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Multistage Sampling

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**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

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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

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**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.

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**4-stage Sampling (example)**

Villages EAs Dwelling Persons A B C A

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**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

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**Example: Maldives HIES 2002**

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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

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**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

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**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

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**Example: Two-Stage Samples**

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**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)

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**SRS – SRS: Estimation of Total**

Estimator of Total Variance of Estimator

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**SRS – SRS: Variance of Estimator**

Sources of Variation = {PSUs} + {SSUs} Total variability = Variability among PSUs + Variability of SSUs

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**SRS-SRS: Estimating Variance**

Estimator of Variance of Estimator for Total

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**SRS-SRS: Estimating a Mean**

Each PSU has same number of elements, B Subsample of b elements is selected where

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**… with variance estimate**

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

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

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

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**SRS-LSS: Estimation of Mean**

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**PPSWR-SRS: Estimation of Total**

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**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.

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**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?

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**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

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**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

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**Optimum Sample Sizes (2)**

Variance function Cost function Minimize V subject to given cost C*

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**Optimum Sample Sizes (3)**

Minimize V subject to given cost C* Optimum a=a* and b=b*

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**Optimum Sample Sizes (4)**

Optimum b=b*

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