Spatial Capture-Recapture

Classic approaches to estimating abundance and density of mammals begin with complete censuses, which are rarely feasible, or design-based estimates (Williams et al. 2001:242) from counts of animals within spatial plots (rectangles or irregular polygons of homogenous habitat) drawn by simple random or stratified designs, as widely practiced to count diurnal large mammals in open grasslands across Africa, Australia, northern Asia, and America. Enlarging the region counted into habitats with more obscuring cover or topographic irregularity or applying complete censuses to mammals smaller in size or more secretive in nature leads to biases in counts, which are successfully eliminated by collecting ancillary information on each detection, such as group size and vegetative cover to apply sightability models (Samuel et al. 1987) or distance from observer to apply distance sampling (Williams et al. 2001:263). Even these methods fail to provide accurate, efficient estimates for most mammals that are difficult to observe because of their smaller size, secretive nature, or nocturnal activity patterns and therefore require the most intensive approaches based on capturing individuals directly with traps or cameras or indirectly through DNA sampling. Well-developed statistical approaches to analyzing this mark–recapture information (Williams et al. 2001:289) have proved remarkably successful in estimating abundance but a question persists of the size of the area sampled for expressing abundance in terms of density. Approaches to solve this problem involving analysis based on nested grids of trap locations or trapping webs (Williams et al. 2001:316) have not been widely applied to date but a new model-based approach to utilize multiple detection locations for individual animals to estimate the number of home ranges within the trapping grid is intuitively appealing. Efford (2004) proposed spatially explicit capture–recapture models as “a model for the spatial distribution of home range centres and a model of distance dependent detection.” Efford and Fewster (2012) pointed out that some individuals must be detected at multiple locations and they summarized approaches to estimation for this problem as based on fitting simulations and inverse prediction (Efford 2004), likelihood maximization (Borchers and Efford 2008), or Markov chain Monte Carlo (MCMC—Royle and Young 2008). Spatial Capture–Recapture by Royle et al. provides a detailed introduction to the concepts and statistical approaches as well as a how-to guide, with extensive interesting examples, for developing spatial capture–recapture models and applying them to animal populations. The book is broken into 4 sections that take the reader through a logical procession from background and concepts to basic spatial capture–recapture (SCR) models to advanced SCR models to what they call super-advanced SCR models. Extensive use of the free data-processing and analysis programming language R (R Development Core Team 2014) is used throughout the book with references to its manuals and some good introductions to this rapidly growing collection of tools and procedures as well as its extensive community of scientists and programmers continually expanding its capabilities and access to state-of-the-art analysis and data-processing tools. Andy Royle has developed, maintains, and updates extensive scripts in the scrbook R package freely available from his Web site: https://sites.google.com/site/spatialcapturerecapture/scrbook-r-package. Part I Background and Concepts begins with an introductory chapter highlighting the great potential value but challenges for analyzing data gathered on lions, tigers, and bears by camera trapping, noninvasive DNA sampling, acoustic sampling, and search encounter methods such as detections at bait stations along trails for carnivores or from area searches often conducted for rare carnivores using specially trained scat-sniffing dogs. Problems with the historical method of buffering trap locations to estimate density is reviewed as a foundation for introducing Efford's (2004) approach to make closed mark–recapture models explicit by estimating a point process of home-range centers determining both density and probability of encounter with specific trap, bait, or other detection locations. The next 2 chapters introduce key statistical concepts, probability distributions (binomial, Poisson, etc.), and estimation methods (general linear models, maximum likelihood, general linear mixed models, and Bayesian analysis). If you missed or have forgotten your probability and advanced statistics the brief but clearly presented sections on statistical inference and parameter estimation are an essential foundation for the remainder of the book, but sections on joint, marginal, and conditional distributions, with R code to demonstrate, and hierarchical models may take some study to absorb. Bayesian methods are fundamental to analysis of many of the SCR models presented in the book and the authors thoroughly develop the key components of Bayes' rule, prior distributions, and posterior inference while making a strong argument for the value of results of Bayesian analysis producing biologically desirable statements such as, “The true probability of capturing individuals from this population is between 49.2% and 82.1% in this study.” Such statements are not allowed under the more classic frequentist perspective widely taught in statistics courses. The authors develop fully the critical issues concerning the impacts of subjective prior probabilities on final conclusions and ways to insure priors do not exert undue influence on those conclusions. The authors highlight the desirable