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 54.15. The data set is from Stokes, Davis, and Koch (2012), and is also analyzed in the section Generalized Logits Model of Chapter 30: The CATMOD Procedure.
Table 54.15: 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 baseline style (in this case, class). There are separate sets of intercept parameters and regression parameters for each logit, and the vector is the set of explanatory variables for the hith population. Thus, two logits are modeled for each school and program combination: 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. The LINK=GLOGIT option forms the generalized logits. The response variable 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 ODDSRATIO statement produces odds ratios in the presence of interactions, and a graphical display of the requested odds ratios is produced
when ODS Graphics is enabled.
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 ;
ods graphics on; proc logistic data=school; freq Count; class School Program(ref=first); model Style(order=data)=School Program School*Program / link=glogit; oddsratio program; run; ods graphics off;
Summary information about the model, the response variable, and the classification variables are displayed in Output 54.4.1.
Output 54.4.1: Analysis of Saturated Model
Model Information  

Data Set  WORK.SCHOOL 
Response Variable  Style 
Number of Response Levels  3 
Frequency Variable  Count 
Model  generalized logit 
Optimization Technique  NewtonRaphson 
Number of Observations Read  18 

Number of Observations Used  18 
Sum of Frequencies Read  338 
Sum of Frequencies Used  338 
Response Profile  

Ordered Value 
Style  Total Frequency 
1  self  79 
2  team  85 
3  class  174 
Class Level Information  

Class  Value  Design Variables  
School  1  1  0 
2  0  1  
3  1  1  
Program  afternoon  1  
regular  1 
Model Convergence Status 

Convergence criterion (GCONV=1E8) satisfied. 
The “Testing Global Null Hypothesis: BETA=0” table in Output 54.4.2 shows that the parameters are significantly different from zero.
Output 54.4.2: Analysis of Saturated Model
Model Fit Statistics  

Criterion  Intercept Only  Intercept and Covariates 
AIC  699.404  689.156 
SC  707.050  735.033 
2 Log L  695.404  665.156 
Testing Global Null Hypothesis: BETA=0  

Test  ChiSquare  DF  Pr > ChiSq 
Likelihood Ratio  30.2480  10  0.0008 
Score  28.3738  10  0.0016 
Wald  25.6828  10  0.0042 
However, the “Type 3 Analysis of Effects” table in Output 54.4.3 shows that the interaction effect is clearly nonsignificant.
Output 54.4.3: Analysis of Saturated Model
Type 3 Analysis of Effects  

Effect  DF  Wald ChiSquare 
Pr > ChiSq 
School  4  14.5522  0.0057 
Program  2  10.4815  0.0053 
School*Program  4  1.7439  0.7827 
Analysis of Maximum Likelihood Estimates  

Parameter  Style  DF  Estimate  Standard Error 
Wald ChiSquare 
Pr > ChiSq  
Intercept  self  1  0.8097  0.1488  29.5989  <.0001  
Intercept  team  1  0.6585  0.1366  23.2449  <.0001  
School  1  self  1  0.8194  0.2281  12.9066  0.0003  
School  1  team  1  0.2675  0.1881  2.0233  0.1549  
School  2  self  1  0.2974  0.1919  2.4007  0.1213  
School  2  team  1  0.1033  0.1898  0.2961  0.5863  
Program  regular  self  1  0.3985  0.1488  7.1684  0.0074  
Program  regular  team  1  0.3537  0.1366  6.7071  0.0096  
School*Program  1  regular  self  1  0.2751  0.2281  1.4547  0.2278 
School*Program  1  regular  team  1  0.1474  0.1881  0.6143  0.4332 
School*Program  2  regular  self  1  0.0998  0.1919  0.2702  0.6032 
School*Program  2  regular  team  1  0.0168  0.1898  0.0079  0.9293 
The table produced by the ODDSRATIO statement is displayed in Output 54.4.4. The differences between the program preferences are small across all the styles (logits) compared to their variability as displayed by the confidence limits in Output 54.4.5, confirming that the interaction effect is nonsignificant.
Output 54.4.4: Odds Ratios for Style
Odds Ratio Estimates and Wald Confidence Intervals  

