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.     

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.     

Hierarchical Modeling for Estimating Relative Risks of Rare Genetic Variants: Properties of the Pseudo‐Likelihood Method

Summary Many major genes have been identified that strongly influence the risk of cancer. However, there are typically many different mutations that can occur in the gene, each of which may or may not confer increased risk. It is critical to identify which specific mutations are harmful, and which ones are harmless, so that individuals who learn from genetic testing that they have a mutation can be appropriately counseled. This is a challenging task, since new mutations are continually being identified, and there is typically relatively little evidence available about each individual mutation. In an earlier article, we employed hierarchical modeling ( Capanu et al., 2008 , Statistics in Medicine 27 , 1973–1992) using the pseudo‐likelihood and Gibbs sampling methods to estimate the relative risks of individual rare variants using data from a case–control study and showed that one can draw strength from the aggregating power of hierarchical models to distinguish the variants that contribute to cancer risk. However, further research is needed to validate the application of asymptotic methods to such sparse data. In this article, we use simulations to study in detail the properties of the pseudo‐likelihood method for this purpose. We also explore two alternative approaches: pseudo‐likelihood with correction for the variance component estimate as proposed by Lin and Breslow (1996, Journal of the American Statistical Association 91 , 1007–1016) and a hybrid pseudo‐likelihood approach with Bayesian estimation of the variance component. We investigate the validity of these hierarchical modeling techniques by looking at the bias and coverage properties of the estimators as well as at the efficiency of the hierarchical modeling estimates relative to that of the maximum likelihood estimates. The results indicate that the estimates of the relative risks of very sparse variants have small bias, and that the estimated 95% confidence intervals are typically anti‐conservative, though the actual coverage rates are generally above 90%. The widths of the confidence intervals narrow as the residual variance in the second‐stage model is reduced. The results also show that the hierarchical modeling estimates have shorter confidence intervals relative to estimates obtained from conventional logistic regression, and that these relative improvements increase as the variants become more rare.     

Assessment of mark–recapture models to estimate the abundance of a humpback whale feeding aggregation in Southeast Alaska

Aim The aim of this study was to use photographs of the unique pattern on the ventral surface of the flukes to estimate the abundance of humpback whales (Megaptera novaeangliae) in a discrete feeding aggregation in northern Southeast Alaska. Location The study was located in northern Southeast Alaska, USA, in the eastern North Pacific Ocean. Methods This study evaluated mark–recapture models, ranging from the simpler models (pooled and stratified, closed Petersen estimators) to more complex multi‐strata models (closed Darroch and open Hilborn). The Akaike Information Criterion, corrected (AICc) was used as a model comparison statistic. Results Our best estimate of whale abundance in northern Southeast Alaska in 2000 is 961 whales [95% confidence interval (657, 1076)]. This estimate comes from the Hilborn open, multi‐strata approach with constant migration over time, time‐dependent capture probabilities by area, and a fixed survival rate of 0.98. The simpler models were problematic owing to several aspects of whale behaviour, including that (1) the whales did not mix randomly throughout the study area, (2) some whales emigrated temporarily outside the study area and were not available for capture, and (3) whales were not equally identifiable because they did not behave in the same way when they showed their flukes upon diving. This led to heterogeneity in capture probability and a bias in the estimates. The more complex models stratified by area, and using migration movements among areas, compensated for some of these issues when estimating population size. Main conclusions We believe that the Hilborn open, multi‐strata model produced the best estimate because: (1) it incorporated the best information about survival, (2) it used detailed information about the various release groups, (3) the analysis provided an integrated environment in which parameters such as migration and capture probabilities are shared, (4) the three strata encompassed a large portion of the areas used by whales, and (5) the Hilborn model selected was superior in terms of model selection criteria and biological realism. These data provide valuable insights into the numbers and movements of humpback whales in three areas of Southeast Alaska.     

Prediction errors for state occupation and transition probabilities in multi‐state models

In this paper, we consider the estimation of prediction errors for state occupation probabilities and transition probabilities for multistate time‐to‐event data. We study prediction errors based on the Brier score and on the Kullback–Leibler score and prove their properness. In the presence of right‐censored data, two classes of estimators, based on inverse probability weighting and pseudo‐values, respectively, are proposed, and consistency properties of the proposed estimators are investigated. The second part of the paper is devoted to the estimation of dynamic prediction errors for state occupation probabilities for multistate models, conditional on being alive, and for transition probabilities. Cross‐validated versions are proposed. Our methods are illustrated on the CSL1 randomized clinical trial comparing prednisone versus placebo for liver cirrhosis patients.     

Robustness of close‐kin mark–recapture estimators to dispersal limitation and spatially varying sampling probabilities

1. Close‐kin mark–recapture (CKMR) is a method for estimating abundance and vital rates from kinship relationships observed in genetic samples. CKMR inference only requires animals to be sampled once (e.g., lethally), potentially widening the scope of population‐level inference relative to traditional monitoring programs.
2. One assumption of CKMR is that, conditional on individual covariates like age, all animals have an equal probability of being sampled. However, if genetic data are collected opportunistically (e.g., via hunters or fishers), there is potential for spatial variation in sampling probability that can bias CKMR estimators, particularly when genetically related individuals stay in close proximity.
3. We used individual‐based simulation to investigate consequences of dispersal limitation and spatially biased sampling on performance of naive (nonspatial) CKMR estimators of abundance, fecundity, and adult survival. Population dynamics approximated that of a long‐lived mammal species subject to lethal sampling.
4. Naive CKMR abundance estimators were relatively unbiased when dispersal was unconstrained (i.e., complete mixing) or when sampling was random or subject to moderate levels of spatial variation. When dispersal was limited, extreme variation in spatial sampling probabilities negatively biased abundance estimates. Reproductive schedules and survival were well estimated, except for survival when adults could emigrate out of the sampled area. Incomplete mixing was readily detected using Kolmogorov–Smirnov tests.
5. Although CKMR appears promising for estimating abundance and vital rates with opportunistically collected genetic data, care is needed when dispersal limitation is coupled with spatially biased sampling. Fortunately, incomplete mixing is easily detected with adequate sample sizes. In principle, it is possible to devise and fit spatially explicit CKMR models to avoid bias under dispersal limitation, but development of such models necessitates additional complexity (and possibly additional data). We suggest using simulation studies to examine potential bias and precision of proposed modeling approaches prior to implementing a CKMR program.

