### Example 54.4 Nominal Response Data: 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 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

The LOGISTIC Procedure

Model Information
Data Set WORK.SCHOOL
Response Variable Style
Number of Response Levels 3
Frequency Variable Count
Model generalized logit
Optimization Technique Newton-Raphson

 Number of Observations Read 18 18 338 338

Response Profile
Ordered
Value
Style Total
Frequency
1 self 79
2 team 85
3 class 174

Logits modeled use Style='class' as the reference category.

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=1E-8) 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 Chi-Square 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
Chi-Square
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
Chi-Square
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 main-effects 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);
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 Main-Effects Model

The LOGISTIC Procedure

Model Convergence Status
Convergence criterion (GCONV=1E-8) 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 Chi-Square 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
Chi-Square
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
Chi-Square
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: Model-Predicted Probabilities