Agriculture Reference
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definition of sample size depends not only on the variance within each stratum but
also on other parameters (population size and unknown total, among others).
With regard to the HT estimator, we can assume that the size {
n
1
,
n
2
,
...
,
n
h
,
...
,
n
H
} of the samples {
s
1
,
s
2
,
...
,
s
h
,
...
,
s
H
} selected within each stratum is known and
satisfies
X
H
n
h
¼
n
.
h
¼1
If an SRS is drawn within each stratum, the first-order inclusion probability for
unit
k
in the generic stratum
h
is
n
h
N
h
¼
π
k
¼
f
h
8
k
2
U
h
:
ð
6
:
10
Þ
The second-order probabilities are
n
h
N
h
n
h
1
N
h
n
h
N
h
n
m
N
m
8
π
kl
¼
1
8
k
6
¼
l
2
U
h
and
π
kl
¼
k
2
U
h
8
l
2
U
m
,
h
6
¼
m
: ð
6
:
11
Þ
X
k2s
^
k
¼
X
k2s
d
k
y
k
¼
X
h
X
k2s
h
d
k
y
k
¼
X
X
k2s
h
y
k
;
H
N
h
n
h
t
HT
,
STR
¼
ð
6
:
12
Þ
h
¼
1
where
d
k
¼
N
h
/
n
h
are the
direct sampling weights
for any unit
k
belonging to a
generic stratum
h
.
Replacing Eq. (
6.11
) in the general HT variance estimator (see Eq. (1.28) of
Sect.
1.2
), we obtain
X
H
1
f
h
V
HT
t
HT
,
STR
N
h
S
y
,
h
;
ð
Þ
¼
ð
6
:
13
Þ
n
h
h
¼
1
X
k2s
h
2
ð
y
k
y
h
Þ
where
S
y
,
h
¼
. Its efficiency with respect to the SRS design
n
h
1
even if we use the simple proportional allocation (i.e.,
n
h
/
N
h
) from the known
decomposition of the variance within groups (strata) and between groups, the
stratified design (with SRS performed within each stratum) will usually be more
efficient than the classical SRS (S¨rndal et al.
1992
, p. 108).
Stratified sampling has been shown to be more efficient than SRS when the units
within each stratum are as similar as possible, and the units in different strata are as
different as possible (Cochran
1977
).
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