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Double-Observer Line Transect Methods: Levels of Independence Double-Observer Line Transect Methods: Levels of Independence
Stephen T. Buckland
Centre for Research into Ecological and Environmental Modelling, University of St Andrews, The
Observatory, Buchanan Gardens, St Andrews, Fife KY16 9LZ, Scotland
Jeffrey L. Laake
National Marine Mammal Laboratory, Alaska Fisheries Science Center
David L. Borchers
Centre for Research into Ecological and Environmental Modelling, University of St Andrews, The
Observatory, Buchanan Gardens, St Andrews, Fife KY16 9LZ, Scotland
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Buckland, Stephen T.; Laake, Jeffrey L.; and Borchers, David L., "Double-Observer Line Transect Methods:
Levels of Independence" (2010).
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Biometrics 66, 169–177
March 2010
DOI: 10.1111/j.1541-0420.2009.01239.x
Double-Observer Line Transect Methods: Levels of Independence
Stephen T. Buckland,
1, ∗
Jeffrey L. Laake,
2
and David L. Borchers
1
1
Centre for Research into Ecological and Environmental Modelling, University of St Andrews, The Observatory,
Buchanan Gardens, St Andrews, Fife KY16 9LZ, Scotland
2
National Marine Mammal Laboratory, Alaska Fisheries Science Center, NMFS, Seattle, Washington 98115, U.S.A.
∗
email: steve@mcs.st-and.ac.uk
Summary. Double-observer line transect methods are becoming increasingly widespread, especially for the estimation of
marine mammal abundance from aerial and shipboard surveys when detection of animals on the line is uncertain. The
resulting data supplement conventional distance sampling data with two-sample mark–recapture data. Like conventional
mark–recapture data, these have inherent problems for estimating abundance in the presence of heterogeneity. Unlike conven-
tional mark–recapture methods, line transect methods use knowledge of the distribution of a covariate, which affects detection
probability (namely, distance from the transect line) in inference. This knowledge can be used to diagnose unmodeled hetero-
geneity in the mark–recapture component of the data. By modeling the covariance in detection probabilities with distance,
we show how the estimation problem can be formulated in terms of different levels of independence. At one extreme, full
independence is assumed, as in the Petersen estimator (which does not use distance data); at the other extreme, independence
only occurs in the limit as detection probability tends to one. Between the two extremes, there is a range of models, including
those currently in common use, which have intermediate levels of independence. We show how this framework can be used to
provide more reliable analysis of double-observer line transect data. We test the methods by simulation, and by analysis of a
dataset for which true abundance is known. We illustrate the approach through analysis of minke whale sightings data from
the North Sea and adjacent waters.
Key words: Distance sampling; Double-observer methods; Full independence; Limiting independence; Line transect sam-
pling; Point independence.
1. Introduction
Distance sampling (Buckland et al., 2001) is widely used for
estimating animal abundance. In line transect sampling, an
observer travels along each of a number of lines, laid out
according to some randomized (usually systematic random)
scheme, and records each detected animal, together with its
perpendicular distance from the line. One of the key assump-
tions of the method is that animals on the line are certain to
be detected.
A number of authors have considered so-called double-
observer or double-platform methods to extend line tran-
sect sampling to the case that not all animals on the
line are detected (e.g., Buckland and Turnock, 1992; Palka,
1995; Alpizar-Jara and Pollock, 1996; Manly, McDonald, and
Garner, 1996; Quang and Becker, 1997; Chen, 2000; Innes
et al., 2002). The double-observer data can be regarded as
two-sample mark–recapture. However, heterogeneity in de-
tection probabilities generates bias in abundance estimates,
just as heterogeneity in capture probabilities generates bias
in mark–recapture estimates of abundance. Authors have at-
tempted to minimize this bias, for example, by modeling the
effects of covariates (Borchers, Zucchini, and Fewster, 1998;
Borchers et al., 1998, 2006; Borchers, 1999; Schweder et al.,
1999; Laake and Borchers, 2004), or by assuming indepen-
dence in the detections of instantaneous cues (such as whale
blows) rather than of animals (Schweder et al., 1999; Skaug
and Schweder, 1999).
