Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 110

Systematic sampling is also a common design for spatially distributed
populations. If the ordering uses the coordinate system that geo-codes the popula-
tion frame, it has the additional advantage that it has a good spatial coverage. It is an
efficient method for sampling autocorrelated populations (Wolter 2007 ), which is
often needed for spatially distributed populations.
Another common method for performing systematic sampling is to divide the
population into equally spaced clusters. Cochran ( 1977 ) has shown that systematic
sampling has a lower variance than SRS if

ˁ IC is the
intraclass correlation coefficient (the correlation between a pair of units in the same
systematic sample). See Sect. 7.3 for a generalization of this result.
One must be cautious when using systematic sampling for rare, clustered
populations. This is because the groupings of rare units may extend over distances
shorter than the distance between sample sites (see Sects. 6.10 and 6.11 and
Christman 2000 ). In spatial populations, the clusters are typically arranged so that
the units in each cluster are regularly spread over the region, in both dimensions. In
populations with trends or periodicity, the allocation and sample size can have
major effects on whether the two-dimensional systematic design is more or less
efficient than an SRS. Because rare, clustered populations do not typically exhibit
large-scale trends or periodicity, the allocation of the systematic sample over the
geographic region is less critical. It is more important that the sample is sufficiently
large for there to be a high probability of sampling rare units (Christman 2000 ).
Further, the main difficulty when extending its use to a spatial context is because
only a regular grid of points or a set of regularly shaped polygons have a natural
ordering that can be used to spread the sample over the study region (see Chap. 7 ) .
Some functions in the sampling package are available for one-dimensional
systematic sampling with constant or variable probabilities , but there is no defined
procedure for the systematic selection of samples in two or more dimensions. This
is quite easy to implement, but it is difficult to generalize the method for two
coordinates to any spatial structure, population size, and steps.
The point estimates of the total do not represent a problem for the HT estimation
from a systematically selected data sample, as the π k s are not affected by the design.
Its efficiency with respect to the SRS design can be measured using (S¨rndal
et al. 1992 , p. 80)

ˁ IC < n /[ N ( n 1)], where

n 1

;t HT deff SYS ð Þ ¼ 1 þ

f ʴ;

ð 6 : 7 Þ

1

SST , SSW is the variation within systematic samples, SST is the
total variation, and rs is the sampling interval. As stated by S ¨ rndal et al. ( 1992 ), the
homogeneity measure

ʴ ¼

N 1
Nrs

SSW

where 1

ˁ IC .
Variance estimation is much more complicated, because a very large number of

ʴ

is closely related to

the π kl are equal to zero (see S¨rndal et al. 1992 , p. 83 for some possible solutions).

The most widely used solution is to assume reductively that an SRS variance estimate
is an upper bound for the variability of the systematic design and, as a consequence,
may be used as a cautionary first guess. The underlying hypothesis is that the order of