Content uploaded by Len Thomas

Author content

All content in this area was uploaded by Len Thomas on Mar 05, 2016

Content may be subject to copyright.

Available via license: CC BY 4.0

Content may be subject to copyright.

RESEARCH ARTICLE

The Effect of Animal Movement on Line

Transect Estimates of Abundance

Richard Glennie*, Stephen T. Buckland, Len Thomas

Centre for Research into Ecological and Environmental Modelling, University of St Andrews, St Andrews, UK

*rg374@st-andrews.ac.uk

Abstract

Line transect sampling is a distance sampling method for estimating the abundance of wild

animal populations. One key assumption of this method is that all animals are detected at

their initial location. Animal movement independent of the transect and observer can thus

cause substantial bias. We present an analytic expression for this bias when detection with-

in the transect is certain (strip transect sampling) and use simulation to quantify bias when

detection falls off with distance from the line (line transect sampling). We also explore the

non-linear relationship between bias, detection, and animal movement by varying detect-

ability and movement type. We consider animals that move in randomly orientated straight

lines, which provides an upper bound on bias, and animals that are constrained to a home

range of random radius. We find that bias is reduced when animal movement is constrained,

and bias is considerably smaller in line transect sampling than strip transect sampling pro-

vided that mean animal speed is less than observer speed. By contrast, when mean animal

speed exceeds observer speed the bias in line transect sampling becomes comparable

with, and may exceed, that of strip transect sampling. Bias from independent animal move-

ment is reduced by the observer searching further perpendicular to the transect, searching

a shorter distance ahead and by ignoring animals that may overtake the observer from be-

hind. However, when animals move in response to the observer, the standard practice of

searching further ahead should continue as the bias from responsive movement is often

greater than that from independent movement.

Introduction

Line transect sampling is a distance sampling method [1,2], widely used for estimating the

abundance of wild animal populations. Lines are placed randomly across a study region, or

more usually a systematic grid of lines is randomly positioned, and the observer traverses each

line, recording the perpendicular distance from the line to each detected animal. Typically, the

observer fails to detect all animals in the surveyed strip, and these distances are used to model

the probability of detection, and hence to estimate abundance.

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 1/15

a11111

OPEN ACCESS

Citation: Glennie R, Buckland ST, Thomas L (2015)

The Effect of Animal Movement on Line Transect

Estimates of Abundance. PLoS ONE 10(3):

e0121333. doi:10.1371/journal.pone.0121333

Academic Editor: Marco Festa-Bianchet, Université

de Sherbrooke, CANADA

Received: November 18, 2014

Accepted: February 10, 2015

Published: March 23, 2015

Copyright: © 2015 Glennie et al. This is an open

access article distributed under the terms of the

Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any

medium, provided the original author and source are

credited.

Data Availability Statement: All relevant data are

within the paper.

Funding: This work was supported by the University

of St Andrews (http://www.st-andrews.ac.uk/; RG,

STB, LT) and by a summer scholarship and PhD

grant from The Carnegie Trust for the Universities of

Scotland (http://www.carnegie-trust.org/) to RG. The

funders had no role in study design, data collection

and analysis, decision to publish, or preparation of

the manuscript.

Competing Interests: The authors have declared

that no competing interests exist.

The method relies on three key assumptions, and if these are not met then estimates of

abundance can have substantial bias [1] (pp. 29–37). These are:

1. animals on the line are detected with certainty;

2. measurements are exact;

3. animals are detected at their initial location.

When the first assumption is not met, double-observer (or multi-observer) methods, which

combine distance sampling with mark-recapture, may be used [3,4]. Similarly, if the second as-

sumption is violated, measurement error models [5,6] can be adopted. We concentrate on vio-

lations of the third assumption.

Conceptually, line transect sampling is a snapshot method [1](pp.31),thatis,weconsiderthat

theanimalsareatafixedlocationwhilethesurveytakesplace.Thus,wehaveprobabilisticen-

counters between a moving observer and immobile animals [7]. There are models [8,9]foren-

counters between mobile animal populations and observers, but these are limited in use: they rely

upon quantities that are difficult to determine (mean animal speed, encounter radius) and they as-

sume an ideal free gas movement model which can be unrealistic [10]. Making the third assump-

tion avoids such problems. This assumption can be violated in two ways: animals can move in

response to the observer, or move independently of the observer. Responsive movement is a com-

mon problem in distance sampling surveys and the bias it causes can be reduced by modelling the

movement [11,12] or using double-observer methods [13]. In a few specific cases, independent

