Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 124

Note that in output 1 , the design effects are very close to 1, which denotes that
the survey variable is quite independent from the stratification variable
( strataid). If we perform an estimation of a qualitative variable (the absolute
frequencies of the stratifying variable strataid) the standard errors are exactly
equal to 0, because they are fixed by the design (see output 2 ). In output 3 and
output 4, we have estimated the quantitative variable yobs using q1obs and
strataid . Finally, in output 5 , it is worth noting that the situation clearly
changes if the strata boundaries ( strataid3 ) are close to the quantiles of the
survey variable ( yobs ), reducing the variance by up to 67.50 %.
Suppose that the population is divided into equal-sized strata, and the sampling
design takes one observation from each stratum. Then, the total sample size and the
total number of strata is n

H . This is a common design when good spatial
coverage is desirable, for example, consider a square region divided into 10

¼

10

sub-squares.
There are several common methods for selecting a single unit from each stratum.
The most common is a random selection (Christman 2000 ). Undoubtedly, this
maximal stratification shares many features with a spatial systematic design, and
it is not limited by the constraint that the population should lie on a regular grid.
Breidt ( 1995 ) suggested other methods for selecting a single unit within each
stratum, which the author collectively refers to as Markov chain designs. The
code for implementing maximal stratification is given below. The selected sample
is plotted in Fig. 6.5 .

> framepop < - cbind(framepop,strataid2 ¼ floor(framepop$xc*10)*10
+ +floor(framepop$yc*10))
> table(framepop$strataid2)
...
> set.seed(200694)
> str < - strata(framepop,"strataid2",size ¼ rep(1,100),method ¼ "srswor")
> str < - getdata(framepop,str)
> table(str$strataid2)
...
> par(mar ¼ c(1,1,1,1), xaxs ¼ "i",yaxs ¼ "i")
> plot(framepop$xc,framepop$yc, axes ¼ F,cex ¼ 0.5,
+ pch ¼ 19,xlim ¼ c(0,1),ylim ¼ c(0,1))
> for (i in seq(0.1,0.9,0.1))
+{
+

abline(h ¼ i,lty ¼ 2,lwd ¼ 2)

+ abline(v ¼ i,lty ¼ 2,lwd ¼ 2)

+}
> box()
> points(str$xc,str$yc, pch ¼ 1, cex ¼ 2)