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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