In the absence of any heterogeneity in detection probabili-
ties, we might assume that observer j detects any given animal
in the surveyed strip with probability p
j
, j =1,2,andthat
the probability that both observers detect a given animal is
p
12
= p
1
p
2
. This is the “full independence” assumption. How-
ever, in line transect sampling, we allow detection probability
to fall off with distance y from the line so that p
j
= p
j
(y).
Thus it is natural to apply the full independence assumption
at each distance from the line, so that for an animal at y,we
assume p
12
(y)=p
1
(y)p
2
(y).
Laake (1999) introduced the concept of “point indepen-
dence” to reduce the impact of unmodeled heterogeneity in
detection probabilities. Knowledge of the distribution of dis-
tances allows the full independence assumption to be weak-
ened, as outlined below. (For the moment, we ignore variables
other than distance for simplicity.)
A double-observer line transect survey generates both con-
ventional distance sampling data and mark–recapture data.
Under the assumption of uniform animal distribution perpen-
dicular to the transect line (achievable by random line place-
ment or systematic placement with a random start), the shape
of the probability density function of observed distances is the
same as that of the detection function (Buckland et al., 2001,
p. 52–53). The mark–recapture data provide additional infor-
mation on the shape of the detection function based on an
assumption of independence of detection probabilities with-
out any assumption about the distribution of perpendicular
C
2009, The International Biometric Society
169
This article is a U.S. government work, and is not subject to copyright in the United States.
170 Biometrics, March 2010
distances of animals. If we retain the assumption of uni-
form perpendicular distance distribution, discrepancies be-
tween the shapes can be interpreted as failure of the assump-
tion of independence between detection probabilities.
We diagnose dependence by (a) modeling the shape of ob-
server j’s detection function (p
j
(y), j = 1, 2) under the uni-
form perpendicular distance assumption, (b) modeling the
conditional probability p
j |j
(y)thatobserverj detects an an-
imal at y, given that observer j
detected it (j =1,2,j
=
3 − j), and (c) modeling the covariance in the observers’ de-
tection probabilities as a function of y using a function δ(y)
defined below.
For real data, typically p
j |j
(y) does not decline as steeply
as p
j
(y). Hence the full independence assumption (p
j |j
(y)=
p
j
(y)) cannot be made at each distance. The reason for this
is that at greater distances, only the most detectable animals
tend to be recorded, and those that are detected by one ob-
server are therefore more likely to be detected by the other
observer. Laake (1999) argued that heterogeneity is less of a
problem on the line, where probability of detection is rela-
tively high, than away from the line, so that assuming inde-
pendence only on the line should yield less biased estimates
of abundance. The idea was further developed by Laake and
Borchers (2004) and Borchers et al. (2006).
Although we can anticipate less dependence between detec-
tions on the line than at greater distances, unless detection
on the line is certain, it seems possible that some dependence
remains. In this article, we consider levels of independence,
and show that the independence assumption can be weak-
ened further by assuming that, as detection probability tends
to unity, dependence tends to zero (i.e., independence). We
term this “limiting independence.”
We illustrate the methods through analyses of data from
a shipboard survey of minke whales in the North Sea and
adjacent waters.
2. Methods
Suppose detected animals within a strip extending a distance
W either side of the line are recorded. We assume that two
observers search independently from the same platform, or
from two platforms following the same route at almost the
same time. We also assume that duplicate detections can be
correctly classified, based on time and location of animals or
animal cues, for example.
2.1 Independence Assumptions
At the simplest level, we might assume that observer 1 de-
tects animals in this covered strip with probability p
1
, while
observer 2 independently detects animals with probability p
2
.