characteristic of conclusions based on Bayesian methods that the conclusions are in fact valid for any sample size rather than only applying asymptotically for infinite sample sizes as required under frequentist inference. Readers should be aware that application of Bayesian methods is controversial in the ecological sciences (Dennis 1996). In the final chapter of this introductory section the authors relate SCR models to the classic capture–recapture models for closed populations assuming equal probability of capture across individuals and occasions (Mo of Otis et al. [1978]) versus the widely applicable model assuming different probabilities of capture between individuals (Mh), which can be cast as a simple analogy to SCR models if location of capture is treated as a covariate. Some of the concepts in this chapter become quite challenging to understand where 3 alternate modes of analysis are presented for closed population capture–recapture data using “joint” or “full likelihood,” which contains N as an explicit parameter, the conditional likelihood, which does not involve N, or by data augmentation, which replaces N with Ψ, a parameter expressing the proportion of augmented sites added in a data augmentation analysis. As they say, this appears like a “strange and mysterious black box.” Part II Basic SCR Models begins by developing Efford's (2004) idea that SCR data are drawn from an underlying pattern of individual home ranges scattered across a landscape interacting with a sample of trap or detector locations where probability of detection depends upon likelihood of the sample area including home-range centers, for example, a binomial model, interacting with a model of space use or, for example, a bivariate normal likelihood of occurrence at a distance from the home-range center. Specifically, if si is defined as the home-range or activity center for individual i, then the probability of detecting that individual, pij, at location j is a decreasing function of the distance between si and trap location j. In simple terms, this makes the classic model of heterogeneity in probability of capture between individuals, Mh, more specific as a model hypothesizing that probability of capture is a declining function of distance between trap or sample locations and home-range or activity centers. In a sense, they have combined mark–recapture and distance sampling methods into a single comprehensive framework termed SCR. The rest of the book is about how to most effectively estimate the key parameters underlying this model of the sampling process described as SCR, which is going to be challenging because we do not know where the home-range or activity centers are located, they are termed “latent” in the sense of present but not evident. As a means to assist readers in convincing themselves that it is feasible to estimate abundance and density within this framework the authors provide R code and scripts to simulate a landscape with a known number of animals occupying home ranges that can be sampled with a specified grid of traps to generate encounter histories for analysis with a Bayesian Markov chain Monte Carlo model using WINBUGS or other BUGS engines or by maximum likelihood (Efford 2011). Next the authors detail the core SCR assumptions (demographic closure, geographic closure, randomly distributed activity centers, detection as a function of distance, and independence of encounters among and within individuals), pointing out that even though they sound more restrictive than classic mark–recapture assumptions, in fact they relax some of the classic assumptions and provide a framework for questions of robustness to certain assumptions, extensions to more complicated models, and evaluation of their applicability. This chapter concludes with an interesting example from a wolverine camera-trapping study, which is carried through the next chapter illustrating likelihood analysis of SCR models along an illustrative procession beginning with maximum-likelihood estimation of SCR with known N, then with N unknown, next with integrated likelihood under data augmentation, and finally incorporating classic model selection using Akaike information criterion (AIC—Burnham and Anderson 2002) methods. The chapter concludes with a nice introduction to Efford's (2011) R package secr for likelihood analysis of SCR models used throughout the book, its capability to handle stratified populations, its extensive additional capabilities, and its value for more sophisticated and Bayesian analyses that are utilized throughout the remainder of the book. Modeling variation in encounter probability is feasible in SCR using either a Bayesian or classic maximum-likelihood approach by evaluating 3 types of animal-specific or sample-location–specific covariates: fixed covariates that are fully observed (e.g., dates of sampling occasions), partially observed individual covariates not known for all individuals (e.g., sex of animals), and unobserved covariates that we cannot observe (e.g., home-range size of individuals or unspecified individual effects). Bayesian approaches using R, WinBUGS, and JAGS are presented as well as likelihood approaches using secr, which offers access to 12 different probability of encounter models as well as automatic model selection using AIC (Burnham and Anderson 2002). Chapter 8 develops alternative approaches to model selection using the standard likelihood-based information theoretic AIC tool versus 4 less straightforward Bayesian approaches including deviance information criterion (DIC) supplemented by detailed discussions of evaluating goodness of hierarchical model fit for its 2 components: observation model evaluated with encounter frequencies