movement has been modelled (for fish: [14]; for seabirds: [15]) to reduce bias; however, these

methods are ad hoc and specific to their application. We do not explore responsive movement

here; instead, we consider how animal movement independent of the observer affects bias. For

whale surveys, it was concluded that such movement may seriously bias the estimate of mean den-

sity if the speed of the whales approaches that of the observer, if the encounter region is long rather

than wide, or if the probability of sighting a whale is not strongly dependent on the time it spends

within that region [16]. Subsequently, simulation revealed that bias was negligible if mean animal

speed was one quarter of that of the observer, but not if animal speed was one half that of the ob-

server [17]. This conclusion has now been adopted within distance sampling [1] (pp. 131, 173)

and is used to determine if independent animal movement is a problem in particular studies [18].

This is despite these results being pertinent only to the single simulation scenario considered.

In this paper, we quantify the bias caused by independent animal movement in some exam-

ple cases and discuss the non-linear relationship between movement, detection and bias. We

first consider strip transect sampling which is a special case of line transect sampling in which

all animals within a strip are assumed to be detected [1] (pp. 2–3). We derive an analytic ex-

pression for the bias caused when animals move in straight lines, each with the same constant

speed but an individual random direction, within a rectangular study area. We then generalize

to surveys in which detectability falls off with distance from the line, and for which animal

movement is more complex, using simulation to quantify bias. Finally, we discuss what these

results reveal about bias caused by animal movement independent of the observer.

Strip Transect Sampling

Consider estimating the abundance of an animal population using strip transect sampling

where the study area is rectangular of length aand breadth Lwith corners, in (x,y)-coordi-

nates, at (−a/2, 0), (a/2, 0), (−a/2, L), (a/2, L). We assume the number of animals within the

study area is fixed, and so adopt a wrap-around model: for each animal that exits across one

boundary of the study area, a new animal enters immediately at the same distance along the

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 2/15

opposite boundary. Intuitively, we think of the study area as part of a wider region. We assume

the (strip) transect is placed randomly within this region and so may take the transect to be at

x= 0. Let the transect have half-width w, where 2w<a, extending out to either side of x=0

and length L, so it transverses the entire study region in the y-direction.

We assume animal itravels in a straight line at speed uand in a direction θ

i

where θ

i

is a

random deviate from a uniform distribution U(0, 2π](Fig. 1). Thus, the equations of motion of

the ith animal are given by:

yi;t¼yi;0þucos ðyiÞt

xi;t¼xi;0þusin ðyiÞt

where tis time and (x

i,0

,y

i,0

) is the initial (t= 0) location of animal i.

The observer travels along the centerline, x= 0, at speed v>0, starting at y= 0. Thus, the

position of the observer at time tis given by (0, z

t

) where z

t

=vt. We define the encounter region

to be the area around the observer within which every animal is recorded. No animals are re-

corded outside the encounter region. Here, we assume the encounter region is rectangular, ex-

tending a distance weither side of the centerline and a distance l, which we call the lookahead

distance, in front of the observer (Fig. 1). The encounter region is wrapped using the wrap-

around model to fix its size. (There are practical methods of implementing a wrap-around de-

sign with strip transects, using buffer zones [19].)

When estimating abundance using strip transect sampling, it is assumed that the animals

are immobile—this is termed a snapshot method as the survey is assumed to take place instan-

taneously [1] (pp. 31). When true, the expected proportion of animals within the strip is

P1¼2w

a. However, if animals are mobile then they can move into the encounter region and so

be recorded; hence, we expect to record more animals, which causes positive bias in

abundance estimates.

Animals not initially inside the encounter region can enter in three ways: from in front,

from the side and from behind. We shall consider each in turn. First, bias is not caused by ani-

mals travelling toward the observer and entering the encounter region from in front since on

average in any interval of time an equal number of animals, that if immobile would be re-

corded, will move away from the observer and not be recorded. Second, bias is caused by ani-

mals entering the encounter region from the side. Any point along the strip is within the

encounter region of length lahead of the observer for l

vunits of time. In that time, all animals

with direction θand speed utravel lusinðyÞ

vunits of distance perpendicular to the line. Hence the

proportion of all such animals not initially inside the encounter region entering from the side

and so being recorded is given by lujsinðyÞj

av . Given that θis uniform on (0, 2π], the expectation of

this proportion is

P2¼luE½j sin ðyÞj

av ¼2lu

pav

as E½jsinðyÞj ¼ 2

p.

