Does your species have memory? Analyzing capture–recapture data with memory models

We examine memory models for multisite capture–recapture data. This is an important topic, as animals may exhibit behavior that is more complex than simple first‐order Markov movement between sites, when it is necessary to devise and fit appropriate models to data. We consider the Arnason–Schwarz model for multisite capture–recapture data, which incorporates just first‐order Markov movement, and also two alternative models that allow for memory, the Brownie model and the Pradel model. We use simulation to compare two alternative tests which may be undertaken to determine whether models for multisite capture–recapture data need to incorporate memory. Increasing the complexity of models runs the risk of introducing parameters that cannot be estimated, irrespective of how much data are collected, a feature which is known as parameter redundancy. Rouan et al. (JABES, 2009, pp 338–355) suggest a constraint that may be applied to overcome parameter redundancy when it is present in multisite memory models. For this case, we apply symbolic methods to derive a simpler constraint, which allows more parameters to be estimated, and give general results not limited to a particular configuration. We also consider the effect sparse data can have on parameter redundancy and recommend minimum sample sizes. Memory models for multisite capture–recapture data can be highly complex and difficult to fit to data. We emphasize the importance of a structured approach to modeling such data, by considering a priori which parameters can be estimated, which constraints are needed in order for estimation to take place, and how much data need to be collected. We also give guidance on the amount of data needed to use two alternative families of tests for whether models for multisite capture–recapture data need to incorporate memory.     

The Petersen–Lincoln Estimator and its Extension to Estimate the Size of a Shared Population

The Petersen–Lincoln estimator has been used to estimate the size of a population in a single mark release experiment. However, the estimator is not valid when the capture sample and recapture sample are not independent. We provide an intuitive interpretation for “independence” between samples based on 2 × 2 categorical data formed by capture/non‐capture in each of the two samples. From the interpretation, we review a general measure of “dependence” and quantify the correlation bias of the Petersen–Lincoln estimator when two types of dependences (local list dependence and heterogeneity of capture probability) exist. An important implication in the census undercount problem is that instead of using a post enumeration sample to assess the undercount of a census, one should conduct a prior enumeration sample to avoid correlation bias. We extend the Petersen–Lincoln method to the case of two populations. This new estimator of the size of the shared population is proposed and its variance is derived. We discuss a special case where the correlation bias of the proposed estimator due to dependence between samples vanishes. The proposed method is applied to a study of the relapse rate of illicit drug use in Taiwan. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evidence of reduced individual heterogeneity in adult survival of long‐lived species

The canalization hypothesis postulates that the rate at which trait variation generates variation in the average individual fitness in a population determines how buffered traits are against environmental and genetic factors. The ranking of a species on the slow‐fast continuum – the covariation among life‐history traits describing species‐specific life cycles along a gradient going from a long life, slow maturity, and low annual reproductive output, to a short life, fast maturity, and high annual reproductive output – strongly correlates with the relative fitness impact of a given amount of variation in adult survival. Under the canalization hypothesis, long‐lived species are thus expected to display less individual heterogeneity in survival at the onset of adulthood, when reproductive values peak, than short‐lived species. We tested this life‐history prediction by analysing long‐term time series of individual‐based data in nine species of birds and mammals using capture‐recapture models. We found that individual heterogeneity in survival was higher in species with short‐generation time (< 3 years) than in species with long generation time (> 4 years). Our findings provide the first piece of empirical evidence for the canalization hypothesis at the individual level from the wild.     

Estimating Completeness in Cancer Registries – Comparing Capture‐Recapture Methods in a Simulation Study

Completeness of registration is one of the quality indicators usually reported by cancer registries. This allows researchers to assess how useful and representative the data is. Several methods have been suggested to estimate completeness. In this paper a multi‐state model for the process of cancer diagnosis and treatment is presented. In principle, every contact with a doctor during diagnosis, treatment, and aftercare can give rise to a cancer registry notification with a certain probability. Therefore the states included in the model are “incident tumour” and “death” but also contacts with doctors such as consultation of a general practitioner or specialised doctor, diagnostic procedures, therapeutic interventions, and aftercare. In this model transitions between states and possible notifications to a cancer registry after entering a state are simulated. Transition intensities are derived and used in simulation. Several capture‐recapture methods have been applied to the simulated data. Simulated “true” numbers of new cases and simulated numbers of registrations are both available. This allows to assess the validity of the completeness estimates and to compare the relative merits of the methods. In the scenarios investigated here, all capture‐recapture estimators tended to underestimate completeness. While a modified DCN method and one type of log‐linear model yielded quite reasonable estimates other methods exhibited large variability or grossly underestimated completeness. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cormack–Jolly–Seber model with environmental covariates: A P‐spline approach

