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Richard B. Harris
Wildlife Biology Program
University of Montana
Missoula, Montana 59812
1-406-542-6399
rharris@montana.com
On estimating wildlife densities from line transect data
RICHARD B. HARRIS, University of Montana, Missoula, MT USA 59812
KENNETH P. BURNHAM, Colorado State University, Fort Collins, CO USA 80523
[English version: Published in Acta Zoological Sinica (动物学报)] 48: 812-818 (2002)
Abstract: Line transects are one of the best ways to estimate density of wildlife
populations over large areas. However, density estimates will be unreliable if using
mathematical procedures that, although simple and easy to use, do not correspond with
reality. We argue here that using a naïve estimator, in which the mean of observed
perpendicular distances are equated with effective strip width, is unlikely to yield reliable
results. If conducted correctly, density estimates using this equation will most often be
too high. Instead, we urge investigators to use program DISTANCE, and to familiarize
themselves with the underlying theory, by reading Buckland et al. (1993).
------------------------------------------------------------------------------------------Key words: density estimation, detection function, Fourier series, line transect, negative
exponential distribution, program DISTANCE
________________________________________________________________
Acta Zoological Sinica (动物学报) 00:000-000
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It is now well known that estimating the abundance of wildlife populations is
fraught with difficulties. As pointed out by Sheng and Xu (1992), line transect methods
are among the best for medium and large-sized animals when estimation on large-sized
areas is required. Thus, the line transect method has increasingly become used by
Chinese wildlife scientists (e.g., Liu and Yi 1993, State Forestry Administration 1995,
Gao and Yao 1997). However, even this method can produce unreliable results if critical
assumptions are violated in the field, and/or if inappropriate mathematical analyses are
applied afterwards.
It is worthwhile reviewing the underlying assumptions of line-transect estimation
here (Anderson et al. 1979, Burnham et al. 1980, Buckland et al. 1993):
1. Objects on the center line must be observed with probability = 1.0 (i.e., every
object on the line must be detected).
2. Transect lines are placed randomly, or at least objectively, with respect to the
population being studied;
3. Objects (i.e., animals or animal groups) do not move toward or away from the
transect line in response to the observer before distances are measured;
4. Distances from the transect line to each object are measured accurately;
5. Transect line segments are straight;
6. The size of the object (or, if objects occur in groups, the size of the group)
does not affect the probability of observation (if it does, analyses that account for
size-bias must be used); and
7. Objects encountered are independent (i.e., observing an object does not affect the
probability of observing any other object);
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Additionally, sample sizes (number of objects observed) must be sufficient to
provide robust estimates of the detection function and its variance (Burnham et al. 1980
proposed a minimum of 40 for any single estimate). If sample sizes are too small, results can
be accurate in theory, but unreliable in practice. Under field conditions, it is difficult to
comply with all these assumptions and obtain reasonably large sample sizes(Southwell,
1994, Harris 1996).
After data have been appropriately collected, equations or numerical methods are
used to model the detection function, from which density is estimated. A number of
competing models of how detection decreases with distance have been proposed. Common
sense, empirical data and simulation modeling have supported use of detection functions
with a “shoulder” near the center-line, such as the Fourier series (Burnham et al. 1980) and
the half-normal (Buckland et al. 1993). Detection functions with a ‘shoulder’ are likely to
reflect reality better than other shapes, because objects are often only slightly less detectable
when near the center line than on it, whereas detectability often drops off at some distance
from the center line. Equally importantly, modern theory has stressed that detection
functions may differ among taxa, habitats, sighting conditions, and other factors. Thus,
computer programs, such as DISTANCE (Thomas et al. 1998), provide alternative detection
functions as well as metrics comparing the fit of each, allowing the user to chose the most
appropriate based on a priori or empirical information (Burnham and Anderson 1998).
A simple model of declining detection with distance is the negative exponential
(Eberhardt 1968, Gates et al. 1968, Eq. 1).
g(x) = e(-ax)
where
(Eq. 1)
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g(x) = probability of detecting the animal at perpendicular distance x,
assuming that all animals on the transect line are seen, i.e., g(0) = 1
a = parameter fitted to the data
x = perpendicular distance
However, the negative exponential function lacks a shoulder; in fact, the steepest
decline in detection is closest to the center line. An even simpler approach to treating
distance data is to equate the mean of recorded perpendicular distances with the effective
width of a sampled strip, and then to proceed with calculations (Eq. 2) as though a strip
transect had been conducted (Sheng and Xu 1992, State Forestry Administration 1995).