Lastly, animals can only enter the encounter region from behind by overtaking the observer;

hence, no bias is caused when animals travel slower than the observer. Under our movement

model, only animals that are initially behind the transect, y<0, can overtake the observer and

enter the encounter region. By the equations of motion given above, an animal with direction θ

and initial location (x

0

,y

0

) overtakes the observer at time

T¼y0

vucos ðyÞ

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 3/15

Thus, to have 0<T<L

vwhen y

0

<0 and have the animal recorded, we require that the follow-

ing inequalities are satisﬁed:

v

u<cos ðyÞ

b

vðucos ðyÞvÞy00

wxTþkw

for some k2[0, T

δ

] where Td¼l

ucosðyÞvis the time the animal is within the encounter region.

From these inequalities, we see that within the subset of those animals with direction θsuch

Fig 1. Strip transect with encounter region. Transect line (dashed grey line) with the strip extending a

distance wout to either side (solid grey lines). The observer and encounter region at time t−1 (dashed black)

and t(solid black) are shown, together with an animal moving with speed uin direction θ

doi:10.1371/journal.pone.0121333.g001

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 4/15

that cosðyÞ>v

u, the proportion that are recorded is given by

ð2wþujsin ðyÞjTdÞL

vðucos ðyÞvÞ

aL

The expected value of this proportion with respect to θis

2wﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

u2v2

p

cos 1v

u

av 2w

aþ

ul 1v

u

av cos 1v

u

since EjsinðyÞj jcosðyÞ>v

u

¼1v

u

cos1ðv

uÞand EcosðyÞjcosðyÞ>v

u

¼ﬃﬃﬃﬃﬃﬃﬃﬃ

u2v2

p

ucos1v

u

ðÞ

. Finally, since the

proportion of animals with required direction θis given by PcosðyÞ>v

u

¼1

pcos1v

u

, the

total proportion of animals causing this bias is

P3¼2wﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

u2v2

p

apv2w

pacos 1v

u

þlðuvÞ

pav

We are now able to give an expression for the total bias in the abundance estimate. If Pis

the total expected proportion of animals detected in the strip transect, then when uv,P=P

1

+P

2

and when u>v,P=P

1

+P

2

+P

3

. Thus, strip transect sampling produces an abundance

estimate with percentage bias given by

100

ul

pwv uv

ul

pwv þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

u2v2

p

pv1

pcos 1v

u

þluvðÞ

2wpvu>v

8

>

>

>

<

>

>

>

:

In particular, if movement is absent (u= 0) or animal speed is less than observer speed (uv)

and the observer only searches abeam, not ahead (l= 0), then there is no bias. Nevertheless, the

bias caused can be substantial. For example, if we assume that the observer searches as far

ahead as to the side, l=w, and the animals are slower than the observer, u<v, then bias is the

ratio of animal speed to observer speed, divided by π. Thus, if animal speed is one third observ-

er speed, bias is 1/(3π) = 10.6%.

Line Transect Sampling

In line transect sampling, some animals in the covered strip may be undetected. Movement in-

dependent of the observer causes bias in the estimated probability of detection, and this bias

combines with the bias arising from movement of animals into the encounter region, so that

the formulae of the previous section no longer apply.

The detection process can be modelled as either a continuous [20,21] or a discrete [20,22]

hazard-rate process. The discrete model is more appropriate for surveys of whales, when detec-

tion cue is primarily a whale blow, or for songbirds, if the cue is a songburst. Hiby adopted a

discrete hazard-rate model [17]. To avoid introducing another parameter (cue rate), we adopt

a continuous-time model.

By combining animal movement with a hazard-rate detection process, the problem of deriv-

ing an analytic expression for bias appears intractable as it is a non-linear function of the mean

probability of detection. Therefore, a simulation study was performed in order to quantify bias

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 5/15

and investigate the relationship between bias, movement and detection. We first describe the

simulation setup (Section 1) and then present and discuss the results as they relate to bias (Sec-

tion 2), the detection function (Section 2.1), lookahead distance (Section 2.2) and truncation

distance (Section 2.3).

1 Simulation Setup

We consider two animal movement types at a range of speeds relative to the observer. We also

vary the maximal distance the observer searches ahead (the lookahead distance) and the shape

of the detection function: both control the size of the encounter region around the observer.

The study area and transect are defined similarly to the strip transect case given above. The

study area was one square kilometre and a wrap-around model was used. The transect was one

kilometre long and bisected the study area. There were 100 animals distributed, for each simu-

lation, randomly across the study area.

