A market research firm conducts a survey among undergraduate students at a certain university to evaluate three new Web designs for a commercial Web site targeting undergraduate students at the university.
The sample design is a stratified sample where the strata are students’ classes. Within each class, 300 students are randomly selected by using simple random sampling without replacement. The total number of students in each class in the fall semester of 2001 is shown in the following table:
Class 
Enrollment 

1  Freshman 
3,734 
2  Sophomore 
3,565 
3  Junior 
3,903 
4  Senior 
4,196 
This total enrollment information is saved in the SAS data set Enrollment
by using the following SAS statements:
proc format; value Class 1='Freshman' 2='Sophomore' 3='Junior' 4='Senior'; run; data Enrollment; format Class Class.; input Class _TOTAL_; datalines; 1 3734 2 3565 3 3903 4 4196 ;
In the data set Enrollment
, the variable _TOTAL_
contains the enrollment figures for all classes. They are also the population size for each stratum in this example.
Each student selected in the sample evaluates one randomly selected Web design by using the following scale:
1 
Dislike very much 
2 
Dislike 
3 
Neutral 
4 
Like 
5 
Like very much 
The survey results are collected and shown in the following table, with the three different Web designs coded as A, B, and C.
Evaluation of New Web Designs 


Rating Counts 

Strata 
Design 
1 
2 
3 
4 
5 
Freshman 
A 
10 
34 
35 
16 
15 
B 
5 
6 
24 
30 
25 

C 
11 
14 
20 
34 
21 

Sophomore 
A 
19 
12 
26 
18 
25 
B 
10 
18 
32 
23 
26 

C 
15 
22 
34 
9 
20 

Junior 
A 
8 
21 
23 
26 
22 
B 
1 
4 
15 
33 
47 

C 
16 
19 
30 
23 
12 

Senior 
A 
11 
14 
24 
33 
18 
B 
8 
15 
25 
30 
22 

C 
2 
34 
30 
18 
16 
The survey results are stored in a SAS data set WebSurvey
by using the following SAS statements:
proc format; value Design 1='A' 2='B' 3='C'; value Rating 1='dislike very much' 2='dislike' 3='neutral' 4='like' 5='like very much'; run; data WebSurvey; format Class Class. Design Design. Rating Rating.; do Class=1 to 4; do Design=1 to 3; do Rating=1 to 5; input Count @@; output; end; end; end; datalines; 10 34 35 16 15 8 21 23 26 22 5 10 24 30 21 1 14 25 23 37 11 14 20 34 21 16 19 30 23 12 19 12 26 18 25 11 14 24 33 18 10 18 32 23 17 8 15 35 30 12 15 22 34 9 20 2 34 30 18 16 ; data WebSurvey; set WebSurvey; if Class=1 then Weight=3734/300; if Class=2 then Weight=3565/300; if Class=3 then Weight=3903/300; if Class=4 then Weight=4196/300; run;
The data set WebSurvey
contains the variables Class
, Design
, Rating
, Count
, and Weight
. The variable class
is the stratum variable, with four strata: freshman, sophomore, junior, and senior. The variable Design
specifies the three new Web designs: A, B, and C. The variable Rating
contains students’ evaluations of the new Web designs. The variable counts
gives the frequency with which each Web design received each rating within each stratum. The variable weight
contains the sampling weights, which are the reciprocals of selection probabilities in this example.
Output 111.1.1 shows the first 20 observations of the data set.
Output 111.1.1: Web Design Survey Sample (First 20 Observations)
Obs  Class  Design  Rating  Count  Weight 

1  Freshman  A  dislike very much  10  12.4467 
2  Freshman  A  dislike  34  12.4467 
3  Freshman  A  neutral  35  12.4467 
4  Freshman  A  like  16  12.4467 
5  Freshman  A  like very much  15  12.4467 
6  Freshman  B  dislike very much  8  12.4467 
7  Freshman  B  dislike  21  12.4467 
8  Freshman  B  neutral  23  12.4467 
9  Freshman  B  like  26  12.4467 
10  Freshman  B  like very much  22  12.4467 
11  Freshman  C  dislike very much  5  12.4467 
12  Freshman  C  dislike  10  12.4467 
13  Freshman  C  neutral  24  12.4467 
14  Freshman  C  like  30  12.4467 
15  Freshman  C  like very much  21  12.4467 
16  Sophomore  A  dislike very much  1  11.8833 
17  Sophomore  A  dislike  14  11.8833 
18  Sophomore  A  neutral  25  11.8833 
19  Sophomore  A  like  23  11.8833 
20  Sophomore  A  like very much  37  11.8833 
The following SAS statements perform the logistic regression:
proc surveylogistic data=WebSurvey total=Enrollment; stratum Class; freq Count; class Design; model Rating (order=internal) = design; weight Weight; run;
The PROC SURVEYLOGISTIC statement invokes the procedure. The TOTAL= option specifies the data set Enrollment
, which contains the population totals in the strata. The population totals are used to calculate the finite population correction
factor in the variance estimates. The response variable Rating
is in the ordinal scale. A cumulative logit model is used to investigate the responses to the Web designs. In the MODEL statement,
rating
is the response variable, and Design
is the effect in the regression model. The ORDER=INTERNAL option is used for the response variable Rating
to sort the ordinal response levels of Rating
by its internal (numerical) values rather than by the formatted values (for example, 'like very much'). Because the sample
design involves stratified simple random sampling, the STRATA statement is used to specify the stratification variable Class
. The WEIGHT statement specifies the variable Weight
for sampling weights.
The sample and analysis summary is shown in Output 111.1.2. There are five response levels for the Rating
, with 'dislike very much' as the lowest ordered value. The regression model is modeling lower cumulative probabilities by
using logit as the link function. Because the TOTAL= option is used, the finite population correction is included in the variance
estimation. The sampling weight is also used in the analysis.
Output 111.1.2: Web Design Survey, Model Information
Model Information  