In this “full independence” case, an animal is detected by at
least one observer with probability p
•
= p
1
+ p
2
− p
1
p
2
.A
Horvitz–Thompson estimator of N
c
, the number of animals
in the strip, is thus
ˆ
N
c
=
1
p
•
=
n
p
•
where n is the number
of animals detected by at least one observer. Note that n =
n
1
+ n
2
− n
12
,wheren
j
is the number of animals detected by
observer j, j =1,2,andn
12
is the number of animals detected
by both observers. If we estimate p
j
by ˆp
j
= n
12
/n
j
for j =
1, 2, j
=3− j, and substitute in, we find that
ˆ
N
c
=
n
1
n
2
n
12
,
which is the familiar Petersen estimator. This is the full max-
imum likelihood estimator of N
c
(Borchers, Buckland, and
Zucchini, 2002, p. 111), or within a single animal of the
maximum likelihood estimator, if we allow for the fact that
N
c
is integer.
Now suppose that probability of detection is a function
of distance y from the line. There may also be dependence
on additional covariates z
, although we omit this dependence
below, for clarity. Full independence applied at each y gives
p
•
(y)=p
1
(y)+p
2
(y) − p
1
(y)p
2
(y), so that a model is now
needed for p
j
(y), j = 1, 2. We can then proceed to fit the
model, and hence to estimate abundance in the covered strip
(below).
We would like to relax the full independence assump-
tion. Allowing some degree of dependence (δ(y)), the in-
dependence assumption can be expressed more generally
such that p
12
(y)=δ(y)p
1
(y)p
2
(y), p
•
(y)=p
1
(y)+p
2
(y) −
δ(y)p
1
(y)p
2
(y), and p
j | j
(y)=δ(y)p
j
(y), j =1,2,j
=3− j.
The function δ(y) is related to the covariance σ
12
(y) between
detection probabilities p
1
(y)andp
2
(y) as follows: σ
12
(y)=
[δ(y) − 1]p
1
(y)p
2
(y). Various alternative expressions can be
derived for δ(y) including
δ(y)=p
12
(y)/{p
1
(y)p
2
(y)}
= {p
1 | 2
(y)+p
2 | 1
(y) − p
1 | 2
(y)p
2 | 1
(y)}/p
•
(y)
= p
j |j
(y)/p
j
(y)
for j = 1, 2. The latter expressions demonstrate that δ(y)
measures the discrepancy between the conditional detection
functions p
j | j
(y) derived from the mark–recapture data and
the unconditional detection functions p
j
(y), which are derived
from distance sampling data with the requirement that p
j
(y
∗
)
is known for some y
∗
. For distance sampling with a single
observer, the standard assumption is p
j
(0) = 1. With double
observers, this often untenable assumption can be replaced
with the assumption of full independence, δ(y) = 1 for all y,
or point independence, δ(y
∗
)=1ataspecifiedy
∗
, usually
y
∗
= 0 (Laake and Borchers, 2004). Fitting full independence
models to data requires a functional form for p
j
(y)andpoint
independence requires the same and a model for p
j | j
(y). Nei-
ther require a model for δ(y).
We now relax the assumption that δ(y
∗
) = 1 at a specified
y
∗
. Instead we assume that we achieve independence in the
limit as detection probability tends to one. This requires a
model for δ(y) with the following properties to ensure valid
probabilities:
(1) δ(y) ≤ U (y), where U (y)=min{1/p
1
(y), 1/p
2
(y)},
which ensures that p
j |j
(y) ≤ 1.
(2) δ(y) ≥ L(y), where L(y)=max{0,
p
1
(y )+p
2
(y )−1
p
1
(y )p
2
(y )
},which
ensures that p
•
(y) ≤ 1.
If we define δ
0
(y)={δ(y) − L(y)}/{U (y) − L(y)},itisre-
stricted to the unit interval and can be represented by an
appropriate functional form such as a logistic. Note also that
as p
j
(y) → 1, j =1,2thenδ(y) → 1.
Using a logistic formulation for δ
0
(y), we can write
log
e
{
δ
0
(y )
1−δ
0
(y )
} as some linear function of y. Full and point inde-
pendence can be derived as special cases of the limiting inde-
pendence model if we include the following offset log
e
{
1−L (y )
U (y )−1
}
in δ
0
(y) which fixes δ(y) = 1. If we consider the following lo-
gistic model for limiting independence:
log
e
δ
0
(y)
1 − δ
0
(y)
= α + βy +log
e
1 − L(y)
U(y) − 1
, (1)
Double-Observer Line Transect Methods 171
then α = 0 specifies point independence at y
∗
=0,andα =
β = 0 specifies a full independence model. If β =0andα =0,
a model with constant dependence for all y can be specified.