Two types of animal movement were considered. First, the linear movement model, the

same as in the strip transect case, was used, where all animals moved in straight lines at the

same speed but each in a randomly assigned direction. It is particularly useful to quantify the

bias arising from this model: for any other random animal movement, each animal will travel

through a smaller range of perpendicular distances from the line compared with linear move-

ment, and so bias will be less as bias depends critically upon how far an animal travels from its

initial location. Thus, the bias arising from linear movement is an upper bound for the bias

caused by any other type of random animal movement. The second model considered is a

home range model: each animal is contained within a circular home range with a randomly as-

signed radius from a Gamma distribution with mean 20 and variance 11. Inside its home

range, the animal performed a correlated random walk, where the direction of travel was per-

turbed at each time step of the simulation by adding a random deviate from a wrapped normal

with zero mean and 0.1 variance. When an animal reached the boundary of its home range, it

was reflected back by changing its direction of travel by πradians. By restricting the range of

perpendicular distances from the line the animal may travel through, we expect the home

range model to produce smaller bias than the linear movement model; clearly, however, this

will depend critically on the radius of the home ranges. If home range radius greatly exceeds

the strip half-width w, then the constraints to movement would have little effect on the bias. If,

however, the radius is smaller than w, as might occur with many songbird territories, then the

constraints on movement should result in appreciably lower bias. For each movement model

we consider animal speeds from 0 m s

−1

to 20 m s

−1

where all animals have the same speed. If

each animal’s speed were to vary around a population mean, we expect bias would be similar

(as indicated by some preliminary simulation) since bias depends strongly on the mean speed

of the population rather than on an individual’s instantaneous speed.

The observer travels along the line at 10 m s

−1

and the probability of detecting an animal is

modelled as a continuous hazard-rate process. However, as simulation studies are, by nature,

discrete we integrate this hazard over the time step used in the simulation to give the probabili-

ty of being detected within that time step. Thus, the probability of being detected in time step t

with the continuous radial hazard k(r)=cr

−d

where c>0, d>2is

gtðr0;r1Þ¼1exp xt

s2

b

Ix2

t

r2

1

Ix2

t

r2

0

0

B

B

@

1

C

C

A

0

B

B

@

1

C

C

A

where x

t

is the perpendicular distance of the animal from the line at time t,r

0

is the radial

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 6/15

distance of the animal from the observer at the beginning of the time step, r

1

is the radial dis-

tance at the end of the time step, b=d−1, sb¼cGð0:5bÞGð0:5Þ

2Gð0:5dÞand I

r

is the cumulative distribution

function of a Beta random variable evaluated at rwith parameters (b, 0.5). Here, we use a time

step of 0.05s duration. We will also consider different sizes of the encounter region by changing

the shape parameter (b= 2, 3, 5) and the lookahead distance (l= 0, 5, 10, 15, 20, 25, 30). Finally,

we will investigate the effect that truncating the data at the analysis stage has on bias by trun-

cating at distances where the true detection function appoximately equals 0.05, 0.1, 0.2, 0.4, 0.6

and 0.8.

For each animal speed and movement model, we simulated one thousand surveys consisting

of five hundred transects each. We did not consider any model selection: we fit a hazard-rate

model only. The simulated data were analysed using the Multiple-Covariate Distance Sampling

(MCDS) engine that is included in the software Distance [23]. The computer code used to per-

form this simulation study is included as online supporting material (S1 Code).

2 Simulation Results

Fig. 2 gives the bias estimated from the simulation study (with hazard-rate shape parameter

b= 2) for each relative animal speed and movement model. The bias caused in strip transect

sampling, calculated from the analytic expression in Section 1, with the observer searching as

Fig 2. Percentage bias in the abundance estimate for strip and line transects. Bias for strip transect

sampling (solid line) calculated from the analytic expression given and for line transects (hazard-rate b=2)by

averaging over 1000 simulations for linear movement (dashed line) and home range movement (dotted line)

doi:10.1371/journal.pone.0121333.g002

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 7/15

far ahead as to the side, is also included in the figure for comparison. The strip transect bias in-

creases linearly when animal speed is less than observer speed, and then increases nonlinearly

and rapidly thereafter. Bias in line transect sampling under both movement models behaves

similarly as animal speed increases, but the bias at each speed is less than in strip transects. For

strip transect sampling, bias reaches nearly 10% when animal speed is just 30% of observer

speed, while for our simulation of line transect sampling under linear animal movement, bias

does not reach 10% until animal speed is 80% of observer speed. However, for this scenario, at

very large animal speeds (150% –200% observer speed) the bias in line transect sampling is

similar to or exceeds that in strip transect sampling.

