5.10: Stratification after Selection of Sample – Post Stratification

Situations can arise in which we cannot place sampling units into their correct strata until after the sample has been selected. If wish to stratify by gender and we obtain

the sample by telephone survey, respondents

cannot be placed in male or female stratum

until after they have been contacted.An auditor may wish to stratify accounts by whether they are wholesale or retail, but this information may not be available until after the account has been selected for the sample.

5.10: Post Stratification - 2

Sample of n people selected in a poll; can be divided into n1 men and n2 women after the sample has been interviewed.

Instead of using to estimate , we can use

is known for both men and women.

st

i

y y

Nprovided N

Note that n1 and n2 are random b/c they can change from sample to sample even though n is fixed.

5.10 Post Stratification - 3

Since n1, n2,… nL are random, post stratification is not exactly a stratified random sample according to the definition:

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

5.10: Post Stratification - 4

If Ni/N is known (at least approximately) and in ni ≥ 20 for each stratum, then post stratification is nearly as accurate as stratified random sampling with proportional allocation.

post stratification very common for household and population surveys– Census data provide number of persons,

households per area, by age, etc.

5.10 Post Stratification - 5 Post stratification appropriate when a

SRS is not properly balanced according to major groupings of the population.

1 2

1 2

20 80

180 110

n n

y y

Men Women 20 80(180) (110) 124

100 100y

Population is about 50/50 men/women, men underrepresented in sample

1 21 2 0.5(180) 0.5(110) 145st

N Ny y y

N N

Note that Ni/N is known to a good degree of approximation, even though N1, N2, N not known.

5.10 Post Stratification - 6

1 1 2 21

2 2 21 1 2 22

22 2

1 1

1

1ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )

ˆ( ) 1

Li

st post L L ii

st L L

L Li i

i i ii i i i

Ny y N y N y N y y

N N

V y N V y N V y N V yN

n sA V y A

N n

Let ; the are random with ( )ii i i i

NA n E n nA

N

Recall that ni are random

5.10 Post Stratification - 7

2 2

1A good approximation for is

11 1 1. Substituting this for gives

i

i

i i i i

En

AEn nA n A n

2 22

1 1

1 1ˆ ( ) 1 (1 )L L

p st i i i ii i

nV y A s A s

n N n

If we had stratified in advance

Penalty for not stratifying in advance