Models restricted to independence or positive dependence can
be achieved by restricting α ≥ 0, β ≥ 0. Hence this general
formulation provides a model selection framework for a range
of models with varying degrees of independence.
2.2 Likelihood
The full likelihood for double-platform data may be expressed
as L = L
n
L
z
L
y |z
L
ω
where L
n
is the component accounting
for variation in total number of animals n detected by at least
one observer, L
z
corresponds to any observation-specific co-
variates z
, L
y |z
corresponds to the conditional distribution of
distances y, given covariates z
,andL
ω
corresponds to the
mark–recapture data (Laake and Borchers, 2004). L
y |z
incor-
porates the assumption of uniform distribution of animals per-
pendicular to transect lines. We use just two components of
the full likelihood: L
y |z
and L
ω
. By doing this, we can avoid
making distributional assumptions about n and z
,asestima-
tion is not robust to failure of such assumptions. Instead, we
draw inference conditional on n and z
, and use a design-based
approach to allow for variation in n. If there are no covariates
z
, the full likelihood is L = L
n
L
y
L
ω
, and we use the second
and third components only (in this case, L
y
incorporates the
assumption of uniform distribution of animals perpendicular
to transect lines). Again for simplicity we consider this latter
case; the extensions to include covariates z
are straightfor-
ward.
We have
L
y
=
n
i=1
f
•
(y
i
)=
n
i=1
p
•
(y
i
)π(y
i
)
E(p
•
)
,
where f
•
(y
i
) is the probability density function (pdf) of detec-
tion distances y of animals detected by at least one observer,
evaluated at y
i
, p
•
(y
i
)=p
1
(y
i
)+p
2
(y
i
) − δ(y
i
)p
1
(y
i
)p
2
(y
i
)
is the probability that an animal at distance y
i
from the line
is detected by at least one observer, π(y
i
) is the uncondi-
tional pdf of distances y in the population (whether detected
or not), evaluated at y
i
,andE(p
•
)=
w
0
p
•
(y)π(y)dy (Laake
and Borchers, 2004, p. 114). Random positioning of the lines
(or of a systematic grid of lines) ensures that π(y)=1/W .
We also need
L
ω
=
n
i=1
Pr(ω
i
| y
i
)
p
•
(y
i
)
,
where
Pr{ω
i
=(1, 0) | y
i
} = p
1
(y
i
){1 − p
2
(y
i
)δ(y
i
)},
Pr{ω
i
=(0, 1) | y
i
} = p
2
(y
i
){1 − p
1
(y
i
)δ(y
i
)},
Pr{ω
i
=(1, 1) | y
i
} = p
1
(y
i
)p
2
(y
i
)δ(y
i
).
The likelihoods for full, point, and limiting independence
only differ in the definition of δ(y
i
). However, if the full in-
dependence assumption holds then it is only necessary to
use L
ω
(Borchers et al., 1998) and with the point indepen-
dence assumption, L
ω
and L
y
can be maximized indepen-
dently using models for p
j |j
(y)andp
j
(y), which separate
into the two respective likelihood components (Borchers
et al., 2006). When the likelihood is specified in terms of mod-
els for p
j
(y)andδ(y), both components of the likelihood must
be maximized jointly.
