Agriculture Reference
In-Depth Information
0
@
1
A
1
y
11
y
12
...
y
1
g
2
y
21
y
22
...
y
2
g
...
...
...
...
...
:
ð
1
:
4
Þ
k
k
1
y
k
2
...
y
kg
...
...
...
...
...
n
n
1
y
n
2
...
y
ng
For simplicity reasons, we will generally consider one study variable y, so the
second subscript is not needed.
Now, let
be the set of all possible samples,
s
. A sampling design,
p
(
s
), is a
probability distribution on
Ω
Ω
that satisfies
pðÞ
0, all
s 2 Ω
,
X
ð
1
:
5
Þ
pðÞ
¼1
Ω
where
s
is the outcome of a random variable
S
. The function
p
(
.
) plays a central role
in the theory, because it has a one-to-one correspondence with the selection
criterion. For this reason it is called the sampling design.
Consider a particular sampling design,
p
(
s
). The inclusion probability of the unit
k
in a sample is a random event that can be expressed using the indicator random
variable
1if
k 2 S
0 otherwise
I
k
¼
:
ð
1
:
6
Þ
The variable
I
k
is called the sample membership indicator of element
k
, and denotes
whether the
k-
th element is a member of
s
.
The
R
code for generating the indicator random variable when drawing samples
of fixed size
n
is as follows. The most widely used
R
package in this field is
sampling. The reference manual can be downloaded from
http://cran.r-project.
org/web/packages/sampling/sampling.pdf
.
In this example, we consider a popula-
tion with
N
¼5 and a sample with
n
¼3.
>
library(sampling)
>
set.seed(160964)
>
n
<
-3
>
N
<
-5
>
indicator_matrix
<
- writesample(n,N)
>
indicator_matrix
[,1] [,2] [,3] [,4] [,5]
[1,]
0
0
1
1
1
[2,]
0
1
0
1
1
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