Chapter 5Stratified Random Sampling

Advantages of stratified random sampling

How to select stratified random sample Estimating population mean and total Determining sample size, allocation Estimating population proportion; sample

size and allocation Optimal rule for choosing strata

Stratified Random Sampling

The ultimate function of stratification is to organize the population into homogeneous subsets and to select a SRS of the appropriate size from each stratum.

Stratified Random Sampling Often-used option

– May produce smaller BOE than SRS of same size

– Cost per observation may be reduced– Obtain estimates of population parameters

for subgroups Useful when the population is

heterogeneous and it is possible to establish strata which are reasonably homogeneous within each stratum

Chapter 5Stratified Random Sampling

Improved Sampling Designs with Auxiliary

Information

Chapter 6 Ratio and Regression Estimators

Stratified Random Sampling

Stratified Random Sampling: Notation

1 2

:sample mean of data from stratum i,

1, ,:sample size for stratum i

: population mean of stratum i

:population total of stratum i

population total

i

i

i

i

L

y

i Ln

Stratified Random Sampling

1 2

SRS within each stratum, so:

)

ˆ ˆ; ( ) ( ) ( )

ˆ ˆ ˆˆ

(

Estimate population total by summingestimates of

i

i i i i i i i i i i i

i

Lst

i

N y E E N y N E y N

yN

E y

Stratified Random Sampling: Estimate of Mean

1 1 2 2

1

2 2 21 1 2 22

2 2 22 2 21 1 2 2

1 221 1 2 2

1

1

1ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )

11 1 1

st L L

L

i ii

st L L

L LL

L L

y N y N y N yN

N yN

V y N V y N V y N V yN

n s n s n sN N N

N n N n N nN

Stratified Random Sampling: Estimate of Mean , BOE

2 2 2

1 1 2 22

2 2 22 2 21 1 2 2

1 22

1 1 2 2

2 2 22 2 21 1 2 2

1 22

1 1 2 2

1ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )

11 1 1

11.96 1 1 1

st L L

L LL

L L

L LL

L L

V y N V y N V y N V yN

n s n s n sN N N

N n N n N nN

BOE

n s n s n sN N N

N n N n N nN

Stratified Random Sampling: Estimate of Population Total

1 1 2 2

1

2

2 2 21 1 2 2

2 2 22 2 21 1 2 2

1 21 1 2 2

ˆ ˆ( ) ( )

ˆ ˆ ˆ( ) ( ) ( )

1 1 1

st L L

L

i ii

st st

L L

L LL

L L

Ny N y N y N y

N y

V Ny N V y

N V y N V y N V y

n s n s n sN N N

N n N n N n

Stratified Random Sampling: BOE for Mean and Total , t distribution

When stratum sample sizes are small, can use t dist.

2 2 2

2 2 21 1 2 2

1 22

1 1 2 2

2 2

2 21 1 2 2

1 2

1 1 2 2

2

2

1

22

1

:

11 1 1

1 1

( )Satterwaithe where

1

:

L L

L

L L

L

k k

k k k k

k

Lkk k

k k

df

df

BOE for

n s n s n st N N N

N n N n N nN

n s n st N N

N n N n

a sN N n

df ana s

n

BOE for

2

2

1L L

L

L L

n sN

N n

Degrees of Freedom(worksheet cont.)

1 2 3

1 2 3

1 2 3

22 2 2

2 2 22 2 2

Stratified Random Sample Summary:

( ), 20, 8, 12

155, 62, 93,

1046.25, 418.5, 627.75

1046.25 5.95 418.5 15.25 627.75 9.36

1046.25 5.95 418.5 15.25 627.75 9.36

19 7

k k k

k

k

N N na n n n

n

N N N

a a a

df

21.09

11

21.09; 2.08 (see Excel worksheet)t

Compare BOE in Stratified Random Sample and SRS (worksheet cont.)

2

Stratified Random Sample Summary:

ˆ40, 27.7; ( ) 1.97

If observations were from SRS:

40 11.31ˆ11.31, ( ) 1 2.79310 40

st stn y V y

s V y

Strat. random sample has more precision

Approx. Sample Size to Estimate 2

2 2

2

2

1

2 2

2

1

2 2

1

2 ( ) ( )4

Let , prop. of sample from stratum i

11

4

where 4

st st

i i i

L

i i

i

i i i

L

i i i

i

L

i i

i

BV y B V y

n a n a

a n s BN

N a nN

N s aB

n D

N D N s

Approx. Sample Size to Estimate 2

2 2

2

2

1

2 2

2

1

2 2

1

2

2

2

2 ( ) ( )4

Let , prop. of sample from stratum i

11

4

where 4

st st

i i i

L

i i

i

i i i

L

i i i

i

L

i i

i

BV Ny B V y

N

n a n a

a n s BN

N a nN N

N s aB

n DN

N D N s

Summary: Approx. Sample Size to Estimate ,

2 2

1

2 2

1

2

2

2

when estimating 4

when estimating 4

L

i i ii

L

i ii

N s an

N D N s

BD

BD

N

Example: Sample Size to Estimate (worksheet cont.)

