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STATISTICAL REPORTSEcology, 85(6), 2004, pp. 1591–1597q 2004 by the Ecological Society of America

MODELING ABUNDANCE EFFECTS IN DISTANCE SAMPLING

J. ANDREW ROYLE,1,4 DEANNA K. DAWSON,2 AND SCOTT BATES3

1U.S. Fish and Wildlife Service, Division of Migratory Bird Management, 11510 American Holly Drive,Laurel, Maryland 20708 USA

2USGS Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, Maryland 20708 USA3National Park Service, Center for Urban Ecology, 4598 MacArthur Boulevard, NW, Washington, D.C. 20007-4227 USA

Abstract. Distance-sampling methods are commonly used in studies of animal popu-lations to estimate population density. A common objective of such studies is to evaluatethe relationship between abundance or density and covariates that describe animal habitator other environmental influences. However, little attention has been focused on methodsof modeling abundance covariate effects in conventional distance-sampling models. In thispaper we propose a distance-sampling model that accommodates covariate effects on abun-dance. The model is based on specification of the distance-sampling likelihood at the levelof the sample unit in terms of local abundance (for each sampling unit). This model isaugmented with a Poisson regression model for local abundance that is parameterized interms of available covariates. Maximum-likelihood estimation of detection and densityparameters is based on the integrated likelihood, wherein local abundance is removed fromthe likelihood by integration. We provide an example using avian point-transect data ofOvenbirds (Seiurus aurocapillus) collected using a distance-sampling protocol and twomeasures of habitat structure (understory cover and basal area of overstory trees). Themodel yields a sensible description (positive effect of understory cover, negative effect onbasal area) of the relationship between habitat and Ovenbird density that can be used toevaluate the effects of habitat management on Ovenbird populations.

Key words: abundance estimation; avian point counts; distance-sampling methodology; mixturemodels; Ovenbird; random effects.

INTRODUCTION

The use of distance-sampling methods to estimatedensity is widespread in studies of animal populations(Buckland et al. 2001, Williams et al. 2002). Suchmethods use information on observed distances of an-imals from transects or points of observation to char-acterize the detection probability of individuals. Underthe hypothesis that detection probability is related tothe distance between animals and the point of obser-vation, one may obtain an estimate of density that is,in effect, adjusted for nondetection bias.

Distance-sampling methods are attractive in manyanimal-sampling problems because they do not requirethat individuals be uniquely marked and recaptured (orresighted) through time. In avian counting applicationsbased on point counts (Buckland et al. 2001, Rosen-stock et al. 2002), distances are recorded from a pointof observation (instead of a transect), and this is usuallyreferred to as a ‘‘point transect’’ (Buckland et al. 2001).

Manuscript received 9 September 2003; revised 16 January2003; accepted 19 January 2003. Corresponding Editor: M. S.Boyce.

4 E-mail: andy [email protected]

In many field situations, birds are detected only by theirvocalizations and so it can be difficult to obtain precisedistance information. Consequently, distances are fre-quently recorded by grouping into discrete distance in-tervals. We focus subsequent discussion on these point-transect situations, due to our interest in modeling avi-an point-count data described in Application . . . , be-low.

In distance sampling, as with other common sam-pling protocols, considerable modeling effort is fo-cused on describing variation in detection probability(Buckland et al. 2001, Marques and Buckland 2003,Ramsey and Harrison 2004). To this end, very complexmodels of detection probability are often considered.However, many applications have explicit objectivesthat involve understanding mechanisms that affectabundance. In particular, estimation of the effect ofspatial covariates on abundance is fundamental to manyinvestigations of animal populations. For example,landscape or habitat characteristics associated witheach spatial sample unit are often collected, and interestis in modeling the relationship between abundance (ordensity) and these measured covariates. Most frequent-ly, applications that focus on modeling the effect of

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1592 J. ANDREW ROYLE ET AL. Ecology, Vol. 85, No. 6

abundance covariates do not explicitly address the issueof detectability, and use conventional techniques suchas regression or generalized linear models for modelingthe observed counts or detection/nondetection (e.g.,Vincent and Haworth 1983, Robbins et al. 1989, Frie-sen et al. 1995, Hutto 1995, Young and Hutto 1999,2002, Fewster et al. 2000, Brand and Georgea 2001,Vernier et al. 2002, Beadell et al. 2003). This can leadto biased estimators of habitat effects (Gu and Swihart2004). There have been few methods developed foraccommodating covariate effects on abundance intodistance sampling or other procedures that directly ac-count for detectability. Application of conventionaldistance sampling (and related) methods to point-tran-sect data is based on pooling data collected from mul-tiple point counts. This yields an estimate of averagedensity, but results in a loss of information at the levelof sample units (e.g., point locations) due to pooling.

