Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 105

than 1, there is loss of efficiency; if it is less than 1, the estimates are more accurate.
Note that the first and the second-order inclusion probabilities are considered in
both designs; it is their distributions that may differ from one strategy to another
(S ¨ rndal et al. 1992 , p. 54).
A very simple scheme for SRS consists of randomly sorting the population frame
(Till ´ 2006 , algorithm 4.5, p. 50) as follows:
1. An independent uniform variable R k ~ U [0,1] is generated for each unit k of the
population.
2. The population is sorted in ascending (or descending) order according to R k .
3. The first (or last) n units of the sorted population are selected for the sample.

Sunter ( 1977 ) proved that this random sorting results in SRS.
To demonstrate the performance of each sampling strategy, we generated
framepop as an artificial spatial population to be used as a sampling frame of
size N 1,000. It contains the geographical coordinates ( xc,yc )ofeachunit

generated according to a uniform distribution U [0,1], a survey variable yobs that
follows a spatial trend on the coordinates and has some added Gaussian white noise,
and two qualitative variables: the first random with three possible codes ( q1obs)
and the second with five possible codes ( q2obs) , which are based on the quantiles
of the variable yobs . Note that this artificial population can be treated as a realization
of a spatial point process. In addition to the sampling library that was already
presented in Chap. 1 ,wehavealsousedthe survey library . The reference manual
canbedownloadedatfrom http://cran.r-project.org/web/packages/survey/survey.pdf .

¼

> library(sampling)
> library(survey)
> n < - 100
> N < - 1000
> set.seed(160964)
> framepop < - data.frame(id ¼ 1:N, xc ¼ runif(N), yc ¼ runif(N))
> yobs < - (exp((framepop$xc-0.5)^2)+exp((framepop$yc-0.5)^2))
> yobs < - 100-((yobs-min(yobs))/(max(yobs-min(yobs))))*100+
+ (rnorm(N)+5)*5
> q1obs < - sample(1:3,N,replace ¼ T)
> q2obs < - as.numeric(cut(yobs,quantile(yobs,probs ¼
+ seq(0, 1, 0.2))))

> q2obs[is.na(q2obs)] < -1

> sum(yobs)
[1] 92044.83

> table(q1obs)
123
304 354 342