Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 112

+}
> par(mar ¼ c(1,1,1,1), xaxs ¼ "i",yaxs ¼ "i")
> plot(datasys[,1],datasys[,2], 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)

+}
> points(datasys[datasys[,3] ¼¼ 1,1],datasys[datasys[,3] ¼¼ 1,2],
+ pch ¼ 1, cex ¼ 2)

> box()

6.4 Unequal Selection Probabilities

The statistical units in agricultural surveys do not necessarily have the same size,
particularly if we are dealing with legal bodies such as households or farms, which
tend to have a very skewed size distribution. This is true if the size measure is an
economic or business indicator, or the total surface of the farm, its arable land, or
the number of livestock.
With spatial units, by using points and regular polygons, we can guarantee that
this situation is avoided. However, the widely used technique of partitioning the
study area into irregular polygons will almost definitely result in a set of
non-skewed units that are not of the same size, although this requirement is
considered essential for the frame set up (see Chap. 5 ) .
Moreover, in multistage sampling (see Sect. 6.6 ) the aggregates that are used as
primary sampling units are classically defined by administrative boundaries, and
most likely do not enclose the same number of secondary sampling units. Thus, if
we have an equal probability sample from the first stage, it will be impossible to
have the same selection probability for all the secondary sampling units in the
second stage.
The challenge is that the size has a considerable impact on the precision of
survey estimates. Failing to select units using this attribute will most likely intro-
duce serious biases when estimating the population characteristics. Conversely,
when the distribution of the survey variable is concentrated in a few large units, an
appropriate random selection plan that exploits this feature provides a smaller
sample with a higher efficiency.
Assume that the survey variable y is approximately proportional to an auxiliary
variable x, where x plays the role of a size measure of the statistical units (and can
thus be assumed to be strictly positive). A sampling plan that uses probabilities
proportional to size can be applied using two different frameworks: without and with
replacement (with fixed or variable size). These two plans are respectively denoted as