In capture–recapture models, survival and capture probabilities can be modelled as functions of time‐varying covariates, such as temperature or rainfall. The Cormack–Jolly–Seber (CJS) model allows for flexible modelling of these covariates; however, the functional relationship may not be linear. We extend the CJS model by semi‐parametrically modelling capture and survival probabilities using a frequentist approach via P‐splines techniques. We investigate the performance of the estimators by conducting simulation studies. We also apply and compare these models with known semi‐parametric Bayesian approaches on simulated and real data sets.     

Estimating Alpine ibex Capra ibex abundance from photographic sampling

Monitoring procedures for Alpine ibex Capra ibex are limited in habitats with reduced visibility and when physical capture and marking of the animals is not intended. Photographic sampling, involving using camera‐trap data and identifying ibex from natural markings, was adopted with capture‐recapture models to estimate the abundance of ibex in Austria. The software CAPTURE's model produced an average capture probability of 0.44 with an estimate of 34–51 ibex and a mean population size of 38 ibex. This first study showed the applicability of photographic capture‐recapture techniques to estimate the abundance of ibex based on their natural markings.     

Open Capture–Recapture Models with Heterogeneity: II. Jolly–Seber Model

Summary Estimation of abundance is important in both open and closed population capture–recapture analysis, but unmodeled heterogeneity of capture probability leads to negative bias in abundance estimates. This article defines and develops a suite of open population capture–recapture models using finite mixtures to model heterogeneity of capture and survival probabilities. Model comparisons and parameter estimation use likelihood‐based methods. A real example is analyzed, and simulations are used to check the main features of the heterogeneous models, especially the quality of estimation of abundance, survival, recruitment, and turnover. The two major advances in this article are the provision of realistic abundance estimates that take account of heterogenetiy of capture, and an appraisal of the amount of overestimation of survival arising from conditioning on the first capture when heterogeneity of survival is present.     

Augmenting superpopulation capture-recapture models with population assignment data

Summary Ecologists applying capture–recapture models to animal populations sometimes have access to additional information about individuals' populations of origin (e.g., information about genetics, stable isotopes, etc.). Tests that assign an individual's genotype to its most likely source population are increasingly used. Here we show how to augment a superpopulation capture–recapture model with such information. We consider a single superpopulation model without age structure, and split each entry probability into separate components due to births in situ and immigration. We show that it is possible to estimate these two probabilities separately. We first consider the case of perfect information about population of origin, where we can distinguish individuals born in situ from immigrants with certainty. Then we consider the more realistic case of imperfect information, where we use genetic or other information to assign probabilities to each individual's origin as in situ or outside the population. We use a resampling approach to impute the true population of origin from imperfect assignment information. The integration of data on population of origin with capture–recapture data allows us to determine the contributions of immigration and in situ reproduction to the growth of the population, an issue of importance to ecologists. We illustrate our new models with capture–recapture and genetic assignment data from a population of banner‐tailed kangaroo rats Dipodomys spectabilis in Arizona.     

Model selection for integrated recovery/recapture data

Catchpole et al. (1998, Biometrics 54, 33-46) provide a novel scheme for integrating both recovery and recapture data analyses and derive sufficient statistics that facilitate likelihood computations. In this article, we demonstrate how their efficient likelihood expression can facilitate Bayesian analyses of these kinds of data and extend their methodology to provide a formal framework for model determination. We consider in detail the issue of model selection with respect to a set of recapture/recovery histories of shags (Phalacrocorax aristotelis) and determine, from the enormous range of biologically plausible models available, which best describe the data. By using reversible jump Markov chain Monte Carlo methodology, we demonstrate how this enormous model space can be efficiently and effectively explored without having to resort to performing an infeasibly large number of pairwise comparisons or some ad hoc stepwise procedure. We find that the model used by Catchpole et al. (1998) has essentially zero posterior probability and that, of the 477,144 possible models considered, over 60% of the posterior mass is placed on three neighboring models with biologically interesting interpretations.     

