## Example 87.1 Stratified Cluster Sampling

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: Dislike very much Dislike Neutral Like 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 87.1.1 shows the first 20 observations of the data set.

Output 87.1.1 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 87.1.2. 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 87.1.2 Web Design Survey, Model Information
The SURVEYLOGISTIC Procedure

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

Response Profile
Ordered
Value
Rating Total
Frequency
Total
Weight
1 dislike very much 116 1489.0733
2 dislike 227 2933.0433
3 neutral 338 4363.3767
4 like 283 3606.8067
5 like very much 236 3005.7000

Probabilities modeled are cumulated over the lower Ordered Values.

In Output 87.1.3, the score chi-square 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 87.1.3 Web Design Survey, Testing the Proportional Odds Assumption
Score Test for the Proportional
Odds Assumption
Chi-Square DF Pr > ChiSq
98.1957 6 <.0001

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 87.1.4, which indicates that the generalized logit model is used in the analysis.

Output 87.1.4 Web Design Survey, Model Information
The SURVEYLOGISTIC Procedure

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 Newton-Raphson
Variance Adjustment Degrees of Freedom (DF)
Finite Population Correction Used

Response Profile
Ordered
Value
Rating Total
Frequency
Total
Weight
1 dislike 227 2933.0433
2 dislike very much 116 1489.0733
3 like 283 3606.8067
4 like very much 236 3005.7000
5 neutral 338 4363.3767

Logits modeled use Rating='neutral' as the reference category.

Output 87.1.5 shows the parameterization for the main effect Design.

Output 87.1.5 Web Design Survey, Class Level Information
Class Level Information
Class Value Design Variables
Design A 1 0
B 0 1
C -1 -1

The parameter and odds ratio estimates are shown in Output 87.1.6. For each odds ratio estimate, the 95% confidence limits 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 87.1.6 Web Design Survey, Parameter and Odds Ratio Estimates
Analysis of Maximum Likelihood Estimates
Parameter   Rating DF Estimate Standard
Error
Wald
Chi-Square
Pr > ChiSq
Intercept   dislike 1 -0.3964 0.0832 22.7100 <.0001
Intercept   dislike very much 1 -1.0826 0.1045 107.3889 <.0001
Intercept   like 1 -0.1892 0.0780 5.8888 0.0152
Intercept   like very much 1 -0.3767 0.0824 20.9223 <.0001
Design A dislike 1 -0.0942 0.1166 0.6518 0.4195
Design A dislike very much 1 -0.0647 0.1469 0.1940 0.6596
Design A like 1 -0.1370 0.1104 1.5400 0.2146
Design A like very much 1 0.0446 0.1130 0.1555 0.6933
Design B dislike 1 0.0391 0.1201 0.1057 0.7451
Design B dislike very much 1 0.2721 0.1448 3.5294 0.0603
Design B like 1 0.1669 0.1102 2.2954 0.1298
Design B like very much 1 0.1420 0.1174 1.4641 0.2263

Odds Ratio Estimates
Effect Rating Point Estimate 95% Wald
Confidence Limits
Design A vs C dislike 0.861 0.583 1.272
Design A vs C dislike very much 1.153 0.692 1.923
Design A vs C like 0.899 0.618 1.306
Design A vs C like very much 1.260 0.851 1.865
Design B vs C dislike 0.984 0.659 1.471
Design B vs C dislike very much 1.615 0.975 2.675
Design B vs C like 1.218 0.838 1.768
Design B vs C like very much 1.389 0.925 2.086