### Generalized Logits Model

Over the course of one school year, third-graders from three different schools are exposed to three different styles of mathematics instruction: a self-paced computer-learning 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

The CATMOD Procedure

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 main-effects model is fit.

Figure 30.7: Saturated Model: ANOVA Table

Maximum Likelihood Analysis of Variance
Source DF Chi-Square 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 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 30.6. The analysis of variance table is shown in Figure 30.8. The likelihood ratio chi-square statistic is 1.78 with a p-value of 0.7766, indicating a good fit; the Wald chi-square 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: Main-Effects Model: ANOVA Table

Maximum Likelihood Analysis of Variance
Source DF Chi-Square 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.