We propose a model that allows for incorporation ofabundance covariate effects within distance-samplingmodels. We pose the likelihood for data from each pointas a function of ‘‘local abundance’’ in the vicinity ofthat point. Then local abundance may be related tocovariates using a Poisson or other generalized linearregression model. Local abundance is regarded as arandom effect, and analysis is based on the integratedlikelihood, which is a function of parameters of thedetection function, density, and relevant density co-variates. We apply the proposed model to avian point-transect data that are typical of those to which distance-sampling methods are often applied.

DATA AND MODEL

We suppose that point counts of a fixed radius areconducted at i 5 1, 2, . . . , R points in some region(e.g., a park or forest). Each point count represents asample of fixed area, which we will refer to as a ‘‘sam-ple unit’’ or ‘‘site.’’ We consider grouped data herewherein distances are recorded in discrete intervalsfrom the central point of observation for each site. Letk 5 1, 2, . . . , K index the distance classes, with endpoints (c1, c2), (c2, c3), . . . , (cK, cK11). Here, cK11 is themaximum distance at which birds were counted, or theradius of the point count. Let yik be the observed count ofindividuals in distance class k for site i 5 1, 2, . . . , R.

Let g(x; u) denote some function that describes therelationship between detection probability and dis-tance, x, from the point of observation. This detectionprobability function is parameterized by the (possiblyvector-valued) parameter u. See Buckland et al. (2001)for a discussion of possible detection functions andunderlying assumptions.

The key departure from the traditional formulationof distance-sampling estimators of density that we en-tertain here is that we consider estimation based on theunconditional likelihood of the observed counts as op-posed to the likelihood which is ‘‘conditioned on de-tection’’ (see Sanathanan [1972] for a discussion of the

distinction). This allows direct parameterization of thelikelihood for the data from each site in terms of theabundance at that site, Ni. The main reason for con-sideration of the unconditional likelihood here is thatmodeling spatial variation in Ni (or density) is straight-forward under the unconditional formulation, as dem-onstrated subsequently. Under the conventional con-ditional formulation, there is no site-specific quantitythat can be regarded as abundance or density, and thatcan be related to site-specific covariates. However,Hedley and Buckland (2004) recently have developedan approach based on the conditional likelihood (seeDiscussion, below).

In terms of Ni, the site-specific likelihood for data yi

5 (yi1, yi2, . . . , yiK) is

N !i yikf (y z u) 5 p (u)Pi k[ ]k

y ! (N 2 y )!P ik i i .1 2k

N 2yi i.

3 1 2 p (u) (1)O k[ ]k

where yi· 5 Sk yik.Construction of cell probabilities, pk(u), is based on

conventional considerations (e.g., Buckland et al. 2001:chapter 3). Specifically, in Eq. 1, pk(u) is the probabilitythat an individual occurs and is detected in distanceclass k, which depends on the detection function underconsideration. For point transects, this can be computedby integrating g(x; u) over the area of the circle betweenck and ck11 (e.g., see Buckland et al. 2001:54). To beprecise,

ck11 2pxg(x; u)p (u) 5 dx.k E 2pcK11ck

We emphasize that pk(u) here are unconditional cellprobabilities, in contrast to those considered in mostdistance-sampling situations (e.g., the p’s given inBuckland et al. [2001: chapter 3]).

Modeling variation in abundance among sites

Fundamental to the situation under considerationhere is that abundance may vary among sites due tomeasurable environmental characteristics that are alsosite specific (e.g., habitat). Thus, we require an exten-sion of Eq. 1 that makes this idea precise. For that, weaugment the model with an additional model containingpotential sources of variation in Ni. A natural choiceis based on conventional generalized-linear-modeling(GLM) ideas. Suppose Ni were observed, then considerthe Poisson regression model specified by

N ; Poisson(l )i i (2)

where li is the expected value of Ni, typically assumedto be linearly related to available covariates accordingto the following (for a single covariate):

June 2004 1593ABUNDANCE EFFECTS IN DISTANCE SAMPLING

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TABLE 1. Results of fitting the four candidate models of mean density to the Ovenbird data,using the Poisson and negative-binomial abundance models.