D = ns/2LW
(Eq. 2)
where
D = estimated density of animals (or animal groups)
n = number of animals (or animal groups) seen
s = mean group size
L = length of transect line(s)
W = mean perpendicular distance of animals (or groups) seen
However, the point estimate of density obtained using Eq. 2 will only be accurate if
the underlying (i.e., true) detection function is negative exponential. As well, Eq. 2 lacks a
theoretical basis and a method to estimate its variance.
Our objective here is to examine the use of Eq. 2 (and the conceptually similar Eq.
1) to estimating density from distance data, and to encourage Chinese scientists to use
alternative methods that have been found superior.
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PROBLEMS IN USING EQUATION 2
Equation 2 is inflexible and will usually show a positive bias
If the true decline in detectability with distance follows a negative exponential
distribution, point estimates produced by either Eq. 1 or Eq. 2 will be approximately
correct. However, they will be unreliable if other detection functions characterize the
data. Buckland et al. (1993) recommend using a modified half-normal parametric
detection function. If the half-normal detection function is true, and the “mean distances”
approach is used, a positive bias in the resulting density of 57% can be expected.
Burnham et al. (1980) conducted simulations to assess the performance of
alternative detection functions when the true, underlying detection function was known.
Table 1 reprints a portion of their results, comparing the negative exponential distribution
(Eq. 1) with the much more flexible Fourier series. As can easily be seen, the negative
exponential model performed well when the detection probability did, in fact, decline
exponentially. Under these conditions, the Fourier series produced a negative bias of
about 12-16%. However, when any other detection probability was simulated, the
negative exponential function produced highly biased results, from 10 to almost 66% too
high, while the Fourier series returned relatively unbiased results. Thus, Burnham eta al.
(1980) recommended using the Fourier series because it was more robust to varied
underlying detection functions.
One obvious way to compare the appropriateness of alternative detection
functions is to apply all of them in a situation in which density is already known. In a test
of various sighting methods performed prior to the development of rigorous line-transect
theory, Robinette et al. (1974) demonstrated that Eq. 2 produced positive proportional
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biases of 19% to 89%, with a mean proportional bias of 48%. Similarly, Parmenter et al.
(1989) showed that modeling detectability using the negative exponential model always
resulted in an upward biased.
Laake (1978) conducted experiments in which observers documented
perpendicular distances to wooden stakes placed in the ground at a known density
(37.5/ha). Even in this well-controlled experiment, observers often failed to record all
objects directly on the line, violating a critical assumption. An example of the detection
function for one of the experiments is shown in Figure 1a, where we have corrected for
the fact that, in this case, g(0) = 0.82 (rather than g(0) = 1.0). In this example, the Fourier
series estimator using program DISTANCE estimated the density as 42.5, about 13%
from the true value. Had Eq. 2 been used with these data (as illustrated in Figure 1b), the
estimated density would have been 67.9/ha (biased positively by 81%), and there would
have been no way to assess the amount of uncertainty in this estimate. This example is
not an isolated case. In a recent experimental survey of the desert tortoise in the
southwestern United States, Anderson et al. (submitted) had 12 teams estimate the
abundance of artificial tortoises in which the true number was known. The 12 estimates
varied from a negative bias of 7% to a positive bias of 13%, with a mean bias of –4%
(Table 2). Had Eq. 2 been used instead, biases would have varied from 62% to 93%, with
a mean of 70% (Table 2).
Equation 2 allows calculation without inspecting the data
Most published applications of line transects in the Chinese literature lack raw
data with which to compare competing mathematical approaches. However, Gao and Yao
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(1997) displayed their raw data on line-transect surveys of argali (Ovis ammon) in
Xinjiang. They calculated densities from line transects with sample sizes of
4,1,1,3,3,2,2,2, and 3 argali groups/transect. Even if transects from each study area had
been combined (the more appropriate procedure), total sample sizes for the 2 study areas
would have been 9 and 14, both far smaller than the recommended minimum of 40
(Burnham et al. 1980).
A cursory examination of histograms for the 2 study sites suggests that detection
did not decline with distance (Fig. 2). Thus, the fundamental assumptions for fitting any
of the possible detection functions were evidently not met. The only function that is truly
consistent with these (admittedly few) data are that detection probability was
approximately invariant at least as far as the furthest group of argali seen. Thus, for the
Hami study area (Fig. 2a), a more appropriate estimate of the width of strip “effectively”
sampled would not have been the mean perpendicular distance of 327 m, but instead the
largest perpendicular distance of 380 m. Doing so would have reduced their estimated
density of 0.53 argali/km2 to 0.41 argali/km2. Similarly, for the Mulei study area (Fig.