We assume logistic forms for the detection functions:
p
j
(y)=
exp(λ
0j
+ λ
1j
y)
1+exp(λ
0j
+ λ
1j
y)
for j =1or2. (2)
2.3 Diagnostic for Reliable Estimation under Limiting
Independence
When fitting limiting independence models, the Hessian ma-
trix is sometimes nearly singular, due to high correlation be-
tween the estimates of p
j
(y)andδ(y)aty = 0. In these cases,
the models are unstable, typically yielding very large abun-
dance estimates and associated variances. We can still usefully
calculate Akaike’s information criterion (AIC), but if AIC in-
dicates that a limiting independence model is required, then
reliable estimation is not possible. To identify such cases, the
following diagnostic check was found useful. If the magnitude
of the estimated correlation between ˆα of equation (1) and
ˆ
λ
0j
of equation (2) is found to be large, then estimated abundance
should be considered unreliable. The test can be conducted
for each of j =1andj = 2, or by arbitrarily choosing one of
the two; the two correlations tend to be similar when they are
close to ±1. We defined “large” to be greater than 0.99 in Sec-
tion 3 and 0.9 in Section 4; choices in the range of 0.9 to 0.99
were found to be effective. Lowering the correlation criterion
provides a more conservative approach to avoid overestima-
tion with the only cost being potential underestimation due
to the unmodeled dependence.
2.4 Estimating Abundance
Given models for p
1
(y), p
2
(y), and δ(y), the likelihood con-
ditional on n, L
y
L
ω
, can be maximized, which allows us to
estimate E(p
•
). Estimated abundance in the covered area is
then
ˆ
N
c
=
n
i=1
1
ˆ
E(p
•
)
=
n
ˆ
E(p
•
)
. (3)
This is a Horvitz–Thompson estimator in which the inclu-
sion probabilities have been estimated (Laake and Borchers,
2004, p. 116). When covariates z
are present, the simplifica-
tion represented by the second equality does not hold. If the
covered area is of size a, and the entire survey region of size
A, then estimated abundance in the survey region is
ˆ
N =
A
a
ˆ
N
c
=
A
a
n
ˆ
E(p
•
)
, (4)
where a =2wL and L is the total length of transect line.
For our limiting independence model, we cannot use
vˆar(
ˆ
N
c
) as defined in Borchers et al. (2006), because the con-
ditional and unconditional detection functions share param-
eters under the above formulation. Adapting their result, we
have
vˆar(
ˆ
N
c
)=S
2
(
ˆ
θ)+
ˆ
d
T
ˆ
I
−1
ˆ
d
,
172 Biometrics, March 2010
where S
2
(
ˆ
θ)=
n
i=1
1−
ˆ
E (p
•
)
{
ˆ
E (p
•
)}
2
=
n {1−
ˆ
E (p
•
)}
{
ˆ
E (p
•
)}
2
,
ˆ
d =
d
ˆ
N
c
dθ
|
ˆ
θ
,and−
ˆ
I
is the matrix of second derivatives of ln(L
y
)+ln(L
ω
), evalu-
ated at
ˆ
θ
, the vector of parameter estimates.
Adapting equation (11) of Marques and Buckland (2003),
vˆar(
ˆ
N )=
A
a
2
L
K
k =1
l
k
(
ˆ
N
ck
/l
k
−
ˆ
N
c
/L)
2
K − 1
+
ˆ
d
T
ˆ
I
−1
ˆ
d
, (5)
where
ˆ
N
ck
=
n
k
i=1
1
ˆ
E (p
•
)
=
n
k
ˆ
E (p
•
)
is estimated abundance for
strip k, which has half-width w and length l
k
,where
K
k =1
l
k
=
L.
An alternative to the above is to use the bootstrap, in which
bootstrap resamples are generated by sampling the lines with
replacement.
If animals occur in clusters, with s
i
animals in the ith de-
tected cluster, then the above formula gives estimated clus-
ter abundance, and estimated animal abundance is given
by
ˆ
N =
A
a
n
i=1
s
i
ˆ
E(p
•
)
.
Variance can be estimated as before, except that now,
S
2
(
ˆ
θ)=
{1−
ˆ
E (p
•
)}
{
ˆ
E (p
•
)}
2
n
i=1
s
2
i
,
ˆ
d =
d
ˆ
N
c
dθ
|
ˆ
θ
is evaluated using
ˆ
N
c
=
n
i =1
s
i
ˆ
E (p
•
)
, and in the formula for variance of
ˆ
N ,
ˆ
N
c
=
n
i =1
s
i
ˆ
E (p
•
)
and
ˆ
N
ck
=
n
k
i =1
s
i
ˆ
E (p
•
)
.
