This example uses the corn yield data set FARMS
from Example 98.4 to illustrate how to construct a regression estimator for a stratified sample design.
As in Example 98.3, by incorporating auxiliary information into a regression estimator, the procedure can produce more accurate estimates of the population characteristics that are of interest. In this example, the sample design is a stratified sample design. The auxiliary information is the total farm areas in regions of each state, as displayed in Table 98.13. You want to estimate the total corn yield by using this information under the three linear models given in Example 98.4.
Table 98.13: Information for Each Stratum
Number of Farms 


Stratum 
State 
Region 
Population 
Sample 
Total Farm Area 
1 
Iowa 
1 
100 
3 

2 
2 
50 
5 
13,200 

3 
3 
15 
3 

4 
Nebraska 
1 
30 
6 
8,750 
5 
2 
40 
2 

Total 
235 
19 
21,950 
The regression estimator to estimate the total corn yield under Model I can be obtained by using PROC SURVEYREG with an ESTIMATE statement:
title1 'Estimate Corn Yield from Farm Size'; title2 'Model I: Same Intercept and Slope'; proc surveyreg data=Farms total=StratumTotals; strata State Region / list; class State Region; model CornYield = FarmArea State*Region /solution; weight Weight; estimate 'Estimate of CornYield under Model I' INTERCEPT 235 FarmArea 21950 State*Region 100 50 15 30 40 /e; run;
To apply the constraint in each stratum that the weighted total number of farms equals to the total number of farms in the
stratum, you can include the strata as an effect in the MODEL statement, effect State*Region
. Thus, the CLASS statement must list the STRATA variables, State
and Region
, as classification variables. The following ESTIMATE statement specifies the regression estimator, which is a linear function
of the regression parameters:
estimate 'Estimate of CornYield under Model I' INTERCEPT 235 FarmArea 21950 State*Region 100 50 15 30 40 /e;
This linear function contains the total for each explanatory variable in the model. Because the sampling units are farms in
this example, the coefficient for Intercept
in the ESTIMATE statement is the total number of farms (235); the coefficient for FarmArea
is the total farm area listed in Table 98.13 (21950); and the coefficients for effect State*Region
are the total number of farms in each strata (as displayed in Table 98.13).
Output 98.5.1 displays the results of the ESTIMATE statement. The regression estimator for the total of CornYield
in Iowa and Nebraska is 7464 under Model I, with a standard error of 927.
Output 98.5.1: Regression Estimator for the Total of CornYield under Model I
Estimate Corn Yield from Farm Size 
Model I: Same Intercept and Slope 
Estimate  

Label  Estimate  Standard Error  DF  t Value  Pr > t 
Estimate of CornYield under Model I  7463.52  926.84  14  8.05  <.0001 
Under Model II, a regression estimator for totals can be obtained by using the following statements:
title1 'Estimate Corn Yield from Farm Size'; title2 'Model II: Same Intercept, Different Slopes'; proc surveyreg data=FarmsByState total=StratumTotals; strata State Region; class State Region; model CornYield = FarmAreaIA FarmAreaNE state*region /solution; weight Weight; estimate 'Total of CornYield under Model II' INTERCEPT 235 FarmAreaIA 13200 FarmAreaNE 8750 State*Region 100 50 15 30 40 /e; run;
In this model, you also need to include strata as a fixed effect in the MODEL statement. Other regressors are the auxiliary
variables FarmAreaIA
and FarmAreaNE
(defined in Example 98.4). In the following ESTIMATE statement, the coefficient for Intercept
is still the total number of farms; and the coefficients for FarmAreaIA
and FarmAreaNE
are the total farm area in Iowa and Nebraska, respectively, as displayed in Table 98.13. The total number of farms in each strata are the coefficients for the strata effect:
estimate 'Total of CornYield under Model II' INTERCEPT 235 FarmAreaIA 13200 FarmAreaNE 8750 State*Region 100 50 15 30 40 /e;
Output 98.5.2 displays that the results of the regression estimator for the total of corn yield in two states under Model II is 7580 with a standard error of 859. The regression estimator under Model II has a slightly smaller standard error than under Model I.
Output 98.5.2: Regression Estimator for the Total of CornYield under Model II
Estimate Corn Yield from Farm Size 
Model II: Same Intercept, Different Slopes 
Estimate  

Label  Estimate  Standard Error  DF  t Value  Pr > t 
Total of CornYield under Model II  7580.49  859.18  14  8.82  <.0001 
Finally, you can apply Model III to the data and estimate the total corn yield. Under Model III, you can also obtain the regression estimators for the total corn yield for each state. Three ESTIMATE statements are used in the following statements to create the three regression estimators:
title1 'Estimate Corn Yield from Farm Size'; title2 'Model III: Different Intercepts and Slopes'; proc surveyreg data=FarmsByState total=StratumTotals; strata State Region; class State Region; model CornYield = state FarmAreaIA FarmAreaNE State*Region /noint solution; weight Weight; estimate 'Total CornYield in Iowa under Model III' State 165 0 FarmAreaIA 13200 FarmAreaNE 0 State*region 100 50 15 0 0 /e; estimate 'Total CornYield in Nebraska under Model III' State 0 70 FarmAreaIA 0 FarmAreaNE 8750 State*Region 0 0 0 30 40 /e; estimate 'Total CornYield in both states under Model III' State 165 70 FarmAreaIA 13200 FarmAreaNE 8750 State*Region 100 50 15 30 40 /e; run;
The fixed effect State
is added to the MODEL statement to obtain different intercepts in different states, by using the NOINT option. Among the
ESTIMATE statements, the coefficients for explanatory variables are different depending on which regression estimator is estimated.
For example, in the ESTIMATE statement
estimate 'Total CornYield in Iowa under Model III' State 165 0 FarmAreaIA 13200 FarmAreaNE 0 State*region 100 50 15 0 0 /e;
the coefficients for the effect State
are 165 and 0, respectively. This indicates that the total number of farms in Iowa is 165 and the total number of farms in
Nebraska is 0, because the estimation is the total corn yield in Iowa only. Similarly, the total numbers of farms in three
regions in Iowa are used for the coefficients of the strata effect State*Region
, as displayed in Table 98.13.
Output 98.5.3 displays the results from the three regression estimators by using Model III. Since the estimations are independent in each state, the total corn yield from both states is equal to the sum of the estimated total of corn yield in Iowa and Nebraska, . This regression estimator is the same as the one under Model II. The variance of regression estimator of the total corn yield in both states is the sum of variances of regression estimators for total corn yield in each state. Therefore, it is not necessary to use Model III to obtain the regression estimator for the total corn yield unless you need to estimate the total corn yield for each individual state.
Output 98.5.3: Regression Estimator for the Total of CornYield under Model III
Estimate Corn Yield from Farm Size 
Model III: Different Intercepts and Slopes 
Estimate  

Label  Estimate  Standard Error  DF  t Value  Pr > t 
Total CornYield in Iowa under Model III  6246.11  851.27  14  7.34  <.0001 