Candidates np AIC b a0 UFC BA k

PoissonModel 0Model 1 (UFC)Model 2 (BA)Model 3 (both)

2334

340.82340.27339.30340.67

14.2314.2314.2314.23

20.55120.56720.57420.578

1.859

1.04220.82920.643

Negative binomialModel 0Model 1 (UFC)Model 2 (BA)Model 3 (both)

3445

342.73342.26341.30342.67

14.2314.2314.2314.23

20.55120.56720.57420.578

1.866

1.04220.83220.643

0.0530.0160.0100.000

Notes: Explanation of column heads: np, number of parameters; AIC, Akaike informationcriterion; b, scale parameter of distance function; a0, intercept of the abundance model; UFC,understory foliage cover coefficient estimate; BA, basal area coefficient estimate; k, negativebinomial over-dispersion parameter.

log(l ) 5 a 1 a zi 0 1 i (3)

where zi is the value of the covariate measured at sitei. Note that in the usual terminology of Poisson models,li is density per ‘‘sample unit’’ (assuming the sampleunits are of the same area), and so this model nowincludes a parameter that can be interpreted in a mannerconsistent with traditional distance-sampling ideas, butwhere density is now spatially explicit. If covariatesare centered to have mean zero, then exp(a0) is the‘‘mean density.’’ One referee suggested incorporatingthe term log(p ) as an additive offset in Eq. 3 so2cK11

that li has the more precise interpretation of individualsper unit area (recall that ck11 is the outer limit of ob-servation beyond which birds are not counted). Also,if point-count areas are not constant, then this shouldbe accounted for in a similar manner.

We note that alternative abundance models may alsobe considered. An appealing and common model fordescribing variation in counts (e.g., animal abundance)in the presence of over-dispersion (or ‘‘excess-Poissonvariation’’) is the negative-binomial (Boyce et al. 2001)model. A common parameterization of the negative bi-nomial (we use this parameterization in Application. . . , below) is that in which E[N] 5 l and Var[N] 5l 1 kl2 where k . 0 is the ‘‘over-dispersion’’ param-eter. Thus, Var[N] $ E[N] and in limit (as k goes to 0)the negative binomial is equivalent to the Poisson dis-tribution. As with the Poisson, a log-linear model re-lating covariates to mean abundance may be used.Abundance models other than Poisson or negative bi-nomial that allow for more flexibility in modeling over-dispersion may also be considered (e.g., see Bhatta-charya and Holla 1965, Agarwal et al. 2002, Puig2003). In practice it may be desirable to formallychoose among several plausible models of abundancegiven observational data. When inference is based onthe integrated likelihood described subsequently, mod-el selection may be carried out using AIC (Akaike in-formation criterion; Burnham and Anderson 1998).

In contrast to many conventional applications ofPoisson regression, the Ni are not observable in dis-tance-sampling problems. A solution that yields a pre-cise treatment of the likelihood (Eq. 1) in the contextof the Poisson regression model (expression 2) is toregard Ni as unobserved random effects with distri-bution given by expression 2 and, following conven-tional notions of the treatment of random effects (e.g.,Laird and Ware 1982, Robinson 1991), integrate themfrom the likelihood (Eq. 1). This yields a marginallikelihood (often referred to as the ‘‘integrated likeli-hood’’) that is only a function of detection parameters(contained in g(x; u)), and density parameters a0 anda1.

The integrated likelihood

The integrated likelihood for the data from site i isformulated by integrating Eq. 1 over the random-effectsdistribution (expression 2):

` N !i y ikL(a, u z y ) 5 p (u)O Pi k[ ]N 5y ki i.

y ! (N 2 y )! P ik i i .1 2k

N 2yi i. 2l (a) Ni ie l (a)i3 1 2 p (u) .O k[ ] N !k i

(4)

Note that the dependence of li on the parameters a 5(a0, a1) is emphasized in this expression.

One benefit of the Poisson abundance model is thatthe summation in Eq. 4 can be done analytically, yield-ing the product Poisson likelihood:

K

L(a, u z y ) 5 Poisson[ y ; l (a)p (u)]. (5)Pi ik i kk51

The simple likelihood obtained under the Poisson abun-dance model is more computationally efficient than the

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general mixture form given by Eq. 4. For other abun-dance models (such as the negative binomial) Eq. 4does not simplify in any meaningful way.

