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Basic and Applied Ecology 34 (2019) 25–35
n the identification of mortality hotspots in
linearnfrastructures
uís Borda-de-Águaa,b,∗, Fernando Ascensãoa,b, Manuel
Sapagec,afael Barrientosa,b, Henrique M. Pereiraa,b,d
Theoretical Ecology and Biodiversity Group, and Infraestruturas
de Portugal Biodiversity Chair, CIBIO/InBio,entro de Investigação
em Biodiversidade e Recursos Genéticos, Laboratório Associado,
Universidade do Porto,ampus Agrário de Vairão, 4485-661 Vairão,
Portugal
CEABN/InBio, Centro de Ecologia Aplicada “Professor Baeta
Neves”, Instituto Superior de Agronomia,niversidade de Lisboa,
Tapada da Ajuda, 1349-017 Lisboa, Portugal
cE3c — Centre for Ecology, Evolution and Environmental Changes,
Faculdade de Ciências, Universidade deisboa, Campo Grande, 1749-016
Lisboa, PortugalGerman Centre for Integrative Biodiversity Research
(iDiv) Halle-Jena-Leipzig, Deutscher Platz 5e, 04103
Leipzig,ermany
eceived 8 June 2018; accepted 10 November 2018vailable online 16
November 2018
bstract
One of the main tasks when dealing with the impacts of
infrastructures on wildlife is to identify hotspots of high
mortalityo one can devise and implement mitigation measures. A
common strategy to identify hotspots is to divide an
infrastructurento several segments and determine when the number of
collisions in a segment is above a given threshold, reflecting a
desiredignificance level that is obtained assuming a probability
distribution for the number of collisions, which is often the
Poissonistribution. The problem with this approach, when applied to
each segment individually, is that the probability of
identifyingalse hotspots (Type I error) is potentially high. The
way to solve this problem is to recognize that it requires multiple
testingorrections or a Bayesian approach. Here, we apply three
different methods that implement the required corrections to
thedentification of hotspots: (i) the familywise error rate
correction, (ii) the false discovery rate, and (iii) a Bayesian
hierarchicalrocedure. We illustrate the application of these
methods with data on two bird species collected on a road in
Brazil. The proposedethods provide practitioners with procedures
that are reliable and simple to use in real situations and, in
addition, can reflect
practitioner’s concerns towards identifying false positive or
missing true hotspots. Although one may argue that an overly
autionary approach (reducing the probability of type I error)
may be beneficial from a biological conservation perspective,
it
y raise
ay lead to a waste of resources and, probably worse, it ma
f those suggesting it.
2018 The Authors. Published by Elsevier GmbH on behalf of Gehe
CC BY license (http://creativecommons.org/licenses/by/4.0/).
eywords: Bayesian hierarchical model; False discovery rate
correction;
∗Corresponding author at: CIBIO/InBio, Centro de Investigação
em Bio-iversidade e Recursos Genéticos, Laboratório Associado,
Universidade doorto, Campus Agrário de Vairão, 4485-661 Vairão,
Portugal.
E-mail address: [email protected] (L. Borda-de-Água).
ttps://doi.org/10.1016/j.baae.2018.11.001439-1791/© 2018 The
Authors. Published by Elsevier GmbH on behalf of
Gesellshttp://creativecommons.org/licenses/by/4.0/).
doubts about the methodology adopted and the credibility
sellschaft für Ökologie. This is an open access article
under
Familywise error rate correction; Hotspot; Spatial
auto-correlation
chaft für Ökologie. This is an open access article under the
CC BY license
https://doi.org/10.1016/j.baae.2018.11.001http://crossmark.crossref.org/dialog/?doi=10.1016/j.baae.2018.11.001&domain=pdfhttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/mailto:[email protected]://doi.org/10.1016/j.baae.2018.11.001http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/
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6 L. Borda-de-Água et al. / Basic
ntroduction
Linear infrastructures, such as power lines, roads or rail-ays,
are among the most ubiquitous man-made featuresorldwide, and are
known to have negative impacts on nat-ral habitats and ecosystems
(van der Ree, Smith, & Grilo,015; Borda-de-Água, Barrientos,
Beja & Pereira 2017).A main concern when evaluating the impact
of linear
nfrastructures on wild animal populations is the identifica-ion
of sections with high mortality (Quinn, Alexander, Heck,
Chernoff, 2011; Gunson & Teixeira, 2015). Sections with
large number of animal collisions (“mortality hotspots”)re
generally prioritized when mitigation measures are to beaken,
because this is where such measures are more likelyo be
cost-effective in reducing the excessive animal mortal-ty or in
increasing human safety (Gunson & Teixeira, 2015;arrientos,
Alonso, Ponce, & Palacin, 2011). Mitigation mea-
ures are often expensive (Barrientos et al., 2011), therefore,
judicious identification of the sections that represent a
moreerious threat to wildlife populations may lead to
significantconomic savings while still achieving the goal of
reducinghe impacts of the infrastructure. On the other hand,
mitigat-ng false hotspots results in a waste of resources and
mayndermine the credibility of conservation efforts. Our objec-ive
here is to develop procedures that compromise between
inimizing the erroneous identification of false hotspots andhe
correct identification of true cases of high mortality.