2b), the estimated density of 0.82 argali/km2 would have been more appropriately
estimated as 0.54 argali/km2. By using Eq. 2, Gao and Yao (1997) had no need to
examine histograms of their data, or to consider the implications of assuming the negative
exponential.
AN EXAMPLE FROM FIELD WORK IN CHINA
Field work in China is particularly difficult, and many of the means for
conducting line transects used in the West (e.g., aircraft) are not available. However,
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sufficient observations can sometimes be obtained even in the difficult conditions
Chinese scientists usually find themselves in. From these, detection functions can be
estimated using program DISTANCE, rather than using Eq. 2. For example, Harris
(1996, see also Harris and Miller 1995, Harris et al.1996) walked randomly placed line
transects in Qinghai to estimate densities of Tibetan gazelle (Procapra picticaudata). The
sample sizes obtained (N=64 groups) allowed the estimation of the detection function
using the Fourier series (Fig. 3). It appears that even here, some “heaping” occurred in
those distance categories closest to the center-line, which should be avoided if possible.
RECOMMENDATIONS
It may be tempting to apply Eq. 2 because it is so easy to calculate. However, the
accumulated experience in western countries, illustrated briefly by the examples provided
here, is that it forms an unreliable basis for estimating density from distance data.
Considerable effort has gone into providing user-friendly computer software (Thomas et
al. 1998) and explanatory text material (Burnham et al. 1980, Buckland et al. 1993) for
methods that are known to be more reliable. Both the software (program DISTANCE)
and the accompanying text book (Buckland et al. 1993) are available at no cost over the
internet from site http://ruwpa.st-and.ac.uk/distance. Now that computers and internet
access are becoming more common in China, the methods provided by program
DISTANCE should be used whenever possible. It is true that program DISTANCE is not
a panacea; users can (and will) choose differing ways of treating data, resulting in slightly
different density estimates. As well, the most robust and precise estimators often
underestimate true density slightly because real data rarely match the ideal perfectly.
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However, recent work has shown that, given an introduction to the important concepts,
conscientious investigators will produce results varying by less than 10% from one
another using program DISTANCE (Anderson and Southwell 1995). Thus, even if a
small negative bias cannot be avoided, results will be fairly consistent from survey to
survey.
However, no detection function will perform well when sample sizes are very
small. For example, Gao and Yao (1997) reported density estimates from lines surveyed
in which only a single group of animals (i.e., n=1) was observed. Such an estimate is, of
course, theoretically possible using Eq. 2. However, investigators should not be fooled
into thinking that they really know much about the density of animals in an area when
they have only a single group with which to model detection. Unless there are persuasive
reasons to avoid doing so, it is appropriate to combine data from replicate surveys, or
portions of a study area, to achieve reasonable sample sizes in estimating a detection
function. However, if, after having combined similar lines, sample sizes are still quite
small (e.g., 10-20), it is then best to avoid estimating densities from distances. Instead, it
is more prudent to simply report the number and type of animals observed (as well as
thoroughly documenting methods used), and treating the results as an index to
abundance. This index cannot be used to determine absolute density or abundance, but
might still be useful if repeated periodically to obtain a rough idea of population trends.
Similarly, estimates using program DISTANCE are invalidated to the degree that
field procedures violate the fundamental assumptions of line-transect sampling. If
sampling cannot be conducted in a way that minimizes assumption violations, it is again
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advisable to simply report the raw data and methods used, rather than attempt to derive a
density when no model exists to do so.
Finally, the importance of a rigorous, objective sampling regime cannot be
stressed enough. Extrapolations of density are only valid if the transects sampled truly
represent an unbiased selection of all possible transects within the area of interest.
ACKNOWLEDGEMENTS
Work in China was funded by the Robert M. Lee Foundation and the Liu Guo Lit
Charitable Trust. S. T. Buckland and J. L. Laake provided suggestions to improve the
manuscript.
LITERATURE CITED
Anderson, D.R., Laake, J.L., Crain, B.R., and Burnham, K.P. (1979). Guidelines for line
transect sampling of biological populations. Journal of Wildlife Management.
43:70-78.
Anderson, D.R. and Southwell, C. (1995). Estimates of macropod density from line
transect surveys relative to analyst expertise. Journal of Wildlife Management.
59:852-857.
Anderson, D. R., K. P. Burnham, B. C. Lubow, L. Thomas, P. S. Corn, P. A. Medica, and
R. W. Marlow. 2001. Field trials of line transect methods applied to estimation of
desert tortoise abundance. Journal of Wildlife Management 65: 583-597.