We now consider these results. First, the difference in bias between linear movement and

home range movement confirms that bias is related closely with the range of perpendicular dis-

tances from the line that an animal can travel through. At smaller speeds, the animals move

through a similar range of perpendicular distances under each movement model which ex-

plains why bias is similar between the two. However, at larger speeds, animals under the linear

movement model travel through a large range of perpendicular distances compared with the

home range model where animals are constrained to a local territory; hence, linear movement

causes more bias at these speeds. This shows that bias from animal movement is related to the

distance an animal of interest may travel—the smaller that distance, the less the bias. Second,

these results show that any difference in the bias in strip and line transect sampling is due to

the detection process as line transect sampling is a generalisation of strip transect sampling

where detection of animals is uncertain.

2.1 Detection Function. Fig. 3 shows that as speed increases, the estimated detection

probability falls off more rapidly and the shoulder (range of distances from the line for which

probability of detecting an animal is one) narrows. This can also be seen in the distribution of

observed distances in Fig. 4. The effect on bias of changing the shape parameter of the detection

function is shown in Fig. 5. Here, we discuss how the bias in the estimated detection function

occurs, how it relates to bias in abundance estimates, and how the shape parameter of the func-

tion affects bias.

Bias in the estimated detection function is a result of animal movement and the shape of the

true detection function. As speed increases, the number of detections close to the line increases

rapidly since more animals are able to travel very close to the line during the survey and so be

detected. The number of detections at distances further from the line does not increase as rap-

idly since some of the animals travelling toward the line that are detected replace those that re-

main undetected by travelling away from the line. This does not occur very near to the line

since there the probability of detection is close to one and so the animals travelling both toward

and away from the line are detected. The rapid fall of detections as distance from the line in-

creases is the reason the detection function falls more rapidly at greater speeds. This bias in the

estimated detection function generates positive bias in the abundance estimate as the probabili-

ty of detection is underestimated.

Nevertheless, bias in line transect estimates of abundance is less than that for strip transects

when animal speed is less than 200% of observer speed. This is due to the detection process. In

strip transects, animals entering the strip are detected at the boundary at which they enter

since detection probability is one inside the strip and zero outside. In line transects, the animals

that would be detected at the boundary in strip transects are unlikely to be detected as they ap-

proach from a distance with small detection probability. Furthermore, as detection probability

falls off gradually from the line some of these animals, if detected, may replace those that

moved further from the line that would, if immobile, have been recorded. So, the reduction in

the number of detections by the detection process reduces bias by a greater amount than the

bias in the estimated detection function augments it when animal speeds are slow. Conversely,

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 8/15

at animal speeds close to 200% of observer speed, bias in line transect sampling becomes simi-

lar to that in strip transect sampling due to the biased estimated detection function. At these

speeds, the detection function falls off very rapidly and many animals are assumed to be unde-

tected resulting in very high bias in abundance estimates which may exceed the bias in strip

transect sampling.

Bias is also related to the shape of the detection function (Fig. 5). At slow animal speeds,

bias is similar for all shape parameters considered (b= 2, 3, 5) which is unsurprising as the bias

arises from too many detections close to the observer and all three functions here have similar

shapes in this region (they all approximately equal one). At animal speeds exceeding observer

speed, the hazard rate with shape parameter b= 5 consistently produces smaller bias than the

other two shape parameters (b= 2, 3). This may be due to the assumed detection function

being isotropic: animals overtaking the observer are less likely to be detected ahead of the ob-

server when b= 5 than b= 2, 3 since the shoulder of the detection function is narrower. How-

ever, here we not only change the shape of the detection function but also the probability of

detection and so the bias may be a result of both effects.

2.2 Lookahead Distance. Fig. 6 shows that bias, at each speed, increases toward a plateau

as lookahead distance increases for a fixed detection function. The bias reaches a plateau since

the probability of detection at a large lookahead distance is small and so further increasing

Fig 3. Estimated detection function for line transect sampling for increasing animal speeds. Estimated

detection function averaged over 1000 simulations for line transect sampling (hazard-rate b= 2) with the

linear animal movement model at animal speeds 0% (black solid line), 20% (red dashed line), 40% (blue

dotted line), 80% (green dotdash line), 150%(purple longdash line) and 200% (grey twodash line)

doi:10.1371/journal.pone.0121333.g003

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 9/15

lookahead distance results in a smaller increase in detections—and so bias increases by less.

