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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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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.

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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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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


. 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.

L. Borda-de-Água et al. / Basic and Applied Ecology 34 (2019) 25–35 29

F dels. P n. Ploa the spa

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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

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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

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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).

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.)

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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

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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

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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

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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

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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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On the identification of mortality hotspots in linear ......Basic and Applied Ecology 34 (2019)...

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