Agriculture Reference
In-Depth Information
The optimal allocation commands are as follows:
>
var_yobs
¼
tapply(framepop$yobs,framepop$q1obs,var)
>
numvar
<
- var_yobs*num_units
>
stratsize
<
- pmax(ceiling(sampsize*numvar/sum(numvar)),
+ rep(5,length(num_units)))
>
c(stratsize,sum(stratsize))
123
79
88
88 255
However, if we fix the coefficient of variation using the threshold
c
, the optimal
allocation (i.e., obtained by minimizing the overall sample size
n
)is
!
X
H
N
h
S
y
,
Uh
N
h
S
y
,
Uh
h
¼1
n
h
¼
:
ð
8
:
13
Þ
þ
X
H
N
h
S
y
,
Uh
c
2
t
2
h
¼1
Generally, the main objective of sampling is to acquire as much information as
possible at a minimum cost. A simple cost function can be defined as
K
¼
k
0
þ
X
H
k
h
n
h
;
ð
8
:
14
Þ
h
¼1
where
k
0
is the fixed cost, and
k
h
represents the cost of observing a spatial unit in
stratum
h
. In this case, the goal is to allocate observations to strata so that we
minimize
Var t
HT
,
STR
ð Þ
for a given total cost
K
or, alternatively, minimize
K
for a
fixed
Var t
HT
,
ST
ð Þ
. Minimizing the variance for a fixed cost
K
, the optimal sample
size in stratum
h
is (Lohr
2010
)
N
h
S
y
,
Uh
=
p
k
h
n
h
¼
n
:
ð
8
:
15
Þ
X
H
p
k
h
N
h
S
y
,
Uh
=
h
¼1
This equation gives
n
h
in terms of a fixed
n
. Note that Eq. (
8.15
) reduces to
Eq. (
8.12
) with
k
h
constant.
The solution of the optimal allocation problem can be also specified as follows.
The solution is obviously different if the sample is chosen to address a specified
total cost
K
or a specified coefficient of variation. In the first case, for a fixed
K
the
solution is
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