Combining capture–recapture data and known ages allows estimation of age‐dependent survival rates

In many animal populations, demographic parameters such as survival and recruitment vary markedly with age, as do parameters related to sampling, such as capture probability. Failing to account for such variation can result in biased estimates of population‐level rates. However, estimating age‐dependent survival rates can be challenging because ages of individuals are rarely known unless tagging is done at birth. For many species, it is possible to infer age based on size. In capture–recapture studies of such species, it is possible to use a growth model to infer the age at first capture of individuals. We show how to build estimates of age‐dependent survival into a capture–mark–recapture model based on data obtained in a capture–recapture study. We first show how estimates of age based on length increments closely match those based on definitive aging methods. In simulated analyses, we show that both individual ages and age‐dependent survival rates estimated from simulated data closely match true values. With our approach, we are able to estimate the age‐specific apparent survival rates of Murray and trout cod in the Murray River, Australia. Our model structure provides a flexible framework within which to investigate various aspects of how survival varies with age and will have extensions within a wide range of ecological studies of animals where age can be estimated based on size.     

A Comparison of Population Size Estimators under the Truncated Count Model With and Without Allowance for Contaminations

The purpose of the study is to estimate the population size under a homogeneous truncated count model and under model contaminations via the Horvitz‐Thompson approach on the basis of a count capture‐recapture experiment. The proposed estimator is based on a mixture of zero‐truncated Poisson distributions. The benefit of using the proposed model is statistical inference of the long‐tailed or skewed distributions and the concavity of the likelihood function with strong results available on the nonparametric maximum likelihood estimator (NPMLE). The results of comparisons, for finding the appropriate estimator among McKendrick's, Mantel‐Haenszel's, Zelterman's, Chao's, the maximum likelihood, and the proposed methods in a simulation study, reveal that under model contaminations the proposed estimator provides the best choice according to its smallest bias and smallest mean square error for a situation of sufficiently large population sizes and the further results show that the proposed estimator performs well even for a homogeneous situation. The empirical examples, containing the cholera epidemic in India based on homogeneity and the heroin user data in Bangkok 2002 based on heterogeneity, are fitted with an excellent goodness‐of‐fit of the models and the confidence interval estimations may also be of considerable interest. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inferences about population dynamics from count data using multistate models: a comparison to capture–recapture approaches

Wildlife populations consist of individuals that contribute disproportionately to growth and viability. Understanding a population's spatial and temporal dynamics requires estimates of abundance and demographic rates that account for this heterogeneity. Estimating these quantities can be difficult, requiring years of intensive data collection. Often, this is accomplished through the capture and recapture of individual animals, which is generally only feasible at a limited number of locations. In contrast, N‐mixture models allow for the estimation of abundance, and spatial variation in abundance, from count data alone. We extend recently developed multistate, open population N‐mixture models, which can additionally estimate demographic rates based on an organism's life history characteristics. In our extension, we develop an approach to account for the case where not all individuals can be assigned to a state during sampling. Using only state‐specific count data, we show how our model can be used to estimate local population abundance, as well as density‐dependent recruitment rates and state‐specific survival. We apply our model to a population of black‐throated blue warblers (Setophaga caerulescens) that have been surveyed for 25 years on their breeding grounds at the Hubbard Brook Experimental Forest in New Hampshire, USA. The intensive data collection efforts allow us to compare our estimates to estimates derived from capture–recapture data. Our model performed well in estimating population abundance and density‐dependent rates of annual recruitment/immigration. Estimates of local carrying capacity and per capita recruitment of yearlings were consistent with those published in other studies. However, our model moderately underestimated annual survival probability of yearling and adult females and severely underestimates survival probabilities for both of these male stages. The most accurate and precise estimates will necessarily require some amount of intensive data collection efforts (such as capture–recapture). Integrated population models that combine data from both intensive and extensive sources are likely to be the most efficient approach for estimating demographic rates at large spatial and temporal scales.     