Assuming the data are independent across sites, thejoint likelihood for all data is then

R

L(a, u z y , . . . , y ) 5 L(a, u z y ). (6)P1 R ii51

Practically speaking, independence here implies thatpoint counts are made sufficiently far apart so as toensure that the same individuals are not being countedat multiple sites. Consequently, this may only be rea-sonable for species that can be sampled at a time whenindividuals are territorial or sedentary. Note that in-dependence underlies all of the conventional distance-sampling methods and so it should not be viewed asan assumption that is specific to the proposed model.

The integrated likelihood (Eq. 6) can be maximizedusing conventional numerical methods. For the analysesreported in the following section, we developed our ownroutines using the free software R (Ihaka and Gentleman1996). Note that for abundance models other than thePoisson, the upper bound appearing in the summationof Eq. 4 must be truncated at some finite integer. Onecan check that the chosen cut-off was sufficient by in-creasing its value and verifying that the MLEs (maxi-mum-likelihood estimators) are unchanged.

As a final comment, note that it is possible to esti-mate individual Ni’s using what is usually referred toas best unbiased prediction (BUP), although there isoften no obvious need to do so since most inferentialproblems focus on density (‘‘average abundance’’) ef-fects. See Royle (2004) for details.

Evaluating model goodness of fit

Under the Poisson model, the conventional deviancestatistic for Poisson data (Agresti 2002) may be usedin sufficiently large samples. However, small countssuch as those generated from our study (and perhapsmost studies based on avian point-count surveys) ren-der the asymptotic null distribution of this statistic in-valid. Also, a convenient deviance statistic (that is, witha known asymptotic null distribution) under otherabundance distributions has not been developed. Con-sequently, we considered assessment of goodness of fitusing conventional parametric bootstrap procedures(Dixon 2002).

APPLICATION TO AVIAN POINT-COUNT DATA

Here we consider avian point-count data collectedin late May and June of 2002 at 70 sites in the CatoctinMountains, Frederick County, Maryland, USA. To as-sess the possible impacts of white-tailed deer (Odo-coileus virginianus) on natural resources in the Catoc-tin Mountain Park, a unit of the National Park Service,breeding-bird populations and vegetation were sampledin the park and in the nearby Frederick City WatershedCooperative Wildlife Management Area, where deer

numbers are controlled by hunting. Thirty-five sites ineach area were randomly selected for sampling from agrid of points, spaced at 250-m intervals, generated inArcInfo (ESRI, Redlands, California, USA).

At each site, a 12-min count was made of all birdsseen or heard. Distances to detected birds were re-corded in 25-m distance classes out to 100 m from thepoint. Observations of two covariates thought to influ-ence local bird abundance were also collected: percentunderstory-foliage cover (UFC) and basal area of over-story trees (BA; in square meters). Understory vege-tation was sampled by counting the number of heightintervals (0–0.1 m, .0.1–0.3 m, .0.3–0.5 m, .0.5–1.0 m, .1.0–1.5 m, .1.5–2.0 m, .2.0–2.5 m, and.2.5–3.0 m aboveground) in which live vegetation in-tersected a pole planted at 3-m intervals along 25-mtransects that radiated from the point in the cardinaldirections (north, south, east, west). For each point,UFC is the percentage of the height-interval samplesin which vegetation was present. A forestry prism (bas-al area factor: 10) was used to sample overstory basalarea at each point (Hovind and Rieck 1970).

The goal is to evaluate the relationship between birddensity and these habitat characteristics. In this illus-tration, we consider counts of male Ovenbirds (Seiurusaurocapillus), which are detected primarily by song.Ovenbirds are a ground-nesting species, and so it issuspected that understory vegetation serves to providenesting cover. Conversely, excessive overstory (mea-sured by BA) impedes the development of understorycover. Consequently, we expect a positive relationshipbetween Ni (abundance at site i) and UFC and a neg-ative relationship between Ni and BA. These covariateswere standardized to have mean zero, and incorporatedinto the abundance model as in Eq. 3, with the additiveoffset log(p). Because the point-count radius was 100m, li is interpreted as the number of individuals perhectare (10 000 m2). We used a ‘‘half-normal’’ detec-tion function to model the relationship between detec-tion probability and distance:

2xg(x; u) 5 exp 2 .1 225b

Note that the denominator is scaled so that the (equallyspaced) distance intervals are bounded by the integers(0, 1), (1, 2), (2, 3), and (3, 4).