Also we see for small animal speeds, the bias increases slowly with lookahead distance, while

for animal speeds that exceed observer speed, bias increases rapidly. As for strip transect sam-

pling, this rapid increase is due to animals overtaking the observer from behind. For larger loo-

kahead distances, such animals remain in the encounter region for longer and are thus more

likely to be detected. Non-zero bias when the observer searches abeam (lookahead distance is

zero) is another consequence of animal speed exceeding observer speed, when animals travel

slower than the observer no bias occurs.

2.3 Truncation Distance. When analysing line transect data, it is common to truncate the

recorded distances at some truncation distance to prevent outliers unduly influencing the

abundance estimate [1] (pp. 103–108). We explored the effect of truncation distance on bias

with the linear animal movement model and b= 2 hazard-rate. We assumed that observers

were actively searching out to distances beyond the truncation distance, and recorded distances

of detected animals from the line when they were first detected. Thus, truncation was assumed

to be applied by the analyst, not in the field.

Results indicate little if any relationship between bias and truncation distance (Fig. 7). This

is unsurprising as bias is caused by too many detections near to the line which truncation does

not affect.

Fig 4. Frequency of detections at each perpendicular distance from the line. Estimated histogram of

recorded distances averaged over 1000 simulations for line transect sampling (hazard-rate b= 2) with linear

animal movement at animal speeds 0% (black solid), 20% (red dashed) 80%(green dotted) and 150% (purple

dotdash)

doi:10.1371/journal.pone.0121333.g004

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 10 / 15

Discussion

Animal movement independent of the observer generates bias in abundance estimates from

line transect surveys. The degree of bias depends both on animal movement and dectectability.

For a given mean animal speed, the analytical strip transect expression provides an upper

bound for this bias across all possible detection functions and movement types. The expression

shows bias to be reduced when the observer searches further to the side and a shorter distance

ahead (smaller lookahead). Also, the result shows the significant bias caused when animals

travel faster than the observer [1] (pp. 131). To reduce bias in strip transect sampling, observers

should record only those animals that were initially inside the strip and ignore those animals

that enter from the side or behind [24]. As line transects are often much wider than strip tran-

sects and detection of animals is uncertain [25], it is not possible to exclude animals that enter

the strip from the side, but often animals that overtake the observer can (and should)

be excluded.

Bias in line transect sampling for most animal speeds is considerably less than that of strip

transect sampling. The magnitude of the bias depends on the behaviour of the surveyed animal:

those that quickly travel long distances cause more bias than those that travel slowly or remain

within a small area [16]. Once again, survey procedure can reduce bias in line transect surveys

by searching further to the side and a shorter distance ahead; however, this may increase the

Fig 5. Percentage bias in abundance estimate for different shape parameter values. Percentage bias in

abundance estimate for linear movement against animal speed as a percentage of observer speed for shape

parameters b= 2 (black solid), b= 3 (blue dashed) and b= 5 (red dotted) with percentage bias for a strip

transect (grey solid) estimated from 1000 simulations included for comparison.

doi:10.1371/journal.pone.0121333.g005

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 11 / 15

chances of failing to spot an animal before it moves in response to the observer [26] (pp. 129–

131). If the animals to be surveyed tend to flee from the observer or be attracted, searching fur-

ther along the line will reduce the bias caused by this responsive movement [1] (pp. 32). Clear-

ly, this opposes our advice concerning independent animal movement. Often the bias caused

by independent animal movement will be small compared to that caused by movement in re-

sponse to the observer [27].

How to reduce bias from both independent and responsive movement will depend on the

type of survey and species. For aircraft surveys, animal speed is typically slow relative to observ-

er speed [28], and search effort is generally concentrated to the side rather than ahead. Thus in-

dependent animal movement is unlikely to cause bias. For shipboard surveys, observers tend to

concentrate more effort on searching ahead rather than to the side. This has the advantage that

animals are less likely to respond to the ship before detection, but greater bias will be generated

by independent animal movement. Therefore ship speed should be such that bias from inde-

pendent animal movement is low. For fast-moving animals, such as seabirds in flight, this may

not be an option; then methods for correcting for movement should be considered. For exam-

ple, seabirds on the sea might be surveyed, then a correction adopted based on the proportion

of time birds are in flight. If seabirds don’t respond to the ship, then one strategy to eliminate

bias is to simply record distances of detected birds when they pass abeam; if they do not pass

abeam while visible, then they are not recorded [1] (pp. 198–203). Corrections have also been

Fig 6. Percentage bias in abundance estimate against distance observer looks ahead. Percentage bias

against distance observer looks ahead with hazard rate b= 2 and animal speeds 50% (red dotted), 100%

(blue dashed) and 150% (black solid) of observer speed

doi:10.1371/journal.pone.0121333.g006

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 12 / 15

developed for seabirds in flight [15]. For terrestrial surveys of birds, surveying perched birds,

and separately estimating the proportion of time birds are in flight, can be effective [29]. For

surveys on foot of ground or arboreal animals, an assessment should be made of whether it is

feasible for observers to travel sufficiently fast without compromising other assumptions, to en-

sure bias from independent animal movement is low. If not, corrections for animal movement

should be considered.

