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
yi 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( ) ( ) ( ) ( )
1 1 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 21 1 2 22
2 2 22 2 21 1 2 2
1 221 1 2 2
2 2 22 2 21 1 2 2
1 221 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 kk k k k
kL kk 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
1 L 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
21
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
Li 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 DN D N s
Approx. Sample Size to Estimate 2
2 2
2
21
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
Li i
i
i i i
L
i i i
i
L
i i
i
BV Ny B V y
Nn a n a
a n s BN
N a nN N
N s aB
n DNN 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
BDN
Example: Sample Size to Estimate (worksheet cont.)
2 22
1
2 2
1
4
L
i i ii
L
i ii
N s aBn D
N 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
6,991,275 56.7 5796,100 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 aBn DNN D N s
222 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
6,991,275 104.2 10540,000 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 stratum2. Variability in each stratum3. 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
hN
n nN
In general this is not the optimum choice for the stratum sample sizes.
5.5 Optimal {min ( )} allocationof 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 sg i L n n i Ln n N s
22
21
ˆ ( )1 1
Li i
st ii i i
V yn s
NN nN
Directly proportional tostratum size and stratum variability
1 2This method of choosing , , ,called
Ln n nNeyman allocation
5.5 Optimal {min ( )} allocationof 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 sn
N D N s
2 2
1
2 2
1
L
i i ii
L
i ii
N s an
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 ( )} allocationof the sample for fixed cost C: cost/obsin stratum i.
i
stV yc
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 cg i L n n i Ln n N s c
22
21
ˆ ( )1 1
Li i
st ii i i
V yn s
NN nN
Directly proportional tostratum size and stratum variability
Inversely proportional to stratum cost/obs
5.5 Optimal {min ( )} allocationof the sample: cost/obsin stratum
i
stV yc
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 cn
N D N s
2 2
1
2 2
1
L
i i ii
L
i ii
N s an
N D N s
Worksheet 12
=
5.5 Optimal {min ( )} allocation of the sample: cost/obs in each stratumi
stV yc