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Buckland, S. T., D. R. Anderson, K. P. Burnham, and J. L. Laake. 1993. Distance
sampling: Estimating abundance of biological populations. Chapman and Hall,
London. 446 pages. Available from http://ruwpa.st-and.ac.uk/distance.
Burnham, K. P., and D. R. Anderson. 1998. Model selection and inference: A practical
information-theoretical approach. Springer-Verlag, New York. NY.
Burnham, K.P., Anderson, D.R., and Laake, J.L. (1980). Estimation of density from line
transect sampling of biological populations. Wildlife Monographs. 72:1-202.
Eberhardt, L.L. (1968). A preliminary appraisal of line transects. Journal of Wildlife
Management. 32:82-88.
Gao X. Y. and J. Yao. 1997. Argali of the eastern Tianshan, Xianjiang. Chinese Wildlife
(Yesheng Dongwu) 18(4):38-40. (in Chinese).
Gates, C. E., W. H. Marshall, and D. P. Olson 1968. Line transect method of estimating
grouse population densities. Biometrics 24(1):135-145.
Harris, R. B., D. J. Miller, Cai G. Q., and D. H. Pletscher. 1996. Wildlife status and
conservation in Yeniugou, Qinghai. Acta Theriologica Sinica 16:113-118 (in
Chinese, English version available).
Harris, R. B. 1996. Wild ungulate surveys in grassland habitats: Satisfying
methodological assumptions. Chinese Journal of Zoology 31(2):16-21. (in
Chinese, English version available).
Harris, R. B. and D. J. Miller. 1995. Overlap in summer habitats and diets of Tibetan
plateau ungulates. Mammalia 59:197-212.
Laake, J. L. 1978. Line transect estimators robust to animal movement. M.S. Thesis, Utah
State Univ., Logan. 55 pp.
Page 12
Liu, W. L. and B. G. Yi. 1993. Wildlife protection in Tibet. China Forestry Press,
Beijing. (in Chinese).
Parmenter, R. R., J. A. MacMahon, and D. R. Anderson. 1989. Animal density estimation
using a trapping web design: field validation experiments. Ecology 70:169-179.
Robinette, W.L., Loveless, C.M., and Jones, D.A. (1974). Field tests of strip census
methods. Journal of Wildlife Management. 38:81-96.
Sheng, H. L, and H. F. Xu. 1992. Field Research Methods for Mammals. China Forestry
Press, Beijing. (in Chinese).
Southwell, C. (1994). Evaluation of walked line transect counts for estimating macropod
density. Journal of Wildlife Management. 58:348-356.
State Wildlife Protection Office. 1995. National terrestrial wildlife resource survey and
monitoring methods. Ministry of Forestry, Beijing. December, 1995. (in Chinese).
Thomas, L., J. L. Laake, J. F. Durry, S. T. Buckland, D. L. Borchers, D. R. Anderson, K.
P. Burnham, S. Stringberg, S. L. Hedley, M. L. Burt, F. Marques, J. H. Pollard,
and R. M. Fewster. 1998. Program DISTANCE 3.5. Research Unit for Wildlife
Population Assessment, University of St. Andrews, U.K. Available from
http://ruwpa.st-and.ac.uk/distance.
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Table 1. Mean percent relative bias of the negative exponential function ( Eq. 1)
and the more flexible Fourier series, when applied to simulated data using
underlying negative exponential, half-normal, and modified beta detection
functions. Values in each case are means from 25 simulations, the sample size
of distances in each simulation was 100. Data taken from Burnham et al.
(1980:158).
________________________________________________________________
Method Used
-----------------Negative Exponential
Fourier Series
----------------------------
-------------------
Underlying Detection Function
--------------------------------------------------------------------------------------------------Negative Exponential
- Severely truncated
- 0.3
- 13.7
- Moderately truncated
- 1.8
- 12.5
- Untruncated
+ 1.9
- 16.5
- Severely truncated
+ 10.4
+ 0.1
- Moderately truncated
+ 26.5
+ 1.1
- Untruncated
+ 59.7
+ 2.5
- Shoulder
+ 65.9
- 5.3
- Linear
+ 47.7
- 7.1
- Spiked
+ 32.8
- 5.3
Half-Normal
Modified Beta
___________________________________________________________
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Table 2. Abundance estimates of artificial tortoises using Eq. 2 and the Fourier
series method (Burnham et al. 1980) from 12 different survey teams. Data are
taken from Anderson et al. (submitted). The true density of tortoises in all 12
cases was 76.
