multistage sampling

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Multistage Sampling

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Outline

Features of Multi-stage Sample DesignsSelection probabilities in multi-stage samplingEstimation of parametersCalculation of standard errorsEfficiency of multi-stage samples

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Introduction

Multi-stage sampling means what its name suggests -> there are multiple stages in the sampling processThe number of stages can be numerous, although it is rare to have more than 3For 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

B

C

AA

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Your Examples

Estimation DomainsStratificationNumber of stagesSampling units for each stageSample selection scheme in each stageSampling 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,

LSSStage 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 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

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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

• 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

a

1

a

1s yB

a

AY

a

AY

2

Variance of Estimator

2

1

222 1111

S

BbB

a

AS

AaAYV

A

b

2

1

2

1

1

A

b YYA

S

2

1

2

1

1

B

YYB

S

Estimator of Total

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SRS – SRS: Variance of Estimator

2

1

222 1111

S

BbB

a

AS

AaAYV

A

b

Total variability = Variability among PSUs + Variability of SSUs

Sources of Variation = {PSUs} + {SSUs}

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SRS-SRS: Estimating Variance

2 2 2 2

1

1 1 1 1ˆˆa

b

Av Y A s B s

a A a b B

2

1 1

2 1

1

1

a a

b Ya

Ya

s

Estimator of Variance of Estimator for Total

2

2

1 1

1 1

1

b b

s y yb b

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Each PSU has same number of elements, BSubsample of b elements is selected

a

ya

Y1

1

2 2

ˆ 1 1b wS Sa bV Y

A a B ab

where 2

2

1

1

1

A

bS Y YA

SRS-SRS: Estimating a Mean

22 2 2

1 1

1 1;

1

A B

wS S S YA B

Y

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2

2

1

1 ˆ1

a

bs y Ya

… with variance estimate

2 2ˆˆ 1 1b ws sa b

v YA a B ab

22 2 2

1 1

1 1;

1

a b

ws s s y ya b

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

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1ˆa

Y B yaB

2

2 21 1 2

1

1 1 1 1 1ˆ( ) ( ) ( )A

b

BV Y S S

a A aA B b B

2

2 21 1 2

1

1 1 1 1 1ˆˆ( ) ( ) ( )a

b

Bv Y s s

a A aA B b B

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

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1ˆa

Y ya

21

1ˆ( ) ( )A

bias Y B B YAB

2 22 2

1

1 1 1 1 1ˆ( ) ( ) ( )A

bV Y S Sa A aA b B

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

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1

ˆ

a

a

B yY

B

2 2 23 3 2

1

1 1 1 1 1ˆ( ) ( ) ( )A

bV Y S B Sa A B aA b B

21

1

1ˆa

Y ya

2 2

1

1 1 1 1 1ˆ 1 1 1A

bV Y s S ba A ab B A

2ˆˆ bsv Y

a

SRS-LSS: Estimation of Mean

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1

ˆ1ˆ ˆ;a Y

Y Y B ya p

2

1

ˆ1ˆ ˆˆ1

a Yv Y Y

a a p

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.

ˆ( )ˆ( ) 1 ( 1)ˆ( )

cluclu

srs

V YDEFF Y B

V Y

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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 choiceLarger 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 costIllustrative example: PSUs with equal sizes, SRSWOR at both stages

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Optimum Sample Sizes (2)

Variance function2 2b wS S

Va ab

0 1 2C C ac abc 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* 2 21 2* /w bb c S c S

2*

01

2 21 2

( )

*

b

b w

SC C

ca

c S c S

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Optimum Sample Sizes (4)Optimum b=b*

21 1

22 2

1* w

b

Sc c rohb

c S c roh