There is a vast body of literature using different meth-ds to
identify hotspots, such as, generalized linear modelsGomes, Grilo,
Silva, & Mira, 2009), ecological niche factornalysis (Hirzel,
Hausser, Chessel, & Perrin, 2002), kernelensity estimation
(Gitman & Levine, 1970; Bíl, Andrašik,
Janoška, 2013; Bíl, Andrašik, Svoboda, & Sedoník,
2016),earest neighbour hierarchical clustering (Levine,
2004),istance-based approaches (Kolowski & Nielsen, 2008),
andethods based on modelling the number of collisions in
segment assuming a Poisson distribution (Malo, Suárez, Díez,
2004; Grilo, Ascensão, Santos-Reis, & Bissonette,
011; Planillo, Kramer-Schadt, & Malo, 2015; Santos et
al.,015). The latter methods based on the Poisson distributionor
any other), if applied without additional information, areikely to
lead to the identification of hotspots that are meretatistical
artefacts. This happens because researchers do notorrect for
multiple testing when performing simultaneousests of hypotheses
(e.g. Gelman, Hill, & Yajima, 2012).
We first illustrate the consequences of not correcting
forultiple testing and then describe different approaches to
eal with the problem. We chose two frequentist methods,
theamilywise error rate correction (FWER) and the false dis-overy
rate (FDR), because they are easy to implement andheir
interpretation is straightforward, and chose a Bayesian
ethod because it forces the explicit statement of the
assump-
ions and it has the potential to be extended to a wide rangef
situations. Similar versions of the Bayesian models weevelop here
have been applied extensively in traffic acci-
P
plied Ecology 34 (2019) 25–35
ent research and the knowledge gained there can easilye extended
to our purposes (e.g., Heydecker & Wu, 2001;hmed, Huang,
Abdel-Aty, & Guevara, 2011).For simplicity, throughout the text
we assume that the
etectability of carcasses is perfect, as other works
haveddressed the effects of detectability and persistence on
mor-ality estimates but not on hotspot identification bias
(e.g.once, Alonso, Argandona, García Fernandez, & Carrasco,010;
Santos, Carvalho, & Mira, 2011). Therefore, weoncentrate solely
on the statistical aspects of identifyingortality hotspots given a
certain distribution of collisions.oreover, we assume that the data
has been collected in
uch a way that there is only information on the number
ofollisions per segment, that is, the exact location of the
col-isions is unknown, as often occurs in road surveys
(Gunson,levenger, Ford, Bissonette, & Hardy, 2009).Finally,
this paper deals solely with the statistical aspects
f identifying hotspots, but these methods do not
substitutetudies on the road mortality effect on population
viability,or the discussion on the societal choices underlying the
needo identify hotspots. These choices are implicitly or
explicitlyncorporated in the thresholds adopted in each method.
How-ver, we highlight that the Bayesian methods we present forcehe
researcher to explicitly state the perceived costs associ-ted with
missing hotspots or identifying false ones, thushey help state and
confront different strategies to addressoad mortality.
This paper does not follow the typical structure of papers
incology. First, we illustrate the problems that can arise
whenerforming multiple tests. Then we present three
differentrocedures that address the problems arising from
multipleests: the first two use a frequentist approach and the
thirdevelops a Bayesian approach. We exemplify the applica-ion of
these methods using data on road mortality of twoird species, the
burrowing owl (Athene cunicularia) andhe blue–black grassquit
(Volatinia jacarina). We finish byiscussing the merits of the three
approaches.
ultiple hypothesis tests and the need fororrections
When assessing which segments of a linear infrastructurere
hotspots, it is often assumed that the number of colli-ions in each
segment follows a Poisson distribution and thathis distribution
describes equally well the collisions in allegments (e.g. Malo et
al., 2004; Grilo et al., 2011; Planillot al., 2015; Santos et al.,
2015; Costa, Ascensão, & Bager,015; Garriga, Franch, Santos,
Montori, & Llorente, 2017).ccordingly, the probability of n
occurrences pertaining to
segment of size l, in a linear infrastructure of length L,
isescribed by a Poisson distribution
(n) = e−λl(λl)n
n!(1)
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L. Borda-de-Água et al. / Basic
here λ is the rate of collisions per unit of length
estimateduring a certain period of time.If collisions are uniformly
randomly distributed along the
nfrastructures there should be no hotspots, however,
theccasional appearance of regions with a high concentrationf
collisions is to be expected due to the inherent stochastic-ty of
the process. Naturally, one wants to safeguard againsthe
identification of such spurious occurrences as hotspots.owever, if
we use Eq. (1) to establish a threshold to testhether a segment is
a hotspot, and then applying such a test
o all segments simultaneously, we are likely to obtain a
largeumber of false hotspots. For instance, assume that λ =
0.635ollisions km−1 and define that a segment with 3 or more
col-isions is a hotspot. Using Eq. (1) this leads to the
probability(n ≥ 3) ≈3% (Malo et al., 2004). In the typical
terminologyf hypothesis testing, the probability of a Type I error
is 3%.his may look a small probability, but if the infrastructureas
a large number of segments and we apply the test to allhe segments,
the probability of incorrectly identifying seg-ents as hotspots
increases considerably. For example, if a
oad consists of 100 segments we expect that on average 3egments
(i.e., 3%) will be incorrectly identified; we furtherlaborate on
this example in Appendix A in Supplementaryaterial.
