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the additional site should be placed. Replacement is difficult because any
placement of fewer than a full grid reduces the spatial balance of the sample;
however, the replacement of systematically chosen sites does not cause infer-
ential problems any more than replacement of sites under other designs.
Spatial balance can also suffer if budget and time run out before the sys-
tematic sample is completed. If budget or time constraints preclude comple-
tion of a full systematic design, it is possible for large sections of the study
area to go unsampled unless researchers purposefully order site visits to pre-
serve the spatial balance under all realized sample sizes. Guarding against
spatial imbalance under foreshortened field seasons means researchers can-
not visit sites in a systematic order, and this reduces logistical efficiency.
This difficulty in maintaining spatial balance under addition or subtraction
of sites is not present in BAS samples, but BAS sites cannot be visited in an
arbitrary order.
10.4.2.2 General Random Samples in One Dimension
GRSs allow multiple approaches to sampling one-dimensional resources.
GRSs can be used to draw equal probability or unequal probability, ordered
or unordered, simple random or systematic samples over one-dimensional
resources. Samples drawn by the algorithm have a fixed size. Although GRS
does not generalize easily to two dimensions, one-dimensional resources
encompass a large number of natural resource types, such as streams,
beaches, lake edges, forest edges, unordered frames, and so on. Both finite
and infinite one-dimensional resources (such as points on a line) can be
sampled using the GRS algorithm, assuming the infinite resources can be
discretized into a finite list.
GRSs of size
n
are drawn as follows: Let a vector of auxiliary variables
associated with each sample unit in the population be denoted
x
. The vec-
tor
x
could be constant or could contain values such as the size of the unit,
the distance from a particular location, or the anticipated level of a target
variable. When
x
is constant, the design is equiprobable. When
x
is not con-
stant, the sample is drawn with probabilities proportional to
x
. Drawing
the GRS involves first scaling the values in
x
to sum to
n
by dividing by
their sum. If any scaled values are greater than 1, they are set equal to 1,
and the remaining values are rescaled to sum to
n
−
k
, where
k
is the num-
ber of values greater than 1. A random start between 0 and 1 is then cho-
sen and used as the starting point for a systematic sample (step size = 1) of
units associated with the scaled values. A heuristic pictorial presentation
of the GRS algorithm is given in Figure 10.7. R code for drawing a GRS is
provided on the topic's website (https://sites.google.com/a/west-inc.com/
introduction-to-ecological-sampling-supplementary-materials/).
Note that the GRS algorithm is silent regarding the ordering of units in
the population. If all elements in
x
are equal and the order of units in the
population is randomized prior to selecting the GRS, the resulting sample