2 22

1

2 2

1

4

L

i i ii

L

i ii

N s aB

n DN D N s

2

2 2 2

1 2 3

1 2 3

2 24

3155 (25) 62 (225) 93 (100)2 2

1 3 1 3 1 31

2

Prior survey: 5, 15, 10.

Estimate to within 2 hrs with 95% conf.

allocation proportions are 1 3.

2 1; 310 96,100

6,991,275

B

i i ii

i ii

a a a

B D N D

N s a

N s

3

1

11 2 3 3

155(25) 62(225) 93(100) 27,125

6,991,27556.7 57

96,100 27,125

so (57) 19

n

n n n

Example: Sample Size to Estimate (worksheet cont.)

2 22

12

2 2

1

4

L

i i ii

L

i ii

N s aB

n DN

N D N s

22

2 2 2 2

2 2 2

1 2 3

1 2 3

160,00 40,000 24004 4 4

3155 (25) 62 (225) 93 (102 2

1 3 1 31

Prior survey: 5, 15, 10.

Estimate to within 400 hrs with 95% conf.

allocation proportions are 1 3.

; 40,000BN N N N

i i ii

a a a

D N D

N s a

0)1 3

32

1

11 2 3 3

6,991,275

155(25) 62(225) 93(100) 27,125

6,991,275104.2 105

40,000 27,125

so (105) 35

i ii

N s

n

n n n

5.5 Allocation of the Sample

Objective: obtain estimators with small variance at lowest cost.

Allocation affected by 3 factors:1. Total number of elements in each stratum

2. Variability in each stratum

3. Cost per observation in each stratum

5.5 Allocation of the Sample: Proportional Allocation

If don’t have variability and cost information for the strata, can use proportional allocation.

Sample size for stratum :

hh

h

Nn n

N

In general this is not the optimum choice for the stratum sample sizes.

5.5 Optimal {min ( )} allocation

of the sample: same cost/obsin each stratum

stV y

1 21 2, , ,

1 2 1 2

1

ˆ ( ), subject to (

where (

ˆ ( )

min , , , ) 0,

, , , )

Use Lagrange multipliers:

0, 1, , , 1, ,

Lst

st

Ln n n

L L

i ii L

i ik k

k

V y g n

g n n

V y

n n

n n n n n

N sgi L n n i L

n nN s

22

21

ˆ ( )1

1L

i ist i

i i i

V yn s

NN nN

Directly proportional tostratum size and stratum variability

1 2This method of choosing , , ,

called Ln n n

Neyman allocation

5.5 Optimal {min ( )} allocation

of the sample: same cost/obsin each stratum

stV y

1

2

1

2 2

1

From previous slide

, 1, ,

substitute for above gives

i ii L

k kk

ii

L

i ii

L

i ii

N sn n i L

N s

na

n

N s

nN D N s

2 2

1

2 2

1

L

i i i

i

L

i i

i

N s a

n

N D N s

Worksheet 11

5.5 Optimal {min ( )} allocation of the sample:same cost/obs in each stratum

stV y

=

5.5 Optimal {min ( )} allocation

of the sample for fixed cost C: cost/obs

in stratum i.i

stV y

c

1 21 2, , ,

1 2 1 1 2 2

1

ˆ ( ), subject to (

where (

ˆ ( )

min , , , ) 0,

, , , )

Use Lagrange multipliers:

0, 1, , , 1, ,

Lst

st

Ln n n

L L L

i i ii L

i ik k k

k

V y g n

g n n

V y

n n

n n c c n c n C

N s cgi L n n i L

n nN s c

22

21

ˆ ( )1

1L

i ist i

i i i

V yn s

NN nN

Directly proportional tostratum size and stratum variability

Inversely proportional to stratum cost/obs

5.5 Optimal {min ( )} allocation

of the sample: cost/obs

in stratum i

stV y

c

i

1

1 1

2 2

1

From previous slide

, 1, ,

substitute for above gives

i i ii L

k k kk

ii

L L

k k k i i ik i

L

i ii

N s cn n i L

N s c

na

n

N s c N s c

nN D N s

2 2

1

2 2

1

L

i i i

i

L

i i

i

N s a

n

N D N s

Worksheet 12

=

5.5 Optimal {min ( )} allocation of the sample: cost/obs in each stratumi

stV yc