orrecting for multiple testing
he familywise error rate correction (FWER)Statisticians have for
long considered several correc-
ions when conducting multiple tests. One of them is
theunn–Šidák correction (Sokal & Rohlf, 1995). In our
case,
t is important to distinguish the probability of
identifyingncorrectly one or more segments in the entire
infrastructure,hich we want to be smaller than or equal to PL, and
therobability of incorrectly identifying one segment, which weant
to be smaller than or equal to PS . With these
definitionsunn–Šidák correction is given by
S ≤ 1 − (1 − PL)1/M (2)here M is the total number of segments.
In most cases Eq.
2) can be approximated by the Bonferroni correction:
S ≤ PLM
(3)
see Appendix A in Supplementary material). For example,f PL =
0.05 and M = 100, then we should use PS = 0.0005 athe segment
level.
he false discovery rate (FDR)The FWER correction reduces the
probability of identi-
ying false positive hotspots (Type I error) at the expense
of
ncreasing the probability of missing true hotspots (false
neg-tives, Type II error). If we are also concerned with Type
IIrrors then, under the frequentist approach, we can considerhe FDR
method.
hbah
plied Ecology 34 (2019) 25–35 27
The FDR procedure was introduced by Benjamini andochberg (1995).
These authors suggested a method that
onsists of sorting all p-values from M hypotheses testedn
ascending order and then determine whether a k-ordered-values
(denoted pvalue) obeys to
value >kPS
M(4)
here PS is a threshold imposed by the researcher. If avalue
obeys to (4) then it is declared non-significant or, inhe present
context, it is not a hotspot. (See Appendix A inupplementary
material for more details.)For instance, consider again an
infrastructure with M = 100
egments, and a threshold PS = 0.027. Calculate the pvalue forach
segment, and order these values from the smallest tohe largest;
call this ordered vector P. Suppose that the pvaluef the segment in
position k = 15 of the vector P is 0.136.nequality (4) leads to kPS
/M = 15 × 0.027/100 = 0.0039,ence, because pvalue = 0.136 >
0.0039, we conclude that thisegment is not a hotspot. Assume now a
segment withvalue = 5.3 × 10−5 corresponding to position k = 1 of
vec-or P. Then kPS /M = 1 × 0.027/100 = 2.7 × 10−4. Becausevalue =
5.3 × 10−5 < 2.7 × 10−4, we identify the segment as
hotspot. In the Appendix B in Supplementary material werovide an
R (R Core Team, 2016) function that implementshis procedure.
ayesian hierarchical models (BHM)The analysis of a dataset under
the Bayesian paradigm
onsists of combining the information contained in the data,hat
is, the likelihood given a parametric model, P(D|M), withrior
information, P(M), using Bayes Rule (e.g. Gelman et al.,014). The
resulting distribution is the posterior, P(M|D),
(M|D) ∝ P (D|M) P (M) ,rom where all conclusions are derived. In
our case, the pos-erior of interest is the distribution of the
parameter λ of theoisson distribution (Eq. (1)) given the observed
number ofollisions, n, in each segment. As we will see, under
theayesian framework multiple testing corrections are taken
nto account implicitly.We start by assuming that the number of
collisions, ni, in
segment i is a random variable with a Poisson distributionith
parameter λi,
i|λi ∼ Poisson(λi).This is an important departure from the
previous methods,
ecause now each segment has its own λi, while before aingle
parameter λ applied to all segments. Next we assumehat the set of
parameters λi are realizations of another distri-ution, called a
prior distribution. For the models developed
ere the parameters of the prior distribution will have
proba-ility distributions, the so-called hyperpriors, and these
willlso have associated distributions, and so on, i.e., there is
aierarchy of distributions.
