Chapter 5 Stratified Random Sampling
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Transcript Chapter 5 Stratified Random Sampling
Chapter 5
Stratified 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 5
Stratified Random Sampling
Improved Sampling
Designs with Auxiliary
Information
Stratified Random
Sampling
Chapter 6 Ratio and
Regression
Estimators
Stratified Random Sampling:
Notation
y i :sam ple m ean of data from stratum i,
i 1,
,L
n i :sam ple size for stratum i
i : population m ean of stratum i
i : population total of stratum i
population total 1 2
L
Stratified Random Sampling
S R S w ithin each stratum , so:
E ( yi ) i
ˆi N i y i ; E (ˆi ) E ( N i y i ) N i E ( y i ) N i i i
E stim ate population total by sum m ing
estim ates of i
ˆ
ˆ1 ˆ2
N
ˆ L
y st
Stratified Random Sampling: Estimate
of Mean
y st
1
N
1
N
N 1 y1 N 2 y 2
N L yL
L
N
i
yi
i 1
1
ˆ
N 12Vˆ ( y1 ) N 22Vˆ ( y 2 )
V ( y st )
2
N
1 2
N1
2
N
2
N L Vˆ ( y L )
2
2
n 1 s1
n
s
2
2
2
1
N
1
2
N 1 n1
N 2 n2
2
n
s
2
L
L
N L 1
N L n L
Stratified Random Sampling:
Estimate of Mean , BOE
Vˆ ( y st )
1
N
N 1 Vˆ ( y 1 ) N 2 Vˆ ( y 2 )
2
2
1
2
2
N L Vˆ ( y L )
n1 s1
n2 s2
2
2
N1 1
N 2 1
2
N 1 n1
N 2 n2
N
2
2
nL sL
2
N L 1
N L nL
2
BOE
1.96
1
n1 s1
n2 s2
2
2
N1 1
N 2 1
2
N 1 n1
N 2 n2
N
2
2
nL sL
2
N L 1
N L nL
2
Stratified Random
Sampling: Estimate of
Population Total
N y st N 1 y1 N 2 y 2
N L yL
L
N
i
yi
i 1
2
Vˆ ( N y st ) N Vˆ ( y st )
2
2
N 1 Vˆ ( y1 ) N 2 Vˆ ( y 2 )
2
N1
2
N L Vˆ ( y L )
2
2
n 1 s1
n
s
2
2
2
1
N
1
2
N 1 n1
N 2 n2
2
n
s
2
L
L
N L 1
N L n L
Stratified Random Sampling: BOE for Mean
and Total , t distribution
When stratum sample sizes are small, can use t dist.
L
2
2
ak sk
k 1
S atterw aith e d f
a
L
2
sk
k
N k ( N k nk )
w h ere a k
2
nk
nk 1
k 1
B O E fo r :
N 1
1
t df
2
n
1
1
2
N
N
1
2
s
1
n
N
1
2
2
1
n
N
2
2
2
s
n
2
N
2
2
L
1
n
N
L
L
B O E fo r :
N 1
2
t df
n
1
1
N
1
2
s
1
n
1
N
2
2
1
n
N
2
2
2
s
n
2
2
N
2
L
1
n
N
L
L
2
s
n
L
L
2
s
n
L
L
Degrees of Freedom(worksheet cont.)
S tratified R andom S am ple S um m ary:
a
k
N (N
k
1
k
155 , N
2
8 , n 12
3
k
2
62 , N
a 1046.25 , a
1
df
, n 20 , n
1
n
N
n )
k
2
3
93,
418.5 , a
1046.25 5.95
1046.25 5.95
2
19
21.09; t 21.09
2
2
3
627.75
2
418.5 15.25 627.75 9.36
418.5 15.25
2
2
2
2
627.75 9.36
2
7
2.08 (see E xcel w orksheet)
11
2
Compare BOE in Stratified Random
Sample and SRS (worksheet cont.)
S tratified R andom S am ple S um m ary:
n 40, y 27.7; Vˆ ( y st ) 1.97.
Strat. random
sample has
more precision
If observations w ere from S R S :
2
40
11.31
s 11.31, Vˆ ( y ) 1
2.79
310 40
Approx. Sample Size to Estimate
V ( y st ) B V ( y st )
2
B
2
4
L et n i a i n , a i prop. of sam ple from stratum i
1
N
a n s
B
N 1
N a n
4
2
L
2
2
i
2
i
i
i 1
i
i
L
n
2
2
N i si
ai
i 1
w here D
L
N D
2
i 1
2
N i si
B
2
4
Approx. Sample Size to Estimate
B
V ( N y st ) B V ( y st )
2
2
4N
2
L et n i a i n , a i prop. of sam ple from stratum i
1
N
a n s
B
N 1
2
N a n
4N
2
L
2
2
i
2
i
i
i 1
i
i
L
n
2
2
N i si
ai
i 1
w here D
L
N D
2
i 1
2
N i si
B
2
4N
2
Summary: Approx. Sample Size to
Estimate ,
L
N
2
i
s
2
i
ai
i 1
n
L
N D
2
N
i
s
2
i
i 1
D
B
2
w hen estim ating
4
D
B
2
4N
2
w hen estim ating
Example: Sample Size to
Estimate (worksheet
cont.)