Once the survey is completed there is, at present, no clear way to reduce bias in abundance

estimates using the standard methods. For example, truncating the data [1] (pp. 103–108) has

no discernible effect on the bias.

Here we have shown that bias can reach up to 10% for animal speeds of 50% –80% of ob-

server speed. This is useful for providing an idea of the severity of violating the assumptions of

distance sampling. However, the quantities of bias presented here are dependent on the type of

movement and survey protocol assumed. The expression for bias in strip transect sampling as-

sumes a linear movement of animals, which is unrealistic and causes greater bias compared to

other movement types [10]. Similarly, within the line transect simulation study we have not

considered how variable animal speed or heterogeneous home range radius may affect bias.

Furthermore, we only considered how the shape of the detection function affected bias by vary-

ing the shape parameter, which not only determines the function’s shape, but also the average

probability of detection; it may be more informative to describe the relationship between

Fig 7. Percentage bias in abundance estimate for different truncation distances. Percentage bias in

abundance estimate averaged over 1000 simulations for hazard-rate b= 2 against truncation distance

applied by the analyst for animal speeds 20% (red solid), 80% (green dashed) and 150% (purple dotdash)

doi:10.1371/journal.pone.0121333.g007

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 13 / 15

detection function shape and bias for a fixed probability of detection. Also, in this study only

the hazard-rate detection function was fit to the simulated data, while in practice other detec-

tion functions may be chosen by model selection [1] (pp. 110), which may affect bias.

In conclusion, independent animal movement can cause substantial bias in abundance esti-

mates and its possible effects should be considered in line transect surveys. It arises from both

the detection process inherent in line transect sampling and from the movement characteristics

of the surveyed animal. There are, at present, no analytical methods to deal with this reality;

however, bias is reduced by searching further perpendicularly to the line, searching less ahead

and ignoring animals that overtake the observer.

Supporting Information

S1 Code. Computer Code for Simulation Study.

(ZIP)

Acknowledgments

The manuscript was peer-reviewed by two helpful and insightful anonymous reviewers.

Author Contributions

Conceived and designed the experiments: RG STB LT. Performed the experiments: RG. Ana-

lyzed the data: RG STB LT. Contributed reagents/materials/analysis tools: RG STB LT. Wrote

the manuscript: RG STB LT.

References

1. Buckland ST, Anderson DR, Burnham KP, Laake JL, Borchers DL, Thomas L. Introduction to Distance

Sampling. Oxford University Press, Oxford. 2001.

2. Buckland ST, Anderson DR, Burnham KP, Laake JL, Borchers DL, Thomas L. Advanced Distance

Sampling. Oxford University Press, Oxford. 2004.

3. Borchers DL, Laake JL, Southwell C, Paxton CGM. Accommodating unmodeled heterogeneity in dou-

ble-observer distance sampling surveys. Biometrics. 2006; 62: 372–378. doi: 10.1111/j.1541-0420.

2005.00493.x PMID: 16918901

4. Buckland ST, Laake JL, Borchers DL. Double-observer line transect methods: levels of independence.

Biometrics. 2010; 66: 169–177 doi: 10.1111/j.1541-0420.2009.01239.x PMID: 19432793.

5. Marques TA. Predicting and correcting bias caused by measurement error in line transect sampling

using multiplicative error models. Biometrics. 2004; 60: 757–763. doi: 10.1111/j.0006-341X.2004.