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8 L. Borda-de-Água et al. / Basic
Assuming no spatial autocorrelation, we develop twoierarchical
models: the Poisson-lognormal model and aoisson-gamma model. This
is because when the rate of fatal-
ties λ is small, a Poisson-lognormal is to be preferred tohe
Poisson-gamma but for larger λ the latter offers somenalytical
advantages and performs better (Lord & Miranda-oreno, 2008).For
the Poisson-lognormal model the prior for λi is
i = exp(εi),nd the hyperprior for the parameter εi is a normal
distribu-ion, such as
i | φ ∼ Normal(0,φ)here φ is the precision (the inverse of the
variance). For theyperparameter, φ, we assume
| c1, c2 ∼ Gamma(c1, c2),here c1, and c2 are constants.The
development of the Poisson-gamma model is similar,
xcept that for λi we assume a gamma distribution instead of
lognormal. Thus,
i | α,β ∼ Gamma(α,β),here α and β are the shape and scale
parameters of theamma distribution. The last step is to specify the
distribu-ions of the hyperparameters α and β,
| a1, a2 ∼ Gamma(a1,a2),
| b1, b2 ∼ Gamma(b1,b2)here a1, a2, b1, and b2 are constants.
Fig. 1A illustrates
he structure of the hierarchical model for the Poisson-amma
model and Fig. 1B for the Poisson-lognormal model;ppendix C in
Supplementary material provides codes for the
mplementation of the above two models using the softwareackage
JAGS (Plummer, 2003).
Until now we have disregarded spatial autocorrelationmong the
segments in order to provide very simple meth-ds that give a first
correction to the more serious problemsf not considering multiple
testing. Introducing spatial auto-orrelation means we take into
account the positions ofhe segments relative to each other when
determining theirelative influence. Because of the mathematical
problemsf dealing with a multivariate gamma distribution
(Zhang,atthaiou, Karagiannidis, & Dai, 2016), we consider
only
he Poisson-lognormal model.We introduce spatial dependence in
the Poisson-lognormalodel by changing expression εi | φ ∼
Normal(0,φ) to
Banerjee, Carlin, & Gelfand, 2014)
i|δ, φ, εj,i /= j∼ Normal(δM∑
bijεj, φ)
j=1
here 0 ≤ δ ≤ 1 controls the overall spatial dependence and ≤ bij
≤ 1 controls the dependence between segments i and
iict
plied Ecology 34 (2019) 25–35
. The term δM∑
j=1bijεj adds to the formulation the influence
f the neighbourhood of segment i on the estimation of itsi and,
hence, of λi. For instance, if bij = 0 site j is not inhe immediate
neighbourhood of site i and it does not have
direct influence, if bij ≥ 0 then site j influences site i.
Theum of all influences for one segment must sum up to one, soach
bij is equal to the inverse of the number of immediateeighbours of
i. If δ = 0, εi does not take into account
spatialutocorrelation.
It is more convenient to use the above expression in a matri-ial
form (see Jin, Bradley & Sudipto (2005) for details):
| φ, δ, W ∼ Multivariate Normal(0, φ (D − δW)),here φ(D − δW) is
the precision matrix (the inverse of theariance matrix). Vector ε
is of length M and contains ε1, ε2,
. ., εM , and 0 is a vector of zeroes, also of length M. W is
andjacency matrix of size M × M where wij = 1 if i is adjacento j
and wij = 0 otherwise. D is a vector of size M whereach element i
is the sum of the elements of row i in matrix
. This model is known as the conditionally autoregressiveCAR)
model (Jin et al., 2005).
Finally, to complete the models we need to specify
theistributions of the hyperparameters φ and δ. We assume
| c1, c2 ∼ Gamma(c1, c2),
∼ Uniform(0, 1),here c1, c2 are constants that define the
hyperpriors, and maye set by the researcher or be based on previous
related stud-es; Fig. 1C illustrates this model graphically and
Appendix
in Supplementary material provides codes in RStan
(Stanevelopment Team, 2016a) for its implementation. Please
efer to Box 1 for a summary of the steps to build a hierar-hical
Bayesian model.
The important point is that the hierarchical models implic-tly
take into account corrections when performing multipleests since
the posterior of λi of segment i is calculated basedn information
from all segments, and not only from seg-ent i. When we add spatial
autocorrelation, we explicitly
ake into account the position of the segment i relatively tohe
others. The overall result is that the value of a given λi is
oved towards the mean of its distribution and reduces theize of
the confidence intervals, thus reducing the number ofignificant
results (Gelman et al., 2012).
We test whether a segment is a hotspot using a decision-heoretic
approach as in Miranda-Moreno, Labbe and Fu2007). These authors
described three procedures, but we usenly the so-called Bayesian
test with weights, for three maineasons: first, it is the most
straightforward to apply; second,
t allows to state explicitly the researchers’ concerns regard-ng
incurring in Type I or II errors; and, third, for a reasonablehoice
of parameters it gives results that are intermediate tohose of the
other two methods.
