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Design of Statistical Investigations

Stephen Senn

Random Sampling 2

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Stratified Random Sampling

A stratified random sample is one obtained by separating the population elements into nonoverlapping groups, called strata, and then selecting a simple random sample from each stratum.

Scheaffer, Mendenhall and Ott,

Elementary Survey Sampling, Fourth Edition

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Why?

• Stratification can be efficient as regards estimation– Lower variances

• Consequently it may be cost-effective

• It may be desired to make statements about subgroups

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General ModelL = number of strata

Ni = number of sampling units in stratum I

N = number of sampling units in population = N1 + N2 +…NL

ni = number is sample from stratum i

etc.

Basic idea of estimation. For any stratum we can estimate the stratum total by multiplying the sample mean by the number in the population in that stratum. We then calculate the population total by summing all strata and so forth

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Estimation

1 1 2 21

22

1

22

21

1 1...

1( ) ( )

1

L

st L L ii

L

st i ii

Li

ii i

y N y N y N y yN N

V y N V yN

NN n

NB Ignoring FPCF

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Example Surv_3

• Advertising firm surveying three areas for mean weekly hours television viewing– Town A, 1550 households– Town B, 620 households– Rural area, 930 households

• Samples are taken at random within these three strata.

• Results on next slide

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Town A Town B Rural Town A Town B Rural Overall35 27 8 N 1550 620 930 310028 4 15 n 20 8 1226 49 21 mean 33.9 25.125 1941 10 7 var 35.35789 232.4107 87.6363643 15 14 Total 52545 15577.5 17670 85792.529 41 30 var contr 4247367 11167335 6316391 2173109332 25 2037 30 1136 1225 3229 34 Mean 27.731 24 Var 2.2639 SE 1.538 Bound 3.0404528273534

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Sample Size Case 1 Equal Allocation

22

21

1 2

2 22 2

2 21 1

22

21

1

... /

1 1( )

12

Li

st ii i

L

L Li i

st i ii i

Li

ii

V y NN n

Suppose n n n n L

LV y N N

nN N nL

LN

N n

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2 22 2

2 12 2

1

2 2

12 2

12 ,

4

4

L

i iLi i

ii

L

i ii

N LL

NN n N n

N Ln

N

Suppose that in planning Surv_3 we had suspected the following

2 2 225, 225, 100A B R

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Analysis of sample size determination for Surv_3Use subscripts 1 for A, 2 for B, 3 for R

ORIGIN 1

L 3 N1 1550 N2 620 N3 930

2

1 25 2 225 3 100

NT1

L

i

Ni

NT 3100

Equal allocation solution

nT

4L

1

L

i

Ni 2 i 2

NT2

2

ceilnTL

25nT 72.75

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Sample Size Case 2 Equal Proportions

2

22

1

2 22

2 21 1

2

21

22 2

2 2 21 1

1

, 1

1 1( )

12

1 14 , 4

Li

st ii i

i i

L Li i

st i ii ii

Li

ii

L Li

i i ii i

V y NN n

Suppose n rN i L

V y N NN rN N r

NN r

N r NN r N

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nT 72.75

r

4

1

L

i

Ni i 2

NT2

2

r 0.028

i 1 3

ni ceil r Ni n

44

18

27

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Sample Size Case 3 Optimal allocation

Approximate allocation that minimises cost for a given variance or minimises variance for a given cost. (ci is the cost per observation sampled in stratum i)

1

/

/

i i ii L

i i ii

N cn n

N c

This is set as an exercise to prove in the coursework

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22

21

1

2

2 12

1

1 12

1( )

/

/

/1

/

/1

Li

st ii

i i iL

i i ii

L

i i i iLi

ii i i i

L L

i i i i i ii i

V y NN

N cn

N c

N cN

N nN c

N c N c

N n

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1 1 1 122 2 2

/ /

4 , 4

L L L L

i i i i i i i i i i i ii i i i

N c N c N c N c

nN n N

(Again this is ignoring FPCF)

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Now consider information on costs

c

9

9

16

nT

4

1

L

i

Ni i ci

1

L

i

Ni i

ci

NT2

2

ni ceil nT

Ni i

ci

1

L

i

Ni i

ci

n

24

29

22

nT1

L

i

ni

nT 75

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Cluster Sampling

A cluster sample is a probability sample in which each sampling unit is a collection, or cluster, of elements

Schaeffer, Mendenhall and Ott

Example. We wish to obtain a n impression of reading skills amongst year 8 children in the UK. We select a simple random sample of schools and test each year 8 child in the schools chosen for reading skills.

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Cluster Sampling Why and Why Not?

• Why: Less costly than simple or stratified sampling per sampled unit– It may be costly to establish sample frame of

individuals– It may be cheaper to sample units close together

• Why not: For a given number of sampled units, the variance will be higher

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A Model for Cluster Sampling

N = number of clusters in population

n = number of clusters selected in a simple random sample of clusters

mi = number of elements in cluster i, i = 1,……N

1

1

, / ,

n

iNi

ii

mM m M M N m

n

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Minimum Variance EstimationGeneral Theory

Suppose that we have a series of unbiased estimators of a given parameter with known but different variances. What is the linear combination of the estimators with the minimum variance?

1 2

2

1 1

ˆ ˆ ˆ

ˆ

ˆ ˆ , 1

k

i i

k k

w i i ii i

E E E

V

w w

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2 2

1

2 21

1 1

1

1

1

21

22

ˆ

( , , ) 1

( , , )1

1

( , , )2

1/(2 )

2

k

w i ii

k k

k i i ii i

kk

ii

k

ii

ki i

i

i ii

V w

f w w w w

df w ww

d

w

df w ww

dw

w

Setting = 0 yields

Setting = 0 yields

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2

21

2

22 2 2

1 12

1

2

21

2 21 1

1

1

1

ˆ1

1 1 1

1 1

ii k

i i

k ki

w i i iki i

i i

k

k ki i

i ii i

w

V w

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Now suppose that the true cluster means have a variance but that the variance within strata is constant

2

2

b

w

Between cluster variance

Within cluster variance

22

22

2

2

2

( )

1

0,

1,

wi b

i

iw

bi

b i i

bi

w

V ym

w

m

If w m

As wn

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Questions

• In the design and analysis of experiments variance estimates are often based on pooled variances. In sampling theory they generally are not. Why the difference in practice?

• For a given total number of observations how do simple, stratified and cluster sampling compare in terms of variance?