Tabl e 1
Mean (standard error in parentheses) of 100 abundance estimates under full independence (FI), point independence (PI), and
limiting independence (LI) models for the eight simulation scenarios. The expected capture history frequencies are shown for
each scenario. Also shown is p
AIC
, the proportion of times each model was selected by AIC, and model-averaged (MA) estimates,
obtained by taking a weighted average of estimates from the above three models, using AIC weights. Where an LI model was
deemed to be parameter redundant ( | correl(ˆα,
ˆ
λ
01
)| > 0.99), the model was not considered even if it had the best AIC value, and
the weighted average was over the FI and PI models only. The mean and standard error for LI models is across only those runs
for which the model was not deemed to be parameter-redundant. The number of runs (Nr) out of 100 contributing to the LI
results under each scenario is shown. True abundance is 1000.
∗
Bias significant at 5% level.
Exp. capture
history freqs. FI PI LI MA
Scenario 10 01 11 Mean p
AIC
Mean p
AIC
Mean p
AIC
Nr Mean
1 101 101 315 607
∗
0.00 819
∗
0.23 1074
∗
0.77 95 1020
(3) (6) (14) (16)
2 33 33 383 450
∗
0.00 692
∗
0.04 1037
∗
0.96 97 1020
(1) (4) (18) (18)
3 89 89 126 499
∗
0.00 616
∗
0.93 971 0.07 60 685
∗
(4) (6) (21) (7)
4 34 34 180 290
∗
0.00 454
∗
0.54 963 0.46 63 664
∗
(3) (4) (21) (21)
5 87 87 135 512
∗
0.00 650
∗
0.65 1437
∗
0.35 76 1021
(4) (6) (124) (96)
6 34 34 189 296
∗
0.00 465
∗
0.23 1149
∗
0.77 86 962
(3) (5) (42) (41)
7 77 77 579 770
∗
0.00 915
∗
0.55 1064
∗
0.45 82 1001
(2) (3) (15) (14)
8 29 29 629 689
∗
0.00 850
∗
0.12 1057
∗
0.88 94 1032
∗
(2) (4) (11) (12)
3. Simulation Study
Simulations were conducted to evaluate the performance of
the limiting independence model. We simulated a population
of N = 1000 animals that were uniformly distributed in a
strip of width two (w = 1) and undefined length. For each
of 100 simulation replicates, we generated capture histories
for two observers with identical detection probability func-
tions p
1
(y)=p
2
(y). We used four different logistic models
for p
j
(y) and two different logistic models for δ
0
(y)tocreate
eight scenarios. For models with a covariate z,thecovariate
value was generated from a uniform (0, 1) distribution. We
fitted the simulated observed data (10, 01, 11 capture histo-
ries) with the model that generated the data, and with the
equivalent models under the point independence and full in-
dependence restrictions. We computed the AIC for each of
the fitted models. For model fits where the magnitude of the
correlation between ˆα and
ˆ
λ
0j
exceeded 0.99, results are not
reported.
The eight scenarios were as follows. The offset log
e
{
1−L (y )
U (y )−1
}
wasusedineachdependencemodeltosimulateandfitthe
data. The dependence model δ
0
(y)=(1+e
−1−y
)
−1
was
used in scenarios 1, 3, 5, and 7, whereas δ
0
(y)=(1+
e
−2−2y
)
−1
(representing stronger dependence) was used in
scenarios 2, 4, 6, and 8. The detection probability model
p
j
(y)=(1+e
−1.1+3y
)
−1
, j = 1, 2, was used in scenar-
ios 1 and 2, p
j
(y)=(1+e
3y
)
−1
in scenarios 3 and 4,
p
j
(y, z)=(1+e
0.8417+3y −0. 8417z
)
−1
in scenarios 5 and 6, and
p
j
(y, z)=(1+e
3y −5z
)
−1
in scenarios 7 and 8.
Simulation results appear in Table 1. Full independence
and point independence models had substantial negative bias