________________________________________________________________
Team
N
Eq. 2
Percent Bias
Fourier
Percent Bias
Estimate
Estimate
--------------------------------
-------------------------------
________________________________________________________________
1
47
124
+63%
74
- 3%
2
49
128
+68%
78
+ 3%
3
52
126
+66%
75
- 1%
4
55
129
+70%
79
+ 4%
5
57
123
+62%
73
- 4%
6
57
129
+70%
71
- 7%
7
64
147
+93%
86
+ 13%
8
52
132
+74%
79
+ 4%
9
52
127
+67%
75
- 1%
10
60
130
+71%
79
+ 4%
11
59
144
+89%
85
+ 12%
12
55
125
+64%
72
- 5%
---------------------------------------------------------------------------------------------------Mean
129
+70%
73
-
4%
_____________________________________________________________
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Figure Captions.
Figure 1. Histograms and fitted detection functions for an example of distance data. Data
are from wooden stakes placed in the ground at a known density (Burnham et al.
1980:62). A. The Fourier series detection function, which yielded an estimate of 42.5
stakes/ha. B. The negative exponential detection function, which is assumed when using
Eq. 2, which yielded an estimate of 67.9 stakes/ha. The true density of stakes as 37.5ha-2.
Figure 2. Histograms of perpendicular distances of argali (Ovis ammon) observed during
line transect surveys conducted by Gao and Yao (1997). A. Surveys in Hami (Table 1
from Gao and Yao 1997), fitted with the negative exponential function. B. Same as A.,
except fitted using a half-normal function. C. Surveys in Mulei (Table 2 from Gao and
Yao 1997), fitted with the negative exponential distribution. D. Same as C., except fitted
using a half-normal function. Compare the shape of histograms from those in Fig. 1.
Figure 3. Fourier series detection functions superimposed on histograms of perpendicular
distances from a survey of Tibetan gazelles in Qinghai province, using program
DISTANCE, N = 64.
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A.
1.2
DETECTION PROBABILITY
1.0
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
PERPENDICULAR DISTANCE (m)
1.2
DETECTION PROBABILITY
1.0
0.8
0.6
0.4
0.2
0.0
0
5
10
15
20
25
PERPENDICULAR DISTANCE (m)
30
32
34
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Line Transect Methods
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Figure 2.
1 .6 6 5 5 5
1 .8 8 2 6 5
1 .4 9 8 9 9
1 .6 9 4 3 9
1 .3 3 2 4 4
1 .5 0 6 1 2
1 .1 6 5 8 8
1 .3 1 7 8 6
0 .9 9 9 3 2 9
1 .1 2 9 5 9
0 .8 3 2 7 7 4
0 .9 4 1 3 2 6
0 .6 6 6 2 2
0 .7 5 3 0 6
0 .4 9 9 6 6 5
0 .5 6 4 7 9 5
0 .3 3 3 1 1
0 .3 7 6 5 3
0 .1 6 6 5 5 5
0 .1 8 8 2 6 5
0
0
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
P e rp e n d ic u la r d is ta n c e in m e te rs
400
500
600
700
800
900
1000
P e r p e n d ic u la r d is ta n c e in m e te r s
A. Hami data: Negative exponential function
A. ????????? ?????
B. Hami data: Half-normal function
B. ????????? ?????
1 .7 1 1 3
1 .6 8 9 4 5
1 .5 4 0 1 7
1 .5 2 0 5
1 .3 6 9 0 4
1 .3 5 1 5 6
1 .1 9 7 9 1
1 .1 8 2 6 1
1 .0 2 6 7 8
1 .0 1 3 6 7
0 .8 5 5 6 5 1
0 .8 4 4 7 2 3
0 .6 8 4 5 2
0 .6 7 5 7 7 9
0 .5 1 3 3 9
0 .5 0 6 8 3 4
0 .3 4 2 2 6
0 .3 3 7 8 8 9
0 .1 7 1 1 3
0 .1 6 8 9 4 5
0
0
0
100
200
300
400
500
600
P e rp e n d ic u la r d is ta n c e in m e te rs
C. Mulei data: Negative exponential function
C.. ????????? ?????
700
800
900
1000
0
100
200
300
400
500
P e rp e n d ic u la r d is ta n c e in m e te rs
D. Mulei data: Half-normal function
D. ????????? ?????
600
700
800
900
1000
Line Transect Methods
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1.6
DETECTION PROBABILITY
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
200
400
600
800 1000 1200 1400 1600 1800 2000 2200 2400
PERPENDICULAR DISTANCE (m)
Figure 3.