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L. Borda-de-Água et al. / Basic and Applied Ecology 34 (2019)
25–35 29
F dels. P n. Ploa the spa
toe
o
ig. 1. Graphical representation of the Bayesian hierarchical
mooisson-gamma models, respectively, without spatial
autocorrelatioutocorrelation. In the latter model, the term (D −
δW) implements
The purpose of the Bayesian test with weights approach is
o minimize a loss function that takes into account the costsf
making a Type I error, CI , and the costs of making a Type IIrror,
CII . The costs CI and CII , or their relative values, can be
ao
Plots (A) and (B) show the Poisson-lognormal hierarchical andt
(C) shows the Poisson-lognormal hierarchical model with spatialtial
autocorrelation.
btained by expert judgment or can reflect social perceptions
ssociated with different choices. Therefore, the specificationf
CI and CII allows us to state explicitly whether we are more
-
30 L. Borda-de-Água et al. / Basic and Applied Ecology 34 (2019)
25–35
Box 1: The Bayesian approach step-by-stepWe summarize here the
several steps to per-form the Bayesian analysis. For simplicity
wedivide them into two sets of procedures.Building a hierarchical
Bayesian model:
1) Identify a sampling distribution to model thenumber of
collisions in each segment (in ourcase a Poisson distribution);
2) Establish prior distributions for the param-eters of the
distribution of the number ofcollisions in each segment (in our
casea prior to model the λi), and if spatialautocorrelation is
present then introduce aformulation for it;
3) Choose the distributions of the (hyper)parameters of the
prior distribution.
Performing hypothesis tests:
1) Define a threshold tS ;2) Assign the costs CI and CII and
calculate the
threshold tC;3) Based on the posterior of λi calculate the
probability P(λi > tS | ni);4) Assess whether P(λ > t | n
) ≥ t . If so, the
cs
aisut{
mpttidt
t
wt
a
Table 1. Number of hotspots identified using different
methods.“Without corrections” correspond to the results obtained
withouttaking into account multiple testing corrections and
assuming thatthe number of collisions in each segment follows a
Poisson dis-tribution and a significance level of 0.05. “FWER”
stands for the“familywise error rate correction” method, “FDR” for
the “falsediscovery rate” method, and “S.A.” for “spatial
autocorrelation”.
Method Burrowingowl (Athenecunicularia)
Blue–black grassquit(Volatinia jacarina)
Without corrections 10 10FWER 1 6FDR 3 9Bayesian without
S.A.
CI = 1; CII = 2 6 6CI = 1; CII = 1 3 2CI = 2; CII = 1 1 2
Bayesian with S.A.CI = 1; CII = 2 4 6CI = 1; CII = 1 4 4
pmtC|nz(
P
ra
T
ru(aettm
sa
i S i C
segment is a hotspot, otherwise it is not.
oncerned with missing hotspots or incorrectly identifyingegments
as hotspots.
For our analyses we need to establish two thresholds, tS ,nd tC,
the former establishing whether or not a segments a hotspot, and
the latter defining the probability of thategment being a hotspot.
Accordingly, for a given tS andsing the terminology H0i for the
null hypothesis and H1i forhe alternative hypothesis, we have
H0i : λi ≤ ts Not a hotspotH1i : λi > ts Hotspot
Ideally, the choice of tS should be based on expert judge-ent or
some other criteria that reflect our knowledge of the
opulation dynamics and the risks posed by the infrastructureo
its viability. Here, assuming that there is no information onhe
abundance or the dynamics of the targeted population (ast often
happens), we estimate tS from the mean and standardeviation of the
number of collisions n obtained directly fromhe raw data
(Miranda-Moreno et al., 2007):
S = mean(ni) + zS standard deviation(ni). (5)With this
formulation the decision is now in terms of zS ,
hich defines how many standard deviations tS is away fromhe
mean.
Once tS (or zS) has been defined we can calculate the
prob-bilities of the null and alternative hypotheses based on
the
bis
CI = 2; CII = 1 2 1
osterior distribution of λi. Thus, the probability of a seg-ent
being a hotspot is P(H1i | ni) = P(λi > tS | ni), knowing
hat an incorrect classification of a false positive has costII .
Equally, the probability of not being a hotspot is P(H0i
ni) = P(λi ≤ tS | ni), and an incorrect classification of
falseegative has cost CI . The cost of a correct identification
isero. The objective is to minimize costs and it can be
shownBerger, 1985) that this is achieved when
(H1i|ni) = P(λi > tS |ni) ≥ CICI + CII (6)
The ratio CI /(CI + CII ) = tC defines the threshold value
thatejects the null hypothesis and defines whether a segment is
hotspot. We summarize these steps in Box 1.
wo case studies
We illustrate the application of the previous methods
withoadkill data on two bird species, the burrowing owl (A.
cunic-laria, with n = 40 collisions) and the blue–black grassquitV.
jacarina, with n = 475 collisions) collected in Brazil along
50-km road; each segment is 1 km, thus M = 50 (see Santost al.
