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SJS SDI_17 1
Design of Statistical Investigations
Stephen Senn
Random Sampling 2
SJS SDI_17 2
Stratified Random Sampling
A stratified random sample is one obtained by separating the population elements into nonoverlapping groups, called strata, and then selecting a simple random sample from each stratum.
Scheaffer, Mendenhall and Ott,
Elementary Survey Sampling, Fourth Edition
SJS SDI_17 3
Why?
• Stratification can be efficient as regards estimation– Lower variances
• Consequently it may be cost-effective
• It may be desired to make statements about subgroups
SJS SDI_17 4
General ModelL = number of strata
Ni = number of sampling units in stratum I
N = number of sampling units in population = N1 + N2 +…NL
ni = number is sample from stratum i
etc.
Basic idea of estimation. For any stratum we can estimate the stratum total by multiplying the sample mean by the number in the population in that stratum. We then calculate the population total by summing all strata and so forth
SJS SDI_17 5
Estimation
1 1 2 21
22
1
22
21
1 1...
1( ) ( )
1
L
st L L ii
L
st i ii
Li
ii i
y N y N y N y yN N
V y N V yN
NN n
NB Ignoring FPCF
SJS SDI_17 6
Example Surv_3
• Advertising firm surveying three areas for mean weekly hours television viewing– Town A, 1550 households– Town B, 620 households– Rural area, 930 households
• Samples are taken at random within these three strata.
• Results on next slide
SJS SDI_17 7
Town A Town B Rural Town A Town B Rural Overall35 27 8 N 1550 620 930 310028 4 15 n 20 8 1226 49 21 mean 33.9 25.125 1941 10 7 var 35.35789 232.4107 87.6363643 15 14 Total 52545 15577.5 17670 85792.529 41 30 var contr 4247367 11167335 6316391 2173109332 25 2037 30 1136 1225 3229 34 Mean 27.731 24 Var 2.2639 SE 1.538 Bound 3.0404528273534
SJS SDI_17 8
Sample Size Case 1 Equal Allocation
22
21
1 2
2 22 2
2 21 1
22
21
1
... /
1 1( )
12
Li
st ii i
L
L Li i
st i ii i
Li
ii
V y NN n
Suppose n n n n L
LV y N N
nN N nL
LN
N n
SJS SDI_17 9
2 22 2
2 12 2
1
2 2
12 2
12 ,
4
4
L
i iLi i
ii
L
i ii
N LL
NN n N n
N Ln
N
Suppose that in planning Surv_3 we had suspected the following
2 2 225, 225, 100A B R
SJS SDI_17 10
Analysis of sample size determination for Surv_3Use subscripts 1 for A, 2 for B, 3 for R
ORIGIN 1
L 3 N1 1550 N2 620 N3 930
2
1 25 2 225 3 100
NT1
L
i
Ni
NT 3100
Equal allocation solution
nT
4L
1
L
i
Ni 2 i 2
NT2
2
ceilnTL
25nT 72.75
SJS SDI_17 11
Sample Size Case 2 Equal Proportions
2
22
1
2 22
2 21 1
2
21
22 2
2 2 21 1
1
, 1
1 1( )
12
1 14 , 4
Li
st ii i
i i
L Li i
st i ii ii
Li
ii
L Li
i i ii i
V y NN n
Suppose n rN i L
V y N NN rN N r
NN r
N r NN r N
SJS SDI_17 12
nT 72.75
r
4
1
L
i
Ni i 2
NT2
2
r 0.028
i 1 3
ni ceil r Ni n
44
18
27
SJS SDI_17 13
Sample Size Case 3 Optimal allocation
Approximate allocation that minimises cost for a given variance or minimises variance for a given cost. (ci is the cost per observation sampled in stratum i)
1
/
/
i i ii L
i i ii
N cn n
N c
This is set as an exercise to prove in the coursework
SJS SDI_17 14
22
21
1
2
2 12
1
1 12
1( )
/
/
/1
/
/1
Li
st ii
i i iL
i i ii
L
i i i iLi
ii i i i
L L
i i i i i ii i
V y NN
N cn
N c
N cN
N nN c
N c N c
N n
SJS SDI_17 15
1 1 1 122 2 2
/ /
4 , 4
L L L L
i i i i i i i i i i i ii i i i
N c N c N c N c
nN n N
(Again this is ignoring FPCF)
SJS SDI_17 16
Now consider information on costs
c
9
9
16
nT
4
1
L
i
Ni i ci
1
L
i
Ni i
ci
NT2
2
ni ceil nT
Ni i
ci
1
L
i
Ni i
ci
n
24
29
22
nT1
L
i
ni
nT 75
SJS SDI_17 17
Cluster Sampling
A cluster sample is a probability sample in which each sampling unit is a collection, or cluster, of elements
Schaeffer, Mendenhall and Ott
Example. We wish to obtain a n impression of reading skills amongst year 8 children in the UK. We select a simple random sample of schools and test each year 8 child in the schools chosen for reading skills.
SJS SDI_17 18
Cluster Sampling Why and Why Not?
• Why: Less costly than simple or stratified sampling per sampled unit– It may be costly to establish sample frame of
individuals– It may be cheaper to sample units close together
• Why not: For a given number of sampled units, the variance will be higher
SJS SDI_17 19
A Model for Cluster Sampling
N = number of clusters in population
n = number of clusters selected in a simple random sample of clusters
mi = number of elements in cluster i, i = 1,……N
1
1
, / ,
n
iNi
ii
mM m M M N m
n
SJS SDI_17 20
Minimum Variance EstimationGeneral Theory
Suppose that we have a series of unbiased estimators of a given parameter with known but different variances. What is the linear combination of the estimators with the minimum variance?
1 2
2
1 1
ˆ ˆ ˆ
ˆ
ˆ ˆ , 1
k
i i
k k
w i i ii i
E E E
V
w w
SJS SDI_17 21
2 2
1
2 21
1 1
1
1
1
21
22
ˆ
( , , ) 1
( , , )1
1
( , , )2
1/(2 )
2
k
w i ii
k k
k i i ii i
kk
ii
k
ii
ki i
i
i ii
V w
f w w w w
df w ww
d
w
df w ww
dw
w
Setting = 0 yields
Setting = 0 yields
SJS SDI_17 22
2
21
2
22 2 2
1 12
1
2
21
2 21 1
1
1
1
ˆ1
1 1 1
1 1
ii k
i i
k ki
w i i iki i
i i
k
k ki i
i ii i
w
V w
SJS SDI_17 23
Now suppose that the true cluster means have a variance but that the variance within strata is constant
2
2
b
w
Between cluster variance
Within cluster variance
22
22
2
2
2
( )
1
0,
1,
wi b
i
iw
bi
b i i
bi
w
V ym
w
m
If w m
As wn
SJS SDI_17 24
Questions
• In the design and analysis of experiments variance estimates are often based on pooled variances. In sampling theory they generally are not. Why the difference in practice?
• For a given total number of observations how do simple, stratified and cluster sampling compare in terms of variance?