1

STRATIFIED SAMPLING

2

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.

3

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

4

Stratum 1: Northern Sweden

Ex. Regionalstratification

Stratum 2: Mid-Sweden

Stratum 3: Southern Sweden

5

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.

6

• 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?

7

• 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

8

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

9

ESTIMATION OF A TOTALAssume: SRS within all strata.

yNty ˆ111ˆ yNty 333ˆ yNty

444ˆ yNty 555ˆ yNty

222ˆ yNty

10

ESTIMATION OF A TOTALAssume: SRS within all strata.

5544

332211strˆ

yNyN

yNyNyNt

In general:

H

hhhyNt

1strˆ

What is the variance of this estimator?

11

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 )(

12

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

13

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

14

ESTIMATORS FOR A MEAN

Note: Start from the estimators for a total!

H

hh

hH

hhh y

NN

yNN

y11

str1

H

hhhyNt

1strˆ

15

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

16

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

17

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