(2016) for details). We use these two data sets becausehese two
species exhibit very different number of casual-ies, thus allowing
us to illustrate the two different Bayesianodels developed without
spatial autocorrelation.Without corrections, assuming that the
number of colli-
ions in each segment follows a Poisson distribution and
significance level of 0.05, the collision threshold for the
urrowing owl is nT = 2 and for the blue–black grassquits nT = 15
leading, coincidentally, to 10 hotspots for eachpecies (Figs. 2 A
and 3 A, and Table 1).
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L. Borda-de-Água et al. / Basic and Applied Ecology 34 (2019)
25–35 31
Fig. 2. Results for the burrowing owl (Athene cunicularia). The
black vertical lines depict the number of collisions along the 50
1-km roadsegments. Plot (A): the grey dotted line shows the
threshold when no corrections are applied (leading to 10 hotspots),
and the grey dashed lineshows the threshold after application of
the familywise error rate correction (leading to 1 hotspot)
identified by the vertical grey bar. Plot (B):The application of
the false discovery rate correction leads to the identification of
3 hotspots identified by the vertical grey bars. Plot (C):
Theapplication of the Bayesian hierarchical Poisson-lognormal model
without spatial autocorrelation leads to the hotspots identified by
verticalcoloured bars: in red are the hotspots identified when CI =
2 and CII = 1, in red and green when CI = 1 and CII = 1, and in
red, green and bluewhen CI = 1 and CII = 2. The broken line
corresponds to the average value of the posterior of each λi (the
mean rate of collisions per unit ofl lts of tt th plotst
rticle.)
T
Pcttha
T
ibTht
B
zsCCCattetf
ength estimated during a certain period of time). Plot (D): The
resuhe same convention for the hotspots as in plot C. Notice that
in bohis figure legend, the reader is referred to the web version
of this a
he familywise error rate correction (FWER)Assuming PL = 0.05,
the FWER correction leads to
S ≈ 0.001 (Eq. (2) or to its simplified version Eq. (3)).
Theorresponding number of collision thresholds are nT = 5 forhe
burrowing owl and nT = 20 for the blue–black grassquit,hus we
identify 1 hotspot for the burrowing owl and 6otspots for the
blue–black grassquit (Figs. 2 A and 3 A,nd Table 1).
he false discovery rate correction (FDR)Applying the FDR
correction with PS = 0.05 (Eq. (4)), we
dentified 3 and 9 hotspots for the burrowing owl and
thelue–black grassquit, respectively (Figs. 2 B and 3 B, and
able 1). Not surprisingly, the FDR correction identifies
moreotspots than the FWER correction, because the FDR reduceshe
probability of false negatives (Type II errors).
mdg
he Bayesian hierarchical model with spatial autocorrelation,
using (C) and (D) that. (For interpretation of the references to
colour in
ayesian hierarchical modelling (BHM)The Bayesian analysis
requires that we assume values for
S , CI and CII , to establish thresholds according to
expres-ions (5) and (6). We considered three combinations forI and
CII : (CI = 1, CII = 2), (CI = 1, CII = 1), and (CI = 2,II = 1)
(Table 1). For example, the combination (CI = 1,II = 2) means that
the cost of missing a hotspot is twices large as that of
identifying a hotspot incorrectly and leadso tC = 0.333 (see Eq.
(6)). For the value of zS we estimatedhe mean and standard
deviation for λ from the data and thenstimated zS (Eq. (5)) such
that the threshold would be closeo 0.05. These led to zS = 1 for
the burrowing owl and zS = 1.5or the blue–black grassquit.
Concerning the hyperpriors, because we had no extra infor-
ation on these parameters, we tried to have non-informative
istributions. We assessed the impact of three differentamma
distributions: Gamma(0.01,0.01), Gamma(0.1,1),
-
32 L. Borda-de-Água et al. / Basic and Applied Ecology 34 (2019)
25–35
Fig. 3. Results for the blue–black grassquit (Volatinia
jacarina). The black vertical lines depict the number of collisions
along the 50 1-kmroad segments. Plot A: the grey dotted line shows
the threshold when no corrections are applied (leading to 10
hotspots), and the grey dashedline shows the threshold after
application of the familywise error rate correction (leading to 6
hotspots) identified by the vertical grey bar. PlotB: The
application of the false discovery rate correction leads to the
identification of 9 hotspots identified by the vertical grey bars.