Label  Estimate  95% Confidence Limits  
Style self: Program regular vs afternoon at School=1  3.846  1.190  12.435 
Style team: Program regular vs afternoon at School=1  2.724  1.132  6.554 
Style self: Program regular vs afternoon at School=2  1.817  0.798  4.139 
Style team: Program regular vs afternoon at School=2  1.962  0.802  4.799 
Style self: Program regular vs afternoon at School=3  1.562  0.572  4.265 
Style team: Program regular vs afternoon at School=3  1.562  0.572  4.265 
Output 54.4.5: Plot of Odds Ratios for Style
Since the interaction effect is clearly nonsignificant, a maineffects model is fit with the following statements. The EFFECTPLOT statement creates a plot of the predicted values versus the levels of the School
variable at each level of the Program
variables. The CLM option adds confidence bars, and the NOOBS option suppresses the display of the observations.
ods graphics on; proc logistic data=school; freq Count; class School Program(ref=first); model Style(order=data)=School Program / link=glogit; effectplot interaction(plotby=Program) / clm noobs; run; ods graphics off;
All of the global fit tests in Output 54.4.6 suggest the model is significant, and the Type 3 tests show that the school and program effects are also significant.
Output 54.4.6: Analysis of MainEffects Model
Model Convergence Status 

Convergence criterion (GCONV=1E8) satisfied. 
Model Fit Statistics  

Criterion  Intercept Only  Intercept and Covariates 
AIC  699.404  682.934 
SC  707.050  713.518 
2 Log L  695.404  666.934 
Testing Global Null Hypothesis: BETA=0  

Test  ChiSquare  DF  Pr > ChiSq 
Likelihood Ratio  28.4704  6  <.0001 
Score  27.1190  6  0.0001 
Wald  25.5881  6  0.0003 
Type 3 Analysis of Effects  

Effect  DF  Wald ChiSquare 
Pr > ChiSq 
School  4  14.8424  0.0050 
Program  2  10.9160  0.0043 
The parameter estimates, tests for individual parameters, and odds ratios are displayed in Output 54.4.7. The Program
variable has nearly the same effect on both logits, while School
=1 has the largest effect of the schools.
Output 54.4.7: Estimates
Analysis of Maximum Likelihood Estimates  

Parameter  Style  DF  Estimate  Standard Error 
Wald ChiSquare 
Pr > ChiSq  
Intercept  self  1  0.7978  0.1465  29.6502  <.0001  
Intercept  team  1  0.6589  0.1367  23.2300  <.0001  
School  1  self  1  0.7992  0.2198  13.2241  0.0003 
School  1  team  1  0.2786  0.1867  2.2269  0.1356 
School  2  self  1  0.2836  0.1899  2.2316  0.1352 
School  2  team  1  0.0985  0.1892  0.2708  0.6028 
Program  regular  self  1  0.3737  0.1410  7.0272  0.0080 
Program  regular  team  1  0.3713  0.1353  7.5332  0.0061 
Odds Ratio Estimates  

Effect  Style  Point Estimate  95% Wald Confidence Limits 

School 1 vs 3  self  0.269  0.127  0.570 
School 1 vs 3  team  0.519  0.267  1.010 
School 2 vs 3  self  0.793  0.413  1.522 
School 2 vs 3  team  0.622  0.317  1.219 
Program regular vs afternoon  self  2.112  1.215  3.670 
Program regular vs afternoon  team  2.101  1.237  3.571 
The interaction plots in Output 54.4.8 show that School
=1 and Program
=afternoon have a preference for the traditional classroom style. Of course, since these are not simultaneous confidence intervals,
the nonoverlapping 95% confidence limits do not take the place of an actual test.
Output 54.4.8: ModelPredicted Probabilities