Geoscience Reference
In-Depth Information
along with any of the sample units that were in the initial sample but below
the threshold. This design has considerable intuitive appeal for clustered
populations. Once the rare plant or animal is observed in the initial search,
survey effort is then focused on the immediate area surrounding the occu-
pied sample unit where the rare plant or animal is most likely to be found.
The statistical properties of the process for selecting the initial sample unit
and the neighborhood search mean that the final sample will be unbiased.
The term
cluster
in adaptive cluster sampling is used to refer to the col-
lection of contiguous sample units (called a network) and the surrounding
edge units. A network is the set of sample units that surround the unit in the
initial sample that triggered neighborhood searching. All the network units
will have met the condition. The neighborhood units that are searched but
have values below the threshold are the edge units. Calculating the sample
estimators requires the definition that any unit in the initial sample that does
not meet the condition is also considered a network and it has a size of one.
The study area can be divided into distinct networks that do not over-
lap. Some of these networks will be only one unit in size; others will be larger
than one unit. In Figure 3.1, with the condition
y
i
≥ 1, the 200-quadrat study
area can be divided into 187 distinct networks. The top right-hand corner has
a network of size 5, the aggregate to the left of this is a network of size 7, and
there is a network of size 4. All the other quadrats, even those with counts in
them, are considered networks of size 1.
In simple random sampling, the probability of selecting any sample unit is
the same for each unit. With adaptive cluster sampling, the probability of a
unit being selected is more complicated because there can be many different
ways for a unit to appear in the sample. The sample unit could be in the sample
because it was in the initial selection or because any one of its neighboring
units was in the initial sample. The survey design is a form of unequal prob-
ability sampling because not all sample units have the same chance of being
selected. In Figure 3.1, the large size 5 network in the top right-hand corner
will appear in the final sample if any of the five units are in the initial sample,
whereas a size 1 network has only one chance of being in the final sample.
There are two estimators that can be used, the Horvitz-Thompson esti-
mator and the Hansen-Hurwitz estimator, although the Horvitz-Thompson
estimator is preferred (Thompson and Seber, 1996). The Horvitz-Thompson
estimate of the population total is calculated using the equation
∗
∑
K
yz
ˆ
kk
τ=
,
(3.1)
α
k
=
1
k
where
∗
y
k
is the total of the
y
values in network
k
;
z
k
is an indicator variable
equal to one if any unit in the
k
th network is in the initial sample and zero
otherwise; and α
k
is the initial intersection probability. This initial intersec-
tion probability is the probability that at least one of the units in the network