L
N
n
2
i
2
si a i
i 1
N D
2
N
i 1
P rio r su rvey: 1 5, 2 1 5, 3 1 0 .
E stim ate to w ith in 2 h rs w ith 9 5 % co n f.
allo catio n p ro p o rtio n s are a1 a 2 a 3 1 3 .
B 2 D
B
4
3
N
2
i
s
2
i
ai
1; N D 3 1 0 9 6,1 0 0
2
2
2
155 ( 25 )
1 3
2
62 ( 225 )
1 3
2
2
93 (100 )
1 3
6, 9 9 1, 2 7 5
i 1
3
N i s i 1 5 5(2 5) 6 2 (2 2 5) 9 3(1 0 0 ) 2 7 ,1 2 5
2
i 1
n
6, 9 9 1, 2 7 5
9 6,1 0 0 2 7 ,1 2 5
so n1 n 2 n 3
1
3
5 6 .7 5 7
(5 7 ) 1 9
D
L
i
s
2
i
B
2
4
Example: Sample Size to
Estimate (worksheet
cont.)
L
N
2
i
2
si a i
i 1
n
N D
2
N
i 1
P rio r su rvey: 1 5, 2 1 5, 3 1 0 .
E stim ate to w ith in 4 0 0 h rs w ith 9 5 % co n f.
allo catio n p ro p o rtio n s are a1 a 2 a 3 1 3 .
D
B
2
4N
2
400
2
4N
2
160 , 00
2
3
N
2
i
s
2
i
ai
4N
155 ( 25 )
1 3
2
40 , 000
N
2
2
62 ( 225 )
1 3
; N D 4 0, 0 0 0
2
2
93 (10 0 )
1 3
6, 9 9 1, 2 7 5
i 1
3
N i s i 1 5 5(2 5) 6 2 (2 2 5) 9 3(1 0 0 ) 2 7 ,1 2 5
2
i 1
n
6, 9 9 1, 2 7 5
4 0, 0 0 0 2 7 ,1 2 5
so n1 n 2 n 3
1
3
1 0 4 .2 1 0 5
(1 0 5) 3 5
D
L
i
s
2
i
B
2
4N
2
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.
S am ple size for stratum h :
nh n
Nh
N
In general this is not the optimum choice
for the stratum sample sizes.
5.5 O ptim al (m in V ( y st ) allocation
Vˆ ( y st )
of the sam ple: sam e cost/obs
L
1
N
2
i 1
2
n
s
2
N i 1 i i
N i ni
in each stratum
m in Vˆ ( y st ), subject to g ( n1 , n 2 ,
, n L ) 0,
, n L ) n1 n 2
nL n
n1 , n 2 ,
, nL
w here g ( n1 , n 2 ,
Directly proportional to
stratum size and stratum variability
U se L agrange m ultipliers:
Vˆ ( y st )
ni
g
ni
0, i 1,
, L ni n
N i si
N
k 1
T his m ethod of choosing n1 , n 2 ,
called N eym an allocation
, i 1,
L
, nL
k
sk
,L
5.5 O ptim al (m in V ( y st ) allocation
L
of the sam ple: sam e cost/obs
n
in each stratum
ni n
, i 1,
L
N D
k
,L
sk
k 1
substitute
ni
n
for a i above gives
N i si
i 1
L
n
2
L
N D
2
i 1
2
N i si
ai
L
i 1
N i si
N
2
i 1
2
F rom previous slide
2
N i si
2
N i si
5 .5 O p tim al (m in V ( y st ) allo catio n o f th e sa m p le:
sam e co st/o b s in each stratu m
Worksheet 11
5.5 O ptim al (m in V ( y st ) allocation
Vˆ ( y st )
of the sam ple for fixed cost C : c i = cost/ob s
L
1
2
N
i 1
in stratum i.
m in Vˆ ( y st ), subject to g ( n1 , n 2 ,
n1 , n 2 ,
, nL
w here g ( n1 , n 2 ,
Vˆ ( y st )
ni
g
ni
, n L ) 0,
, n L ) c1 n1 c 2 n 2
U se L agrange m ultipliers:
0, i 1,
2
n
s
2
N i 1 i i
N i ni
cL nL C
Directly proportional to
stratum size and stratum variability
, L ni n
N i si
ci
, i 1,
L
N
k
sk
,L
ck
k 1
Inversely proportional
to stratum cost/obs
5.5 O ptim al (m in V ( y st ) allocation
L
of the sam ple: sam e cost/obs
n
in each stratum
2
L
N D
i 1
ni n
N i si
ci
, i 1,
L
N
k
sk
,L
ck
k 1
substitute
ni
n
n
for a i above gives
L
N k sk
k 1
L
c k N i si
i 1
L
N D
2
i 1
2
N i si
ci
ai
i 1
2
From previous slide
2
N i si
2
N i si
5 .5 O p tim al (m in V ( y st ) allo catio n o f th e sa m p le:
c i = co st/o b s in each stratu m
Worksheet 12