1 STRATIFIED SAMPLING. 2 1. Stratification: The elements in the population are divided into...
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Transcript of 1 STRATIFIED SAMPLING. 2 1. Stratification: The elements in the population are divided into...
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STRATIFIED SAMPLING
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STRATIFIED SAMPLING
1. Stratification: The elements in the population are divided into layers/groups/ strata based on their values on one/several auxiliary variables. The strata must be non-overlapping and together constitute the whole population.
2. Sampling within strata: Samples are selected independently from each stratum. Different selection methods can be used in different strata.
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Ex. Stratification of individuals by age group
Stratum Age group
1 17 or younger
2 18-24
3 25-34
4 35-44
5 45-54
6 55-64
7 65 or older
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Stratum 1: Northern Sweden
Ex. Regionalstratification
Stratum 2: Mid-Sweden
Stratum 3: Southern Sweden
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Ex. Stratification of individuals by age group and region
Stratum Age group Region
1 17 or younger Northern
2 17 or younger Mid
3 17 or younger Southern
4 18-24 Northern
5 18-24 Mid
6 18-24 Southern
etc. etc. etc.
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• Gain in precision. If the strata are more homogenous with respect to the study variable(s) than the population as a whole, the precision of the estimates will improve.
• Strata = domains of study. Precision requirements of estimates for certain subpopulations/domains can be assured by using domains as strata.
WHY STRATIFY?
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• Practical reasons. For instance nonresponse rates, method of measurement and the quality of auxiliary information may differ between subpopulations, and can be efficiently handled by stratification.
• Administrative reasons. The survey organization may be divided into geographical districts that makes it natural to let each district be a stratum.
WHY STRATIFY?, cont’d
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ESTIMATIONAssume a population divided into H strata of sizes . Independently, a sample of size nh is selected from each stratum.
Hh NNN ,...,,...,1
hjy = y-value for element j in stratum h
hN
jhjyh yt
1= population total for stratum h
h
Sj hjh n
yy h
= sample mean for stratum h
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ESTIMATION OF A TOTALAssume: SRS within all strata.
yNty ˆ111ˆ yNty 333ˆ yNty
444ˆ yNty 555ˆ yNty
222ˆ yNty
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ESTIMATION OF A TOTALAssume: SRS within all strata.
5544
332211strˆ
yNyN
yNyNyNt
In general:
H
hhhyNt
1strˆ
What is the variance of this estimator?
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VARIANCE OF THE ESTIMATOR OF A TOTAL
Principle: Add the variances of the estimators for each stratum.
A legitimate approach since samples are selected independently from each stratum. Remember:
if X, Y are independent random variables.
YVXVYXV )(
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VARIANCE OF THE ESTIMATOR OF A TOTAL, cont’d
Result:
h
h
h
hH
hh n
SNn
NtV2
1
2str 1ˆ
One term per stratum
Finite population correction(one per stratum!)
where
hN
jhUhj
hh yy
NS
1
22
11
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ESTIMATION OF THE VARIANCE OF THE ESTIMATOR OF A TOTAL
Principle: Estimate what’s unknown in the variance formula.
h
h
h
hH
hh n
SNn
NtV2
1
2str 1ˆ
h
h
h
hH
hh n
sNn
NtV2
1
2str 1ˆˆ
where
hn
jhhj
hh yy
ns
1
22
11
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ESTIMATORS FOR A MEAN
Note: Start from the estimators for a total!
H
hh
hH
hhh y
NN
yNN
y11
str1
H
hhhyNt
1strˆ
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ESTIMATORS FOR A MEAN, cont’dNote: Start from the estimators for a total!
h
h
h
hH
hh n
sNn
NN
yV2
1
22str 1
1ˆ
h
h
h
hH
hh n
sNn
NtV2
1
2str 1ˆˆ
h
h
h
hH
h
hns
Nn
NN 22
11
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ESTIMATORS FOR A PROPORTION
H
hh
h pNN
p1
str ˆˆ
h
h
h
hH
h
hns
Nn
NN
yV2
1
2
str 1ˆ
Note: Like the estimators for a mean, only with y a 0/1-variable!
H
hh
h yNN
y1
str
1
ˆ1ˆ1ˆ
1
2
str
h
hh
h
hH
h
hn
ppNn
NN
yV
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IMPORTANT DESIGN CHOICES IN STRATIFIED SAMPLING
• Stratification variable(s)
• Number of strata
• Sample size in each stratum (allocation)
• Sampling design in each stratum
• Estimator for each stratum