Example 13.1: Stratified Cluster Sample Design

The SURVEYMEANS Procedure

Example 13.1: Stratified Cluster Sample Design

Consider the example in the section "Stratified Sampling". The study population is a junior high school with a total of 4,000 students in grades 7, 8, and 9. Researchers want to know how much these students spend weekly for ice cream, on the average, and what percentage of students spend at least $10 weekly for ice cream.

The example in the section "Stratified Sampling" assumes that the sample of students was selected using a stratified simple random sample design. This example shows analysis based on a more complex sample design.

Suppose that every student belongs to a study group and that study groups are formed within each grade level. Each study group contains between two and four students. Table 13.4 shows the total number of study groups for each grade.

Table 13.4: Study Groups and Students by Grade

Grade	Number of Study Groups	Number of Students
7	608	1,824
8	252	1,025
9	403	1,151
Total	617	4,000

It is quicker and more convenient to collect data from students in the same study group than to collect data from students individually. Therefore, this study uses a stratified clustered sample design. The primary sampling units, or clusters, are study groups. The list of all study groups in the school is stratified by grade level. From each grade level, a sample of study groups is randomly selected, and all students in each selected study group are interviewed. The sample consists of eight study groups from the 7th grade, three groups from the 8th grade, and five groups from the 9th grade.

The SAS data set named IceCreamStudy saves the responses of the selected students:

   data IceCreamStudy;
      input Grade StudyGroup Spending @@; 
      if (Spending < 10) then Group='less';
        else Group='more';
      datalines; 
   7  34  7     7  34  7    7 412  4     9  27 14   
   7  34  2     9 230 15    9  27 15     7 501  2  
   9 230  8     9 230  7    7 501  3     8  59 20  
   7 403  4     7 403 11    8  59 13     8  59 17  
   8 143 12     8 143 16    8  59 18     9 235  9 
   8 143 10     9 312  8    9 235  6     9 235 11
   9 312 10     7 321  6    8 156 19     8 156 14 
   7 321  3     7 321 12    7 489  2     7 489  9 
   7  78  1     7  78 10    7 489  2     7 156  1
   7  78  6     7 412  6    7 156  2     9 301  8
   ;

In the data set IceCreamStudy, the variable Grade contain a student's grade. The variable StudyGroup identifies a student's study group. It is possible for students from different grades to have the same study group number because study groups are sequentially numbered within each grade. The variable Spending contains a student's response to how much he or she spends per week for ice cream, in dollars. The variable GROUP indicates whether a student spends at least $10 weekly for ice cream. It is not necessary to store the data in order of grade and study group.

The SAS data set StudyGroup is created to provide PROC SURVEYMEANS with the sample design information shown in Table 13.4:

   data StudyGroups;
      input Grade _total_; datalines;
   7 608
   8 252
   9 403
   ;

The variable Grade identifies the strata, and the variable _TOTAL_ contains the total number of study groups in each stratum. The population totals stored in the variable _TOTAL_ should be expressed in terms of the primary sampling units (PSUs), which are study groups in this example. Therefore, the variable _TOTAL_ contains the total number of study groups for each grade, rather than the total number of students.

In order to obtain unbiased estimates, you create sampling weights using the following SAS statements:

   data IceCreamStudy; 
      set IceCreamStudy; 
      if Grade=7 then Prob=8/608;
      if Grade=8 then Prob=3/252;
      if Grade=9 then Prob=5/403;
      Weight=1/Prob;

The sampling weights are the reciprocals of the probabilities of selections. The variable Weight contains the sampling weights. Because the sampling design is clustered, and all students from each selected cluster are interviewed, the sampling weights equal the inverse of the cluster (or study group) selection probabilities.

The following SAS statements perform the analysis for this sample design:

   title1 'Analysis of Ice Cream Spending';
   title2 'Stratified Clustered Sample Design';
   proc surveymeans data=IceCreamStudy total=StudyGroups;
      stratum Grade / list; 
      cluster StudyGroup;
      var Spending Group;
      weight Weight;
   run;

Output 13.1.1: Data Summary and Class Information

Analysis of Ice Cream Spending

Stratified Clustered Sample Design

The SURVEYMEANS Procedure

Data Summary
Number of Strata	3
Number of Clusters	16
Number of Observations	40
Sum of Weights	3162.6

Class Level Information
Class Variable	Levels	Values
Group	2	less more

Output 13.1.1 provides information on the sample design and the input data set. There are 3 strata in the sample design, and the sample contains 16 clusters and 40 observations. The variable Group has two levels, `less' and `more'.

Output 13.1.2: Stratum Information

Analysis of Ice Cream Spending

Stratified Clustered Sample Design

The SURVEYMEANS Procedure

Stratum Information
Stratum Index	Grade	Population Total	Sampling Rate	N Obs	Variable	N	Clusters
1	7	608	1.32%	20	Spending Group=less Group=more	20 17 3	8 8 3
2	8	252	1.19%	9	Spending Group=less Group=more	9 0 9	3 0 3
3	9	403	1.24%	11	Spending Group=less Group=more	11 6 5	5 4 4

Output 13.1.2 displays information for each stratum. Since the primary sampling units in this design are study groups, the population totals shown in Output 13.1.2 are the total numbers of study groups for each stratum or grade. Output 13.1.2 also displays the number of clusters for each stratum and analysis variable.

Output 13.1.3: Statistics

Analysis of Ice Cream Spending

Stratified Clustered Sample Design

The SURVEYMEANS Procedure

Statistics
Variable	N	Mean	Std Error of Mean	Lower 95% CL for Mean	Upper 95% CL for Mean
Spending Group=less Group=more	40 23 17	8.923860 0.561437 0.438563	0.650859 0.056368 0.056368	7.517764 0.439661 0.316787	10.329957 0.683213 0.560339

Output 13.1.3 displays the estimates of the average weekly ice cream expense and the percentage of students spending at least $10 weekly for ice cream.

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