of individuals and the state process assuming uniformity or spatial randomness. Chapter 9 offers alternative observation models determined by the type of detector, rather than the rarely applicable, standard independent binary detector including the 3 types: multinomial captures from “multi-catch” and “single-catch” trap systems, and Poisson encounter frequency data such as multiple visits to a camera trap by multiple animals. This 2nd major part is concluded with a chapter on sampling design that is one of the more frustrating chapters in the book, perhaps because of the almost diametrically opposed demands for SCR spatial samples: “good designs should generate a high sample size n (the number of individuals captured) and a large number of spatial recaptures.” To sample a large number of individuals we should space our detectors sparsely such that only 1 or 2 occur within each home range, whereas maximizing spatial recaptures requires denser spacing. On the positive side the authors suggest performing scenario analysis using Monte Carlo simulation of alternative designs to evaluate the trade-offs between available effort, funding, and logistics but their efforts to derive a model-based objective function related to average capture probability and its variance becomes challenging to follow. Although mentioned briefly, more attention to the value of a directed pilot study to evaluate a small set of alternative designs and gather key measures of capture probabilities, variances, and costs in time and dollars might be beneficial. Part III Advanced SCR Models extends earlier discussions substantially through chapters that relax initial assumptions: modeling spatial variation in density to relax an assumption of basic models of spatial homogeneity of activity center density or home ranges; modeling landscape connectivity to relax the assumption that encounter probability is simply a function of Euclidean distance between activity centers and detectors, for example, including habitat covariates in maximum-likelihood models of encounter probability; integrating resource selection with SCR models by building models of resource selection as covariates affecting probability of encounter with traps or sensors, but it is presented in a clever manner to allow simultaneous integrated analysis of SCR data with telemetry data from the same landscape to produce integrated resource selection parameters; stratified populations for multisession and multisite data offering the potential to combine data across sites, periods, or groups of animals (e.g., sexes and age classes) or management treatments to obtain more precise estimates of abundance or density; models for search–encounter data are developed to estimate density for rare, widely dispersed carnivores such as mountain lions treed and tagged by dog-hunters or fishers detected and identified with scat-sniffing dogs; and finally, the substantially more complex, open population models. Open population models are formulated either as spatial Jolly–Seber models that include trap or detector location information into the detection function or as a state–space formulation of Cormack–Jolly–Seber models emphasizing survival estimation that allows incorporation of individual effects to model heterogeneity of capture probability. Example applications of these open-population models come from birds and fish rather than mammals because they require substantial data that might become available in the future for some intensely sampled mammals. Part IV Super-Advanced SCR Models includes 4 chapters beginning with the quite technical aspects of building Markov chain Monte Carlo (MCMC) algorithms for use in the final chapters on unmarked populations, spatial mark–resight models, and emerging topics. The typical mammalogist will want to work with cooperating mathematicians, statisticians, or programmers to contemplate undertaking the process of building their own MCMC sampling algorithms but the chapter provides a fairly readable overview of the challenge and potential improved applications using a compiled language such as C++ and parallel processing, which is becoming more widely available through desktop or specialized multiprocessor microcomputers. Estimating density with unmarked populations consists of spatial count models producing imprecise estimates useful in developing the theory behind analysis of mixtures of marked and unmarked animals termed mark–resight models. Mark–resight models are widely applied to wildlife species (e.g., banded Canada geese visually resighted; and ear-tagged grizzly bears resighted with camera trapping) because these models combine the efficiency of only capturing animals once for marking with multiple nonhandling detection locations and times for resighting. The final chapter provides some hints at the future of SCR from modeling territoriality to explicit movement models followed by an appendix describing useful software utilized in the book. Overall Royle et al. have provided a comprehensive introduction to Spatial Capture–Recapture with this book. They provide excellent treatment of the conceptual basis for this complex, new approach offering great potential to incorporate spatial location data into abundance estimation such that we can obtain unbiased estimates of density in addition to simple numbers of individuals. Unfortunately biologists without extensive experience in use of R will find that the learning curve to apply these methods is quite steep, making classic tools such as program MARK (White and Burnham 1999) appear relatively simple. The book will be most useful to mammalogists with a quantitative bent and graduate students in advanced methods classes in population ecology.