Stratified Sampling

Suppose that the previous student sample is actually drawn using a stratified sample design. The strata are grades in the junior high school: 7, 8, and 9. Within strata, simple random samples are selected. Table 90.1 provides the number of students in each grade.

Table 90.1 Students in Grades

Grade

Number of Students

7

1,824

8

1,025

9

1,151

Total

4,000

In order to analyze this sample by using PROC SURVEYREG, you need to input the stratification information by creating a SAS data set with the information in Table 90.1. The following SAS statements create such a data set called StudentTotals:

```data StudentTotals;
input Grade _TOTAL_;
datalines;
7 1824
8 1025
9 1151
;
```

The variable Grade is the stratification variable, and the variable _TOTAL_ contains the total numbers of students in each stratum in the survey population. PROC SURVEYREG  requires you to use the keyword _TOTAL_ as the name of the variable that contains the population total information.

In a stratified sample design, when the sampling rates in the strata are unequal, you need to use sampling weights to reflect this information. For this example, the appropriate sampling weights are the reciprocals of the probabilities of selection. You can use the following DATA step to create the sampling weights:

```data IceCream;
set IceCream;
if Grade=7 then Prob=20/1824;
if Grade=8 then Prob=9/1025;
if Grade=9 then Prob=11/1151;
Weight=1/Prob;
run;
```

If you use PROC SURVEYSELECT to select your sample, PROC SURVEYSELECT creates these sampling weights for you.

The following statements demonstrate how you can fit a linear model while incorporating the sample design information (stratification):

```title1 'Ice Cream Spending Analysis';
title2 'Stratified Sample Design';
proc surveyreg data=IceCream total=StudentTotals;
strata Grade /list;
class Kids;
model Spending = Income Kids / solution;
weight Weight;
run;
```

Comparing these statements to those in the section Simple Random Sampling, you can see how the TOTAL=StudentTotals option replaces the previous TOTAL=4000 option.

The STRATA statement specifies the stratification variable Grade. The LIST option in the STRATA statement requests that the stratification information be included in the output. The WEIGHT statement specifies the weight variable.

Figure 90.4 summarizes the data information, the sample design information, and the fit information. Note that, due to the stratification, the denominator degrees of freedom for F tests and t tests are 37, which is different from the analysis in Figure 90.1.

Figure 90.4 Summary of the Regression
 Ice Cream Spending Analysis Stratified Sample Design

The SURVEYREG Procedure

Regression Analysis for Dependent Variable Spending

Data Summary
Number of Observations 40
Sum of Weights 4000.0
Weighted Mean of Spending 9.14130
Weighted Sum of Spending 36565.2

Design Summary
Number of Strata 3

Fit Statistics
R-square 0.8219
Root MSE 2.4185
Denominator DF 37

For each stratum, Figure 90.5 displays the value of identifying variables, the number of observations (sample size), the total population size, and the calculated sampling rate or fraction.

Figure 90.5 Stratification and Classification Information
Stratum Information
Stratum
Index
Grade N Obs Population Total Sampling Rate
1 7 20 1824 1.10%
2 8 9 1025 0.88%
3 9 11 1151 0.96%

Class Level Information
Class Variable Levels Values
Kids 4 1 2 3 4

Figure 90.6 displays the tests for the significance of model effects under the stratified sample design. The Income effect is strongly significant, while the Kids effect is not significant at the 5% level.

Figure 90.6 Testing Effects
Tests of Model Effects
Effect Num DF F Value Pr > F
Model 4 124.85 <.0001
Intercept 1 150.95 <.0001
Income 1 326.89 <.0001
Kids 3 0.99 0.4081

 Note: The denominator degrees of freedom for the F tests is 37.

The regression coefficient estimates for the stratified sample, along with their standard errors and associated t tests, are displayed in Figure 90.7.

Figure 90.7 Regression Coefficients
Estimated Regression Coefficients
Parameter Estimate Standard Error t Value Pr > |t|
Intercept -26.086882 2.44108058 -10.69 <.0001
Income 0.776699 0.04295904 18.08 <.0001
Kids 1 0.888631 1.07000634 0.83 0.4116
Kids 2 1.545726 1.20815863 1.28 0.2087
Kids 3 -0.526817 1.32748011 -0.40 0.6938
Kids 4 0.000000 0.00000000 . .

 Note: The denominator degrees of freedom for the t tests is 37.Matrix X'WX is singular and a generalized inverse was used to solve the normal equations. Estimates are not unique.

You can request other statistics and tests by using PROC SURVEYREG. You can also analyze data from a more complex sample design. The remainder of this chapter provides more detailed information.