Data Set  WORK.WEBSURVEY 
Response Variable  Rating 
Number of Response Levels  5 
Frequency Variable  Count 
Stratum Variable  Class 
Number of Strata  4 
Weight Variable  Weight 
Model  Cumulative Logit 
Optimization Technique  Fisher's Scoring 
Variance Adjustment  Degrees of Freedom (DF) 
Finite Population Correction  Used 
In Output 111.1.3, the score chisquare for testing the proportional odds assumption is 98.1957, which is highly significant. This indicates that the cumulative logit model might not adequately fit the data.
Output 111.1.3: Web Design Survey, Testing the Proportional Odds Assumption
An alternative model is to use the generalized logit model with the LINK=GLOGIT option, as shown in the following SAS statements:
proc surveylogistic data=WebSurvey total=Enrollment; stratum Class; freq Count; class Design; model Rating (ref='neutral') = Design /link=glogit; weight Weight; run;
The REF='neutral' option is used for the response variable Rating
to indicate that all other response levels are referenced to the level 'neutral.' The option LINK=GLOGIT option requests
that the procedure fit a generalized logit model.
The summary of the analysis is shown in Output 111.1.4, which indicates that the generalized logit model is used in the analysis.
Output 111.1.4: Web Design Survey, Model Information
Model Information  

Data Set  WORK.WEBSURVEY 
Response Variable  Rating 
Number of Response Levels  5 
Frequency Variable  Count 
Stratum Variable  Class 
Number of Strata  4 
Weight Variable  Weight 
Model  Generalized Logit 
Optimization Technique  NewtonRaphson 
Variance Adjustment  Degrees of Freedom (DF) 
Finite Population Correction  Used 
Output 111.1.5 shows the parameterization for the main effect Design
.
Output 111.1.5: Web Design Survey, Class Level Information
The parameter and odds ratio estimates are shown in Output 111.1.6. For each odds ratio estimate, the 95% confidence intervals shown in the table contain the value 1.0. Therefore, no conclusion about which Web design is preferred can be made based on this survey.
Output 111.1.6: Web Design Survey, Parameter and Odds Ratio Estimates
Analysis of Maximum Likelihood Estimates  

Parameter  Rating  Estimate  Standard Error 
t Value  Pr > t  
Intercept  dislike  0.3964  0.0832  4.77  <.0001  
Intercept  dislike very much  1.0826  0.1045  10.36  <.0001  
Intercept  like  0.1892  0.0780  2.43  0.0154  
Intercept  like very much  0.3767  0.0824  4.57  <.0001  
Design  A  dislike  0.0942  0.1166  0.81  0.4196 
Design  A  dislike very much  0.0647  0.1469  0.44  0.6597 
Design  A  like  0.1370  0.1104  1.24  0.2149 
Design  A  like very much  0.0446  0.1130  0.39  0.6934 
Design  B  dislike  0.0391  0.1201  0.33  0.7451 
Design  B  dislike very much  0.2721  0.1448  1.88  0.0605 
Design  B  like  0.1669  0.1102  1.52  0.1300 
Design  B  like very much  0.1420  0.1174  1.21  0.2265 
NOTE: The degrees of freedom for the t tests is 1196. 
Odds Ratio Estimates  

Effect  Rating  Point Estimate  95% Confidence Limits  
Design A vs C  dislike  0.861  0.583  1.272 
Design A vs C  dislike very much  1.153  0.691  1.924 
Design A vs C  like  0.899  0.618  1.306 
Design A vs C  like very much  1.260  0.851  1.866 
Design B vs C  dislike  0.984  0.658  1.471 
Design B vs C  dislike very much  1.615  0.975  2.677 
Design B vs C  like  1.218  0.838  1.769 
Design B vs C  like very much  1.389  0.924  2.087 
NOTE: The degrees of freedom in computing the confidence limits is 1196. 