Stratified Random Sampling - Survey Sampling Techniques - Lecture Slides, Slides of Survey Sampling Techniques

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Stratified

Random

Sampling

(STS)

STS

is

our

first

design

where

we

start

to

group

SUs

For

STS,

we will still be thinking of SUs as elements

This will change for cluster sampling

Stratified

Random

Sampling

(STS)

DEFN:

A

stratified

random

sample

is

obtained

by

separating

the

population

units

into

non

overlapping

groups,

called

strata,

and

then

selecting

a

random

sample

from

each

stratum

STS

population

framework

Select an independent probability sample from each stratum

Select probability sample from stratum 1 - Select probability sample from stratum 2 - … StratumH h= h= ... ... h= H Stratum 1

STS

population

framework

For now, we will assume that the sampling unit

(SU)

is an element

Population sizes - N h = number of SUs in stratum h in the population - N = N 1 + N + … + NH - Partition sample of size n across strata - n h = number of sample units selected from stratum h - n = n 1 + n 2 + … + n H

Ag

example

Strata:

regions

of

the

US

(indexed

by

h

Northeast

h

North central

h

South

h

West

h

Divide

counties

into

regional

strata

Ag

example

In the sampling frame, must have a variable that indicates the stratum assignment (region) for each county Stratum ( h) SU ( j) 1 1 1 2 1 3 … … 1 220 2 1 2 2 … … 4 421 4 422

Ag

example

Sample

size

of

n

from

N

counties

in

the

US

Proportional

allocation

n

is

of

N

counties

Set each stratum sample size to

of stratum population n h

N

h

Ag

example

Stratum ( h) Stratumsize ( N h ) Samplesize ( n h ) 1 (NE) 220 21 2 (NC) 1054 103 3 (S) 1382 135 4 (W) 422 41 Total 3078 300 n 1 = ( n/ N )* N 1 = 0.0975*220= 21

Inclusion

probability

SRSWOR

of

n

h

SUs

in

stratum

h

are

selected

from

the

N

h

SUs

in

stratum

h

For

SU

j

in

stratum

h

the

inclusion

probability

is

hj

n

h

N

h

Step 4: Collect data (100% response rate)

For

STS,

index

each

measurement

by

stratum

h

SU

j

within the stratum

Characteristic

of

SU

j

in

stratum

h

is

y

hj Note change in notation: for SRS and SYS, we indexed each observation on SU with i and the characteristic of interest for SU i was y i

STS

population

framework

Population

parameters

Stratum population total for stratum h

Whole population total

  h N j hj h y t 1

     H h N j hj H h h h y t t 1 1 1

STS

population

framework

StratumH h=1t 1 h=2t 2

... ... h= H t H Stratum 1

STS

population

framework

Stratum

population

variance

for

stratum

h

    h N j^ h hU hj h N y y S 1 2 2 1

STS estimation framework (using SRS within strata)

SRS

estimators

for

stratum

parameters

  1 ˆ 1 2 2 1 1             h n j h hj h h h n j hj h h h h n j hj h n y y s y N y N n t n y y h h h