7979  0.1465  29.65  <.0001  
2  0.6589  0.1367  23.23  <.0001  
School  1  1  0.7992  0.2198  13.22  0.0003 
1  2  0.2786  0.1867  2.23  0.1356  
2  1  0.2836  0.1899  2.23  0.1352  
2  2  0.0985  0.1892  0.27  0.6028  
Program  regular  1  0.3737  0.1410  7.03  0.0080 
regular  2  0.3713  0.1353  7.53  0.0061 
The Program
variable has nearly the same effect on both logits, while School
=1 has the largest effect of the schools.