Geoscience Reference
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the observed distances (e.g., on the basis of goodness of fit or information cri-
teria) and the corresponding estimate of the density and abundance with the
data at hand. In the classic distance-sampling setting, the user may choose
among four key functions and three series adjustments (see Table 4.1). In addi-
tion, there are different options for estimating the standard error of a density
estimate using line distances. When there is only one stratum level and one
sample, this standard error is estimated in Distance. The standard errors of the
number of objects in the sampled area (the encounter rate) and the detection
probability are combined to give the standard error of the density estimate.
A more general equation for this case, given in Equation (3.68) of Buckland et
al . (2001), involves the expected cluster size as a third component of the stan-
dard error equation. Among these three components, the standard error of the
encounter rate is the most difficult to estimate, and its estimation can be biased
when there are few samples. These equations assume a Poisson distribution
for the number of objects in the sampled area, and that points are randomly
located in the study region. However, it is better if possible not to rely on spe-
cific assumptions like this and instead estimate the standard error directly
from the results obtained from replicating the sampling process.
True replications of the sampling process should be physically distinct and
be located in the study area according to a random procedure that provides an
equal chance of detection to all individuals. Given independent replication of a
line or set of lines, the density should be estimated for each replication and the
standard error of density estimated by the usual standard error of the mean
density (weighted by line length if lines vary appreciably in length). For an
TABLE 4.1
Key Functions and Series Adjustments Implemented in Distance when Right
Truncation Is Applicable to the Distance Data
Key Function
Form
Series Adjustment
Form
m
*
Uniform
1/ w
Cosine
a
cos( /)
j yw
π
j
j
=
2
= ayw
m
1
2
22
y /
σ
e
2
j
Half-normal
Simple polynomial
(/)
j
j
2
−σ
(/) b
= aH yw
m
1
e
Hazard rate
Hermite polynomial
(/)
j
2
j
j
2
Negative exponential
e ay
Source: Adapted from the tables given in Chapter 8 of the Distance User's Guide by Thomas, L.,
Laake, J.L., Rexstad, E., Strindberg, S., Marques, F.F.C., Buckland, S.T., Borchers, D.L.,
Anderson, D.R., Burnham, K.P., Burt, M.L., Hedley, S.L., Pollard, J.H., Bishop, J.R.B., and
Marques, T.A. (2009). Distance 6.0. Release 2. Research Unit for Wildlife Population
Assessment, University of St. Andrews, UK (http://www.ruwpa.st-and.ac.uk/distance/).
Note: Here, y is distance, w is the truncation distance, and σ, a , and b are model parameters.
Hermite polynomial functions H x ( y / w ) are deined in the topic by Stuart and Ord (1987,
pp. 220-227).
*
When a uniform key function is used, the summation is from j = 1 to m .
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