Using Mixtures to Model Heterogeneity in Ecological Capture‐Recapture Studies

Modelling heterogeneity of capture is an important problem in estimating animal abundance from capturerecapture data, with underestimation of abundance occurring if different animals have intrinsically high or low capture probabilities. Mixture models are useful in many cases to model the heterogeneity. We summarise mixture model results for closed populations, using a skink data set for illustration. New mixture models for heterogeneous open populations are discussed, and a closed population model is shown to have new and potentially effective applications in community analysis. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of capture probabilities using generalized estimating equations and mixed effects approaches

Modeling individual heterogeneity in capture probabilities has been one of the most challenging tasks in capture–recapture studies. Heterogeneity in capture probabilities can be modeled as a function of individual covariates, but correlation structure among capture occasions should be taking into account. A proposed generalized estimating equations (GEE) and generalized linear mixed modeling (GLMM) approaches can be used to estimate capture probabilities and population size for capture–recapture closed population models. An example is used for an illustrative application and for comparison with currently used methodology. A simulation study is also conducted to show the performance of the estimation procedures. Our simulation results show that the proposed quasi‐likelihood based on GEE approach provides lower SE than partial likelihood based on either generalized linear models (GLM) or GLMM approaches for estimating population size in a closed capture–recapture experiment. Estimator performance is good if a large proportion of individuals are captured. For cases where only a small proportion of individuals are captured, the estimates become unstable, but the GEE approach outperforms the other methods.     

Recommendations for photo‐identification methods used in capture‐recapture models with cetaceans

Capture‐recapture methods are frequently employed to estimate abundance of cetaceans using photographic techniques and a variety of statistical models. However, there are many unresolved issues regarding the selection and manipulation of images that can potentially impose bias on resulting estimates. To examine the potential impact of these issues we circulated a test data set of dorsal fin images from bottlenose dolphins to several independent research groups. Photo‐identification methods were generally similar, but the selection, scoring, and matching of images varied greatly amongst groups. Based on these results we make the following recommendations. Researchers should: (1) determine the degree of marking, or level of distinctiveness, and use images of sufficient quality to recognize animals of that level of distinctiveness; (2) ensure that markings are sufficiently distinct to eliminate the potential for “twins” to occur; (3) stratify data sets by distinctiveness and generate a series of abundance estimates to investigate the influence of including animals of varying degrees of markings; and (4) strive to examine and incorporate variability among analysts into capture‐recapture estimation. In this paper we summarize these potential sources of bias and provide recommendations for best practices for using natural markings in a capture‐recapture framework.     

Demographic estimation methods for plants with unobservable life-states

Demographic estimation of vital parameters in plants with an unobservable dormant state is complicated, because time of death is not known. Conventional methods assume that death occurs at a particular time after a plant has last been seen aboveground but the consequences of assuming a particular duration of dormancy have never been tested. Capture–recapture methods do not make assumptions about time of death; however, problems with parameter estimability have not yet been resolved. To date, a critical comparative assessment of these methods is lacking. We analysed data from a 10 year study of Cleistes bifaria, a terrestrial orchid with frequent dormancy, and compared demographic estimates obtained by five varieties of the conventional methods, and two capture–recapture methods. All conventional methods produced spurious unity survival estimates for some years or for some states, and estimates of demographic rates sensitive to the time of death assumption. In contrast, capture–recapture methods are more parsimonious in terms of assumptions, are based on well founded theory and did not produce spurious estimates. In Cleistes, dormant episodes lasted for 1–4 years (mean 1.4, SD 0.74). The capture–recapture models estimated ramet survival rate at 0.86 (SE～0.01), ranging from 0.77–0.94 (SEs≤0.1) in any one year. The average fraction dormant was estimated at 30% (SE 1.5), ranging 16–47% (SEs≤5.1) in any one year. Multistate capture–recapture models showed that survival rates were positively related to precipitation in the current year, but transition rates were more strongly related to precipitation in the previous than in the current year, with more ramets going dormant following dry years. Not all capture–recapture models of interest have estimable parameters; for instance, without excavating plants in years when they do not appear aboveground, it is not possible to obtain independent time‐specific survival estimates for dormant plants. We introduce rigorous computer algebra methods to identify the parameters that are estimable in principle. As life‐states are a prominent feature in plant life cycles, multistate capture–recapture models are a natural framework for analysing population dynamics of plants with dormancy.     