Plot C:The application of the Bayesian hierarchical Poisson-gamma
model without spatial autocorrelation leads to the hotspots
identified by verticalcoloured bars: in red are the hotspots
identified when CI = 2 and CII = 1, in red and green when CI = 1
and CII = 1, and in red, green and bluewhen CI = 1 and CII = 2. The
broken line corresponds to the average value of the posterior of
each λi (the mean rate of collisions per unit ofl lts of tt n of
tht
aonfln
yitudwctfigi
w2
rtWfmoTaTaT
ength estimated during a certain period of time). Plot D: The
resuhe same convention for the hotspots as in plot C. (For
interpretatiohe web version of this article.)
nd Gamma(1,1). Although tests revealed that the choicef prior
did not affect the conclusions significantly (resultsot shown), we
used the Gamma(1,1) because it has a moreat distribution for the
range of λ observed, thus closer to aon-informative
distribution.Because the models are easier, we start the Bayesian
anal-
ses assuming no spatial autocorrelation. We ran the analysisn
the R environment and JAGS using 10,000 burn in itera-ions, and
collected information on the parameters of interestsing 50,000
iterations. As expected, the number of hotspotsepends on the
relative values of CI and CII and they areithin the range of the
values obtained with the two previous
orrections (Figs. 2 C and 3 C and Table 1). If we were averseo
missing hotspots (CI = 1,CII = 2), we would have identi-
ed 6 hotspots for both the burrowing owl and the
blue–blackrassquit. At the other extreme, if we are concerned
aboutncorrectly identifying segments as hotspots (CI = 2,CII =
1)
ora
he Bayesian hierarchical model with spatial autocorrelation,
usinge references to colour in this figure legend, the reader is
referred to
e could consider only 1 hotspot for the burrowing owl and for
the blue–black grassquit.
A simple test with the autocorrelation function (not
shown)evealed that both data sets exhibit spatial
autocorrelation,hus we re-analysed the data with the Bayesian CAR
model.
e ran the analyses in the R environment and RStan usingour
chains with 1000 burn in iterations, and collected infor-ation on
the parameters of interest using 1000 iterations
f each chain. We used a delta sampler control of 0.99.his
sampler will make the program run slower but it canvoid divergent
transitions after warmup (Stan Developmenteam, 2016a). To run the
model efficiently in RStan welso used a sparse CAR representation
(Stan Developmenteam, 2016b). As before, the number of hotspots
depends
n the relative values of CI and CII and they are within theange
of the values obtained previously (Figs. 2 D and 3 Dnd Table 1).
Adopting a policy of being averse to missing
-
and Ap
hremf
thttbhmhgttaactiwto
D
ftm(22t
BepriIdutiIii“
aah
pdeiiich
itttArcPsnii(wo(
oshTeatithnfnasostd(ec
apt
L. Borda-de-Água et al. / Basic
otspots (CI = 1,CII = 2), we identified 4 hotspots for the
bur-owing owl and 6 for the blue–black grassquit. At the
otherxtreme, being concerned about incorrectly identifying seg-ents
as hotspots (CI = 2,CII = 1), we identified 2 hotspots
or the burrowing owl and only 1 for the blue–black grassquit.The
neighbourhood of a segment plays a role when spa-
ial autocorrelation is introduced. A segment identified as
aotspot in an analysis without spatial autocorrelation mayurn out
not to be a hotspot when spatial autocorrelation isaken into
account if casualty counts are low in the neigh-ouring segments.
Conversely, a segment not identified as aotspot under the
assumption of no spatial autocorrelation,ay become a hotspot if the
surrounding segments have a
igh count of occurrences. For instance, for the
blue–blackrassquit without spatial autocorrelation, Fig. 3C,
hotspotshat are not surrounded by other hotspots are common. Onhe
other hand, segments surrounded hotspots not identifieds hotspots
in the analysis without spatial autocorrelation,re identified as
hotspots when we consider spatial auto-orrelation, Fig. 3D.
Although the Bayesian analysis alwaysakes into account the
information on all segments, thereforemplicitly taking into account
the problem of multiple testing,hen we include spatial
autocorrelation the information on
he neighbouring segments is also relevant to decide whetherr not
a segment is a hotspot.
iscussion
Our work was motivated by the high probability of identi-ying
false hotspots (that is, of making Type I errors) inherento methods
that do not take into account corrections for
ultiple-testing, a common practice in road ecology researche.g.
Malo et al., 2004; Grilo et al., 2011; Planillo et al.,015; Santos
et al., 2015; Costa et al., 2015; Garriga et al.,017), and we
suggest three different methods to deal withhis problem.
Besides the profound differences between frequentist andayesian
schools, we do not see any method as being inher-ntly better than
the others. Ultimately, the choice of therocedure reflects the set
of concerns and priorities of theesearcher. For example, the FDR
procedure should be usedf a researcher is concerned about missing
true hotspots (TypeI error). On the other hand, if the concern is
the probability ofetecting false hotspots (Type I error) the
researcher shouldse the FWER procedure. Nevertheless, our
preference goeso the Bayesian approach because it allows a very
clear spec-fication of the researcher’s concerns about making Type
I orI errors, and, furthermore, it can take spatial
autocorrelationnto account. Since the two frequentist methods do
not takento account spatial autocorrelation, we may say that they
arewrong” when spatial autocorrelation is present.
