1. Multistage Sampling

2. Outline Features of Multi-stage Sample Designs
Selection probabilities in multi-stage sampling
Estimation of parameters
Calculation of standard errors
Efficiency of multi-stage samples

3. Introduction
Multi-stage sampling means what its name suggests -> there are multiple stages in the sampling process
The number of stages can be numerous, although it is rare to have more than 3
For this topic we will concentrate on two-stage sampling
Also known as subsampling

4. Sampling Units in Multi-stage Sampling
First-stage sampling units are called primary sampling units or PSUs.
Second-stage sampling units are called secondary sampling units or SSUs.
Last-stage sampling units are called ultimate sampling units or USUs.

5. 4-stage Sampling (example)
Villages
EAs
Dwelling
Persons A B C A

6. Your Examples Estimation Domains Stratification Number of stages
Sampling units for each stage
Sample selection scheme in each stage
Sampling frames used in each stage

7. Example: Maldives HIES 2002

8. Two-Stage Sampling
Stage One. Select sample of clusters from population of clusters.
Using any sampling scheme, usually: SRSWOR, PPSWR, LSS
Stage Two. Select sample of elements within each of the sample clusters.
Language: also referred to as ‘subsample’ of elements within a cluster
Subsampling can be done also using any sampling scheme

9. Most Large-Scale Surveys Use Multi-stage Sampling Because …
Sampling frames are available at higher stages but not for the ultmate sampling units. Construction of sampling frames at each lower stage becomes less costly.
Cost efficiency with use of clusters at higher stages of selection
Flexibility in choice of sampling units and methods of selection at different stages
Contributions of different stages towards sampling variance may be estimated separately

10. Probabilities of Selection
Probability that an element in the population is selected in a 2-stage sample is the product of
Probability that the cluster to which it belongs is selected at the first stage
Probability that the element is selected at the second stage given that the cluster to which it belongs is selected at the first stage

11. Example: Two-Stage Samples

12. Estimation Procedures: Illustrations
Multistage Sampling
Estimation Procedures: Illustrations
SRS at stage 1 and SRS at stage 2
SRS at stage 1 and LSS at stage 2 (b from B)
PPSWR at stage 1 and SRS at stage 2 (b from B)

13. SRS – SRS: Estimation of Total
Estimator of Total
Variance of Estimator

14. SRS – SRS: Variance of Estimator
Sources of Variation = {PSUs} + {SSUs}
Total variability = Variability among PSUs + Variability of SSUs

15. SRS-SRS: Estimating Variance
Estimator of Variance of Estimator for Total

16. SRS-SRS: Estimating a Mean
Each PSU has same number of elements, B Subsample of b elements is selected
where

17. … with variance estimate

18. SRS-SRS: Population Mean (1) PSU’s have unequal sizes

19. SRS-SRS: Population Mean (2) PSU’s have unequal sizes

20. SRS-SRS: Population Mean (3) PSU’s have unequal sizes

21. SRS-LSS: Estimation of Mean

22. PPSWR-SRS: Estimation of Total

23. Design Effect for 2-stage Sample
If  is positive, the design effect decreases as the subsample size b decreases.
For fixed n=ab, the smaller the sub-sample size and, hence, the larger the number of clusters included in the sample, the more precise is the sample mean.

24. Designing a Cluster Sample
What overall precision is needed?
What size should the psus be?
How many ssus should be sampled in each psu selected for the sample?
How many psus should be sampled?

25. Choosing psu Size Often a natural unit– not much choice
Larger the psu size, more variability within a psu
ICC is smaller for large psu compared to small psu
but, if psu size is too large, less cost efficient
Need to study relationship between psu sizes and ICC and costs

26. Optimum Sample Sizes (1)
Goal: get the most information (and hence, more statistically efficient) for the least cost
Illustrative example: PSUs with equal sizes, SRSWOR at both stages

27. Optimum Sample Sizes (2)
Variance function
Cost function
Minimize V subject to given cost C*

28. Optimum Sample Sizes (3)
Minimize V subject to given cost C*
Optimum a=a* and b=b*

29. Optimum Sample Sizes (4)
Optimum b=b*