1. Estimation in Stratified Random Sampling
SADC Course in Statistics
Estimation in Stratified Random Sampling
(Session 07)

2. Learning Objectives By the end of this session, you will be able to
explain what is meant by stratification, how a stratified sample is drawn, and its advantages
explain proportional or Neyman’s allocation of sample sizes to each stratum
compute estimates of the population mean and population total from results of a stratified random sample
determine measures of precision for the above estimates

3. Review of stratified sampling
We recall first that stratification is done when it is possible to divide the population into groups (strata) so that the within group variance is small, ideally as small as possible.
From each stratum, a sample of suitable size is drawn, usually using simple random sampling.
The greatest challenge is in defining a suitable stratification variable.
It is useful when information is required for each stratum (e.g. each region in a country) as well as for the whole population.

4. Advantages of stratification
Sampling from each stratum guarantees that the overall sample is more representative of the whole population compared to a simple random sample
If each stratum is more homogeneous, i.e. less variable than the population as a whole with respect to key responses of interest, then estimates will be more precise
Likely to be administratively convenient, e.g. when different sampling procedures need to be applied to different strata (see ELUS example in Practical 2 for large sized estates of >500ha)

5. Sampling with proportional allocation
Suppose there are m strata and a sample of size ni is chosen from the Ni units in stratum i.
Then total population size is N =  Ni , while the sample size is n =  ni .
Often convenient to choose ni so that
This is called proportional allocation

6. Sampling using Neyman’s allocation
If costs of sampling are the same in each stratum, but variability is different (although homogeneous within strata), then sensible to take more samples where there is greater variability, i.e. sample in proportion to the standard deviation.
The appropriate value of ni in this case, see below, is called Neyman’s (or optimum) allocation.

7. Other issues and allocation methods
Above assumes within-stratum variances Si are known. A pilot run or a previous study may give estimates.
But results from a pilot run may give very poor estimates, since they will often be based on very small sample sizes
Also note that Neyman’s allocation may lead to very few units being sampled from some strata – not useful if separate results for each stratum are also needed.
Other methods of allocation exists, e.g. incorporating possible differences in sampling costs

8. Estimating the population mean
First carry out computations for each stratum, i.e. find mean and variance for ith stratum.
The estimate the population mean is then
, with variance

9. Estimating the population total
As with the mean, first find an estimate for the total in ith stratum, i.e.
The estimate the population total is then
, with variance
Note: Use expressions on the previous page in computing these estimates

10. An example
Government agricultural inspectors carry out a survey of cattle ownership in a region divided into 3 administrative areas. Five farms are selected from each area and the number of cattle recorded as shown below. The total number of farms is 636.
Area
Number of farms
No of cattle 1
186
8, 50, 92, 60, 34 2
214
0, 0, 4, 12, 24 3
236
16, 4, 28, 46, 28

11. Questions to answer Ni 1 - fi Note: fi = ni/Ni in ith stratum.
What is the mean number of cattle per farm?
What is the total number of cattle in the region?
First need to compute some summaries:
Area Ni
1 - fi 1
186
48.8
969.2
0.9731 2
214
8.0
104.0
0.9766 3
236
24.4
244.8
0.9788
Note: fi = ni/Ni in ith stratum.

12. Answers for estimating mean
The mean number of cattle per farm is estimated as:
= /636 =
i.e. Approximately 26 cows per farm.
This has variance: =
Hence its std. error = 5.0

13. Answers for estimating total
The total number of cattle in the region is estimated as:
= 636 x = 16547
This has variance:
= (636)2 x
Hence its standard error is
636 x  =

14. Estimating population proportion
As with the mean, first find an estimate for proportion in ith stratum, i.e. pi = ri/ni
The estimate the population proportion is then
, with variance

15. References
Barnett, V. (1974) Elements of Sampling Theory. Edward Arnold. ISBN
Levy, P.S. and Lemeshow, S. (1999) Sampling and Populations: Methods and Applications (3rd edition) Wiley, New York. ISBN
Lohr, S.L. (1999) Sampling: Design and Analysis. International Thomson Publishing. ISBN

16. Practical work follows…