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Multistage Sampling

Multistage Sampling. 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. Introduction.

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Multistage Sampling

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

  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 UseMulti-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 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, BSubsample 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. Variance function Optimum Sample Sizes (2) • Cost function • Minimize V subject to given cost C*

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

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

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