Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 161

analogous to a tessellation cell and its nearest neighbors in the RTS design. The
proposed variance estimator approximates the variance by averaging several con-
trasts over a local neighborhood of each sample point, and is defined

0
@

1
A

2

¼ X
k2s

X

π k X
t2NðÞ

y k

wd kt y t
π t

V NBH t HT , y

wd kl ; ð 7 : 5 Þ

l2NðÞ

where t HT , y is the HT estimator of a spatially balanced design, and N ( k ) is a local
neighborhood of unit k. The wd kl s are weights that decrease as the distance between
unit k and l increases, and are constrained in such a way that

X k wd kl ¼ X l wd kl ¼1.
The efficiency of spatially stratified designs such as GRTS increases as the
number of strata increases and the sample size per-stratum decreases. Maximum
efficiency is obviously obtained using a one-unit per-stratum-design, i.e., in the
maximal stratification . In this case, GRTS has the same efficiency as the maximally
efficient spatial stratification.
GRTS has been, and still is, the most widely used method for designing spatially
balanced samples. It has several advantages. For example, it is a probability-based
sampling technique that maintains good spatial balance. Additionally, it can be used
for sampling not only areas and points, but also linear features or phenomena that
are not contiguous. GRTS supports sampling with unequal selection probabilities,
and produces samples that are much more regularly distributed over space than an
ordinary unequal probability design such as the Sampford s design (Sampford

1967 ). However, the greatest advantage of a GRTS design is not that it is more
efficient than spatial maximal stratification, but that it is very practical, because it
can be applied in a straightforward manner in circumstances where spatial maximal
stratification is difficult. All of the problems that occur in sampling populations
(e.g., poor frame information, inaccessibility, variable probability, irregular spatial
patterns, missing data, and panel structures) can be easily addressed using the
GRTS design. However, GRTS has some disadvantages. In particular, the mapping
is not always very efficient, because units that are close in distance may be far apart
in the one-dimensional space.
There are no theoretical results and not enough empirical evidence on the
efficiency gain from using GRTS with finite populations. However, it is very
applicable to continuous surface sampling, because it provides estimators that are
very accurate and normally distributed for large samples, with a variance conver-
gence rate of order n ʳ with 1

'

<ʳ 3 (Barabesi and Franceschi 2011 ).

7.5 The Balanced Sampling and Cube Method

During the planning phase of a sample design, a practitioner may often ask to check
the quality of the selected sample by verifying how it works on some covariates
X that are known for every unit of the population U . This appears to be disrespectful