The frequentist methods exposed here are clearly easier topply.
Therefore, these are the methods we are most likely topply if we
just plan to check if a previous identification ofotspots is likely
to have omitted any corrections for multi-
t
pt
plied Ecology 34 (2019) 25–35 33
le testing, or if someone reports that in studies conducted
atifferent times hotspots keep changing their locations. How-ver,
we would recommend using a Bayesian approach todentify hotspots.
This may take longer to implement, but its a time investment worth
considering given its advantages,n particular, the possibility of
stating explicitly the perceivedosts of giving more or less weight
to the costs of missingotspots associated with different
policies.Furthermore, the Bayesian hierarchical models can
easily
ncorporate explanatory variables that reflect the
charac-eristics of the roads or of the surrounding
environment,hereby leading to a better understanding of the causal
rela-ionships underlying mortality hotspots (e.g.,
Bartonička,ndrášik, Dul’a, Sedoník, & Bíl, 2018). To this
end, the expe-
ience gained in the field of human accidents and preventionan be
invaluable (e.g. Ahmed et al., 2011; Li, Carriquiry,awlovich, &
Welch, 2008; Huang & Abdel-Aty, 2010). Ithould be noted that
our approach assumed that there waso precise information on the
location of the collisions and,nstead, the number of collisions was
given per segment. Thiss often the case for data on collisions of
birds with power linesF. Moreira, personal communication), but can
also occurith road data (Gunson et al., 2009). If the precise
locationf the collisions is known, then other methods can be
appliede.g. Bíl et al., 2013, 2016).
The methods discussed here deal with the statistical aspectsf
identifying hotspots, yet they do not substitute the discus-ion on
the societal choices underlying the need to identifyotspots or to
adopt measures that mitigate their impacts.hese choices are
incorporated in the thresholds adopted inach method, and are
explicitly stated under the Bayesianpproach. From a strict
conservation perspective, the impor-ant question is not whether
individuals are killed by annfrastructure, but how such additional
(non-natural) mor-ality impacts the viability of a population or of
a species, orow much the depletion of a species affects the entire
commu-ity (Borda-de-Água, Grilo, & Pereira, 2014). For
instance,or the two species analysed here we observed a much
largerumber of casualties among the blue–black grassquit thanmong
the burrowing owl. However, this is likely to reflectolely the
higher abundance of one species relatively to thether, in
particular, in the vicinity of roads, but what our analy-is could
not ascertain is how much the road mortality affectshe viability of
these populations. Both species have wideistributions and their
conservation status is “least concern”IUCN Species Survival
Commission, 2000), but it is nev-rtheless possible that the levels
of road mortality observedould endanger the local populations.
Some may argue that for conservation purposes it is prefer-ble
to identify false hotspots than to miss real ones. Theroblem with
such a view is that efforts aimed at mitigatinghe negative impacts
of an infrastructure are unlikely to wield
he desired results.
In particular, if segments identified as hotspots are
falseositive hotspots then their locations are likely to
con-inuously shift over time in a rather arbitrary way. This
-
3 and Ap
utai
A
t
A
mJTFOGRwC(fganhd
A
ch
R
A
B
B
B
B
B
B
B
B
B
C
G
G
G
G
G
G
G
G
H
H
4 L. Borda-de-Água et al. / Basic
ndermines the credibility of conservation efforts in the longerm
and results in a waste of resources, including economicnd manpower.
In turn, this may discourage well-intentionedndividuals or
organizations due to the lack of clear results.
uthors contributions
LBA conceived the paper. All the authors contributed tohe
analysis of the results and the writing of the paper.
cknowledgements
We thank Rodrigo Lima Augusto for providing the roadortality
datasets and Giovani Silva, Paulo Soares and
uan Malo for reading an earlier version of this paper.his
article is a result of the project NORTE-01-0145-EDER-000007,
supported by Norte Portugal Regionalperational Programme
(NORTE2020), under the PORTU-AL 2020 Partnership Agreement, through
the Europeanegional Development Fund (ERDF). LBA, FA and RBere
funded by Infraestruturas de Portugal Biodiversityhair. FA was also
supported by a FCT postdoctoral grant
SFRH/BPD/115968/2016). MS was funded by a PhD grantrom Fundacão
para a Ciência e a Tecnologia (FCT), Portu-al (ref.
PD/BD/128349/2017). All sources of funding arecknowledged in the
manuscript, and the authors declareo direct financial benefits from
its publication. The fundersad no role in the study design, data
collection and analysis,ecision to publish, or preparation of the
manuscript.
ppendix A. Supplementary data
Supplementary data associated with this arti-le can be found, in
the online version,
atttps://doi.org/10.1016/j.baae.2018.11.001.
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