We considered four possible models of density todescribe the Ovenbird data using both the Poisson andnegative-binomial models for abundance (eight modelsin all). Model 0 contained neither covariate. Models 1and 2 contained only UFC and BA, respectively, andModel 3 contained both covariates. Results of fittingthese models are given in Table 1. To evaluate therelative merits of each model, the AIC score (Burnhamand Anderson 1998) for each model is also given.

We note that the estimated over-dispersion parameterof the negative-binomial model is near 0 in all cases


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FIG. 1. Estimated Ovenbird density as a function of un-derstory foliage cover (UFC; top panel) and basal area (BA;bottom panel).

(Table 1). Consequently, the estimates of the other pa-rameters in the negative-binomial models are essen-tially equivalent to those of the corresponding Poissonmodels. Thus, AIC favors the simpler (more parsi-monious) Poisson abundance model. Among the Pois-son models, those with one or the other covariate (butnot both) are preferred. Note that the two covariatesare negatively correlated with one another (sample cor-relation, r 5 20.49), and thus they contain redundantinformation. This explains why the model with bothcovariates has a higher AIC despite the apparent im-portance of each covariate individually.

The signs of the coefficient estimates are as antici-pated (positive relationship between abundance andUFC, negative for BA). The estimated effect of eachcovariate (on the density scale) is depicted graphicallyin Fig. 1 where l is plotted as a function of UFC andBA using estimates obtained from the correspondingmodels. Thus, for example, as UFC increases from 0to 50%, density increases from approximately 0.4 to1.1 Ovenbirds/ha.

Goodness of fit was evaluated for the best fittingPoisson model using the parametric bootstrap proce-dure based on model deviance. This indicated no sig-nificant lack of fit (P 5 0.726).

Finally, note that the estimated scale parameter (bin Table 1) is the same to within two decimal placesfor all eight models considered. Apparently, it is un-affected by variation in N as might be expected (notethat the Ni’s can be removed by conditioning).

DISCUSSION

We have proposed a modeling approach that allowsfor incorporation of covariate effects on abundance in

distance-sampling models. The key idea underlyingthis model is the formulation of the (multinomial) dis-tance-sampling likelihood in terms of site-specificabundance parameters, Ni. These are regarded as ran-dom effects and assigned a distribution wherein themean depends on the covariates under consideration.The marginal likelihood of the data is constructed byintegrating the multinomial likelihood for the data overthis random-effects distribution.

The model proposed here expands on that consideredby Royle (2004) for estimating abundance from point-count data. That model is based on simple point-countdata (i.e., with no distance information), and also usesan integrated likelihood wherein the binomial likeli-hood for the observed counts is integrated over a Pois-son prior distribution on Ni. See also Royle and Nichols(2003) for a similar strategy for modeling heterogeneityin detection in occupancy surveys that is due to vari-ation in abundance. That idea has been used here tofacilitate modeling structure in site-specific abundanceunder a multinomial distance-sampling model.

Distance sampling is widely used to estimate abun-dance of animal populations, and the underlying theoryis well developed and intricate. Despite this, until re-cently, relatively little attention has been focused onthe development of distance-sampling methods that al-low for modeling covariate effects. Note that Bucklandet al. (2001) contains a single section (3.8) devoted tothe topic. Much of what has been done has focused onmodeling covariates that affect detection probability(e.g., Ramsey et al. 1987, Marques and Buckland 2003,Ramsey and Harrison 2004). To the best of our knowl-edge, only Hedley et al. (1999) and Hedley and Buck-land (2004) consider models that allow for covariateeffects on abundance. These recent efforts focus onformulating distance-sampling models in terms of thespatial-intensity function of a Poisson point processand adopt the more common conditional formulationof the likelihood for the observation locations of in-dividuals. The resulting likelihood is not analyticallytractable and appears unstable for complex intensityfunctions (Hedley and Buckland 2004). This motivatesHedley and Buckland (2004) to suggest a more infor-mal approach that first estimates the detection function,and then uses a ‘‘plug-in’’ type of procedure based onthe partial likelihood of the observed counts. Our ap-proach is similar in the sense that the abundance modelis derived from consideration of a spatially varyingintensity function. However, formulation of the prob-lem in terms of the unconditional multinomial likeli-hood (a function of local abundance) and then inte-grating over a prior distribution for N (containing co-variate effects), yields a well-behaved likelihood. Inthe case where the prior is Poisson, the likelihood isof closed form (a product Poisson likelihood). Gen-erality to other abundance distributions (determiningmean–variance relationships) can also be considered,and it is not clear how this can be accommodated in

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the likelihood considered by Hedley and Buckland(2004).

