*Multistage Sampling

*OutlineCiri-ciri Contoh Reka Bentuk Multi- peringkatKebarangkalian pemilihan dalam persampelan pelbagai peringkatAnggaran parameterPengiraan ralat piawaiKecekapan sampel pelbagai peringkat

*IntroductionPersampelan pelbagai peringkat bermakna apa namanya -> terdapat pelbagai peringkat dalam proses pensampelanBilangan peringkat boleh menjadi banyak, walaupun ia jarang sekali mempunyai lebih daripada 3Untuk topik ini kita akan menumpukan perhatian kepada pensampelan dua peringkatJuga dikenali sebagai subsampling

*Persampelan Unit dalam Multi- peringkat PersampelanFirst-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.

*4-stage Sampling (example)VillagesEAsDwellingPersons

*Your ExamplesEstimation DomainsStratificationNumber of stagesSampling units for each stageSample selection scheme in each stageSampling frames used in each stage

*Example: Maldives HIES 2002

*Two-Stage SamplingStage One. Select sample of clusters from population of clusters.Using any sampling scheme, usually: SRSWOR, PPSWR, LSSStage Two. Select sample of elements within each of the sample clusters.Language: also referred to as subsample of elements within a clusterSubsampling can be done also using any sampling scheme

*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 selectionFlexibility in choice of sampling units and methods of selection at different stagesContributions of different stages towards sampling variance may be estimated separately

*Probabilities of SelectionProbability that an element in the population is selected in a 2-stage sample is the product ofProbability that the cluster to which it belongs is selected at the first stageProbability that the element is selected at the second stage given that the cluster to which it belongs is selected at the first stage

*Example: Two-Stage Samples

*Estimation Procedures: IllustrationsSRS at stage 1 and SRS at stage 2SRS at stage 1 and LSS at stage 2 (b from B)PPSWR at stage 1 and SRS at stage 2 (b from B)

* SRS SRS: Estimation of TotalVariance of EstimatorEstimator of Total

* SRS SRS: Variance of EstimatorTotal variability = Variability among PSUs + Variability of SSUsSources of Variation = {PSUs} + {SSUs}

*SRS-SRS: Estimating VarianceEstimator of Variance of Estimator for Total

*Each PSU has same number of elements, B Subsample of b elements is selectedwhereSRS-SRS: Estimating a Mean

* with variance estimate

*SRS-SRS: Population Mean (1)PSUs have unequal sizes

*SRS-SRS: Population Mean (2)PSUs have unequal sizes

*SRS-SRS: Population Mean (3)PSUs have unequal sizes

*SRS-LSS: Estimation of Mean

*PPSWR-SRS: Estimation of Total

*Design Effect for 2-stage SampleIf 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.

*Designing a Cluster SampleWhat 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?

*Choosing psu SizeOften a natural unit not much choiceLarger the psu size, more variability within a psuICC is smaller for large psu compared to small psubut, if psu size is too large, less cost efficientNeed to study relationship between psu sizes and ICC and costs

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

*Optimum Sample Sizes (2)Variance function

Cost functionMinimize V subject to given cost C*

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

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

Multistage SamplingMultistage Sampling*