Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 8

π 1 ¼ X
s ' ðÞ

pðÞ ¼1 = 9 þ 4 = 9 þ 3 = 9 ¼ 8 = 9 :

It is also possible to compute the second-order inclusion probability for the pair
(1,2),

π 12 ¼ X
s

pðÞ ¼1 = 9 þ 4 = 9 ¼ 5 = 9 :

ðÞ 1 ;

'

We can use the following R code to calculate the first- and second-order inclusion
probabilities for a certain sampling design.

> set.seed(160964)
> ps < - runif(nrow(indicator_matrix))+5
> ps < - ps / sum(ps)
> first_order < - colSums(indicator_matrix *ps)
> second_order < - matrix(0,N,N)
> for (i in 1:(N-1))
{

for (j in (i+1):N)
{

second_order[i,j] < - sum(indicator_matrix [,i]*
indicator_matrix [,j]*ps)
}

}
> second_order < - second_order + t(second_order)
> diag(second_order) < - first_order

> first_order
[1] 0.5919065 0.6069048 0.6069510 0.5975452 0.5966925
> second_order
[,1] [,2] [,3] [,4] [,5]
[1,] 0.5919065 0.2986742 0.2944651 0.2938064 0.2968673
[2,] 0.2986742 0.6069048 0.3079239 0.3113729 0.2958386
[3,] 0.2944651 0.3079239 0.6069510 0.3003725 0.3111405
[4,] 0.2938064 0.3113729 0.3003725 0.5975452 0.2895386
[5,] 0.2968673 0.2958386 0.3111405 0.2895386 0.5966925

The second row of the code concerns a sample design using p ( s ). If the
researcher chooses to use simple random sampling (i.e., p ( s ) is constant), this line
can be replaced with:

> ps < - rep(1/nrow(indicator_matrix),nrow(indicator_matrix)).