A natural result of a spatially explicit formulation ofabundance models is that one may produce maps of thedensity surface over the sample domain, or predictabundance at unsampled sites. This is necessary if char-acterizing the total population size is required, and re-quires that the spatial covariates are measured every-where. Hedley and Buckland (2004) consider spatialmapping of the intensity function in their application,which involved covariates that were functions of spatiallocation (e.g., spatial regression functions).

While our focus here has been on point-transect datacollected according to a distance-sampling protocol,the general notion is clearly applicable to more com-mon distance-sampling situations that make use of tran-sects, when landscape or habitat covariates can be mea-sured for each transect, or segments of each transect.Also note that the general modeling strategy proposedhere can be applied regardless of the form of the mul-tinomial likelihood. In particular, it can be applied toproblems where the multinomial likelihood is based onother sampling protocols such as multiple observer(Nichols et al. 2000), removal (Farnsworth et al. 2002),and even conventional capture–recapture when thereare spatially indexed samples. As with distance sam-pling, modeling covariate effects on abundance hasbeen a deficiency in modeling data collected underthese sampling approaches and, consequently, the pro-posed model can be used to address this objective ina formal manner.

Our application of the proposed model to Ovenbirdcounts provides a compelling illustration of the poten-tial management utility of being able to assess covariateeffects within a distance-sampling framework. In par-ticular, the model provides a concise linkage betweenmeasurable habitat structure and Ovenbird density.Consequently, it allows for evaluation of the effect ofhabitat-management actions on bird density. For ex-ample, the density of Ovenbirds increases as understoryfoliage cover increases (Fig. 1). We note that the du-ration of the point count should be an important con-sideration prior to data collection and the 12-min countduration used in our study is probably excessive. Abun-dance estimation methods based on point counting mustbe regarded as instantaneous in time. As sampling du-ration increases, the effect of ‘‘temporary emigration’’(Kendall et al. 1997) will lead to some bias in meandensity but will not lead to biased covariate effect es-timates.

One referee was concerned about potential bias indensity estimates obtained in our study due to potentialheterogeneity as a result of habitat effects on detect-ability. That is, while we have biological reasons tosuggest that these habitat covariates affect abundance,it is plausible that they also affect detectability. Toevaluate this, we considered the distance class fre-quency of detections for each of three levels (low, mod-

erate, high) of the BA (basal-area) covariate (the modelwith the best AIC score). The frequencies of detectionsin each of the four distance classes were (3,6,16,5),(4,7,11,9), and (0,3,7,6) for low, moderate, and highvalues of the BA covariate. A simple chi-square testof homogeneity yields x2 5 5.097 (df 5 6, P 5 0.531),suggesting no difference between these distance-fre-quency distributions. We note also that our bootstrapgoodness-of-fit procedure indicates that there is no sig-nificant lack of fit, suggesting that there is no additionalheterogeneity, such as due to habitat effects on detec-tion probability. Consistent with this point, note thatthe fitted negative-binomial models are essentiallyequivalent to the Poisson models. Finally, we note thatdetection covariates can be modeled easily in the half-normal distance function (Marques and Buckland2004) and so this idea could conceivably be used toextend the model considered here to include covariateeffects on both detection and density. However, webelieve that the identifiability of habitat effects in bothdetection and abundance (i.e., of the same habitat var-iables) will be highly sensitive to model structure.Thus, when a particular covariate affects both detectionand abundance, we believe that biologists will have tomake a priori judgments about the most sensible wayto partition variance and attempt to design surveys thatminimize this possibility.

ACKNOWLEDGMENTS

The authors would like to thank Prof. Mark Boyce and ananonymous referee for many constructive comments on adraft of this paper. We also thank Prof. Steve Buckland fordirecting us to related work, and his many helpful commentsand suggestions especially with regard to the importance ofheterogeneity in detection probability.

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