00226.x PMID: 15339299

6. Borchers DL, Marques TA, Gunnlaugsson T, Jupp PE. Estimating distance sampling detection func-

tions when distances are measured with errors. Journal of Agricultural, Biological, and Environmental

Statistics. 2010; 15: 346–361 doi: 10.1007/s13253-010-0021-y

7. Gurarie E, Ovaskainen O. Towards a general formalization of encounter rates in ecology. Theoretical

Ecology. 2012; 6: 189–202 doi: 10.1007/s12080-012-0170-4

8. Skellam JG. The mathematical foundations underlying the use of line transects in animal ecology. Bio-

metrics. 1958; 14(3): 385–400. doi: 10.2307/2527881

9. Yapp WB. The theory of line transects. Bird Study. 1956; 3(2): 93–104. doi: 10.1080/

00063655609475840

10. Hutchinson JMC, Waser PM. Use, misuse and extensions of the “ideal gas”models of animal encoun-

ter. Biological Reviews. 2007; 82: 335–359. doi: 10.1111/j.1469-185X.2007.00014.x PMID: 17624958

11. Palka DL, Hammond PS. Accounting for responsive movement in line transect estimates of abun-

dance. Canadian Journal of Fisheries, and Aquatic Sciences. 2001; 58(4): 777–787. doi: 10.1139/f01-

024

12. Smith GEJ. Aspects of line transect sampling when the target population moves. Biometrics. 1979; 35

(1): 325–329. doi: 10.2307/2529953

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 14 / 15

13. Buckland ST, Turnock BJ. A robust line transect method. Biometrics. 1992; 48: 901–909. doi: 10.2307/

2532356

14. Watson RA, Carlos GM, Samoilys MA.Bias introduced by non-random movement of fish in visual tran-

sect surveys. Ecological Modelling. 1995; 77: 205–214. doi: 10.1016/0304-3800(93)E0085-H

15. Spear L, Nadav N, Ainley D. Estimating absolute densities of flying seabirds using analyses of relative

movement. The Auk. 1992; 109(2): 359–389. doi: 10.2307/4088211

16. Hiby AR. The effect of random whale movement on density estimates obtained from whale sighting sur-

veys. Report of the International Whaling Commission. 1982; 32: 791–793.

17. Hiby AR. Results of a hazard rate model relevant to experiments on the 1984/85 IDCR minke whale as-

sessment cruise. Report of the International Whaling Commission. 1986; 36: 497–498.

18. Cañadas A, Hammond PS. Model-based abundance estimates for bottlenose dolphins off southern

Spain: implications for conservation and manangement. Journal of Cetacean Research and Manage-

ment. 2006; 8(1): 13–27.

19. Strindberg S, Buckland ST, Thomas L. Design of distance sampling surveys and Geographic Informa-

tion Systems. In: Buckland ST, Anderson DR, Burnham KP, Laake JL, Borchers DL, Thomas L, editors.

Advanced Distance Sampling. Oxford University Press; 2004. pp. 190–228.

20. Schweder T. Point process models for line transect experiments. In: Barba JR, Brodeau F, Romier G,

Van Cutsem B, editors. Recent Developments in Statistics. North-Holland Publishing Company;

1977. pp. 221–242.

21. Hayes RJ, Buckland ST. Radial distance models for the line transect method. Biometrics. 1983; 39:

29–42. doi: 10.2307/2530804

22. Schweder T. Independent observer experiments to estimate the detection function in line transect sur-

veys of whales. Report of the International Whaling Commission. 1990; 40: 349–355.

23. Thomas L, Buckland ST, Rexstad EA, Laake JL, Strindberg S, Hedley SL, et al. Distance Software:de-

sign and analysis of distance sampling surveys for estimating population size. Journal of Applied Ecolo-

gy. 2010; 47(1): 5–14. doi: 10.1111/j.1365-2664.2009.01737.x PMID: 20383262

24. Granholm SL. Bias in density estimations due to movement of birds. Condor. 1983: 243–248.

25. Burnham KP, Anderson DR, Laake JL. Efficiency and bias in strip and line transect sampling. The Jour-

nal of Wildlife Management. 1985; 49: 1012–1018. doi: 10.2307/3801387

26. Burnham KP, Anderson DR, Laake JL. Estimation of density from line transect sampling of biological

populations. Wildlife monographs. 1980: 3–202.

27. Turnock BJ, Quinn TJ. The effect of responsive movement on abundance estimation using line transect

sampling. Biometrics. 1991; 47: 701–715. doi: 10.2307/2532156

28. Certain G, Bretagnolle V. Monitoring seabirds populations in marine ecosystems: the use of strip-tran-

sect aerial surveys. Remote sensing of environment. 2008; 112: 3314–3322. doi: 10.1016/j.rse.2008.

01.019

29. Marsden SJ. Estimation of parrot and hornbill densities using a point count distance sampling method.

Ibis. 1999; 141: 377–390.

The Effect of Animal Movement on Line Transect Estimates of Abundance

PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 15 / 15