Population Dynamics of the Andean Lizard Anolis heterodermus: Fast‐slow Demographic Strategies in Fragmented Scrubland Landscapes

Habitat fragmentation and loss affect population stability and demographic processes, increasing the extinction risk of species. We studied Anolis heterodermus populations inhabiting large and small Andean scrubland patches in three fragmented landscapes in the Sabana de Bogotá (Colombia) to determine the effect of habitat fragmentation and loss on population dynamics. We used the capture‐mark‐recapture method and multistate models to estimate vital rates for each population. We estimated growth population rate and the most important processes that affect λ by elasticity analysis of vital rates. We tested the effects of habitat fragmentation and loss on vital rates of lizard populations. All six isolated populations showed a positive or an equilibrium growth rate (λ = 1), and the most important demographic process affecting λ was the growth to first reproduction. Populations from landscapes with less scrubland natural cover showed higher stasis of young adults. Populations in highly fragmented landscapes showed highest juvenile survival and growth population rates. Independent of the landscape's habitat configuration and connectivity, populations from larger scrubland patches showed low adult survivorship, but high transition rates. Populations varied from a slow strategy with low growth and delayed maturation in smaller patches to a fast strategy with high growth and early maturation in large patches. This variation was congruent with the fast‐slow continuum hypothesis and has serious implications for Andean lizard conservation and management strategies. We suggest that more stable lizard populations will be maintained if different management strategies are adopted according to patch area and habitat structure.     

Population size estimates based on the frequency of genetically assigned parent–offspring pairs within a subsample

Estimating population density as precise as possible is a key premise for managing wild animal species. This can be a challenging task if the species in question is elusive or, due to high quantities, hard to count. We present a new, mathematically derived estimator for population size, where the estimation is based solely on the frequency of genetically assigned parent–offspring pairs within a subsample of an ungulate population. By use of molecular markers like microsatellites, the number of these parent–offspring pairs can be determined. The study's aim was to clarify whether a classical capture–mark–recapture (CMR) method can be adapted or extended by this genetic element to a genetic‐based capture–mark–recapture (g‐CMR). We numerically validate the presented estimator (and corresponding variance estimates) and provide the R‐code for the computation of estimates of population size including confidence intervals. The presented method provides a new framework to precisely estimate population size based on the genetic analysis of a one‐time subsample. This is especially of value where traditional CMR methods or other DNA‐based (fecal or hair) capture–recapture methods fail or are too difficult to apply. The DNA source used is basically irrelevant, but in the present case the sampling of an annual hunting bag is to serve as data basis. In addition to the high quality of muscle tissue samples, hunting bags provide additional and essential information for wildlife management practices, such as age, weight, or sex. In cases where a g‐CMR method is ecologically and hunting‐wise appropriate, it enables a wide applicability, also through its species‐independent use.     

Estimating repertoire size in a songbird: a comparison of three techniques

Many animals produce multiple types of breeding vocalizations that, together, constitute a vocal repertoire. In some species, the size of an individual’s repertoire is important because it correlates with brain size, territory size or social behaviour. Quantifying repertoire size is challenging because the long recordings needed to sample a repertoire comprehensively are difficult to obtain and analyse. The most basic quantification technique is simple enumeration, where one counts unique vocalization types until no new types are detected. Alternative techniques estimate repertoire size from subsamples, but these techniques are useful only if they are accurate. Using 12 years of acoustic data from a population of rufous-and-white wrens in Costa Rica, we used simple enumeration to measure the repertoire size for 40 males. We then compared these to the estimates generated by three estimation techniques: curve fitting, capture–recapture and a new technique based on the coupon collector’s problem. To understand how sampling effort affects the accuracy and precision of estimates, we applied each technique to six different-sized subsets of data per male. When averaged across subset sizes, the capture–recapture and coupon collector techniques showed the highest accuracy, whereas the curve fitting technique underestimated repertoire size. Precision (the average absolute difference between the estimated and true repertoire size) was significantly better for the capture–recapture technique than the coupon collector and curve fitting techniques. Both accuracy and precision improved as subset size increased. We conclude that capture–recapture is the best technique for estimating the sizes of small repertoires.     

On the design of closed recapture experiments

We propose a method to plan the number of occasions of recapture experiments for population size estimation. We do so by fixing the smallest number of capture occasions so that the expected length of the profile confidence interval is less than or equal to a fixed threshold. In some cases, we solve the optimization problem in closed form. For more complex models we use numerical optimization. We detail models assuming homogeneous, time‐varying, subject‐specific capture probabilities, behavioral response to capture, and combining behavioral response with subject‐specific effects. The principle we propose can be extended to plan any other model specification. We formally show the validity of the approach by proving distributional convergence. We illustrate with simulations and challenging examples in epidemiology and ecology. We report that in many cases adding as few as two sampling occasions may substantially reduce the length of confidence intervals.