     

Detection heterogeneity in underwater visual‐census data

This study shows how capture–mark–recapture (CMR) models can provide robust estimates of detection heterogeneity (sources of bias) in underwater visual‐census data. Detection biases among observers and fish family groups were consistent between fished and unfished reef sites in Kenya, even when the overall level of detection declined between locations. Species characteristics were the greatest source of detection heterogeneity and large, highly mobile species were found to have lower probabilities of detection than smaller, site‐attached species. Fish family and functional‐group detectability were also found to be lower at fished locations, probably due to differences in local abundance. Because robust CMR models deal explicitly with sampling where not all species are detected, their use is encouraged for studies addressing reef‐fish community dynamics.     

Who are we sampling? Apparent survival differs between methods in a secretive species

Survival is a fundamental parameter in population dynamics with increasing importance in the management and conservation strategies of wildlife populations. Survival probability in vertebrates is usually estimated by live‐encounter data obtained by means of physical mark–capture–recapture protocols. Non‐invasive acoustic marking relying on individual‐specific features of signals has been alternatively applied as a marking technique, especially in secretive species. Nevertheless, to date no research has compared survival rate estimates obtained by acoustic and physical marking. We estimated half‐yearly and annual survival and recapture rates of a secretive and threatened passerine, the Dupont's lark Chersophilus duponti, using two separate live‐encounter data sets of males collected simultaneously by physical and acoustic marking in the same study area. The separate analysis of both methods led to different model structures, since transient individuals had to be accounted for in the acoustic marking but not in the physical marking data set. Furthermore, while reencounter probabilities did not differ between methods, survival estimates employing physical marking were lower than those obtained acoustically, especially between the postbreeding and the breeding period when the apparent survival of colour‐banded birds was twice as low as for acoustic marking. The combination of marking methods suggested the existence of different subsets of individuals differentially sampled within the population: whereas colour‐banded males seemed to represent the territorial fraction of the population, both resident and floater individuals were probably detected by acoustic marking. Using traditional mark–recapture methods exclusively could have misled our estimates of survival rates, potentially affecting prospective predictions of population dynamics. Acoustic marking has been poorly applied in mark–recapture studies, but might be a powerful complement to obtain accurate estimates of fundamental demographic parameters such as survival and dispersal.     

Estimation in the progressive illness‐death model: A nonexhaustive review

Multistate models can be successfully used for describing complex event history data, for example, describing stages in the disease progression of a patient. The so‐called “illness‐death” model plays a central role in the theory and practice of these models. Many time‐to‐event datasets from medical studies with multiple end points can be reduced to this generic structure. In these models one important goal is the modeling of transition rates but biomedical researchers are also interested in reporting interpretable results in a simple and summarized manner. These include estimates of predictive probabilities, such as the transition probabilities, occupation probabilities, cumulative incidence functions, and the sojourn time distributions. We will give a review of some of the available methods for estimating such quantities in the progressive illness‐death model conditionally (or not) on covariate measures. For some of these quantities estimators based on subsampling are employed. Subsampling, also referred to as landmarking, leads to small sample sizes and usually to heavily censored data leading to estimators with higher variability. To overcome this issue estimators based on a preliminary estimation (presmoothing) of the probability of censoring may be used. Among these, the presmoothed estimators for the cumulative incidences are new. We also introduce feasible estimation methods for the cumulative incidence function conditionally on covariate measures. The proposed methods are illustrated using real data. A comparative simulation study of several estimation approaches is performed and existing software in the form of R packages is discussed.     

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.     

Use of Occupancy Models to Estimate the Influence of Previous Live Captures on DNA-Based Detection Probabilities of Grizzly Bears

Abstract: Large carnivores potentially change their behavior following physical capture, becoming less responsive to the attractants that resulted in their capture, which can bias population estimates where the change in behavior is not appropriately modeled. We applied occupancy models to efficiently estimate and compare detection probabilities of previously collared grizzly bears (Ursus arctos) with bears captured at DNA hair-snag sites that were not previously collared. We found that previously captured bears had lower detection probabilities, although their detection probabilities were still >0, implying that they were still visible to be sampled via the DNA hair-snag grid, which was able to detect finer differences in capture probabilities of previously collared bears compared with Huggins closed-captures population models. To obtain relatively unbiased population estimates for DNA surveys, heterogeneity caused by previous live capture should be accounted for in the population estimator. (JOURNAL OF WILDLIFE MANAGEMENT 72(3):589–595; 2008)     

Capture–Recapture Estimation Using Finite Mixtures of Arbitrary Dimension

Summary Reversible jump Markov chain Monte Carlo (RJMCMC) methods are used to fit Bayesian capture–recapture models incorporating heterogeneity in individuals and samples. Heterogeneity in capture probabilities comes from finite mixtures and/or fixed sample effects allowing for interactions. Estimation by RJMCMC allows automatic model selection and/or model averaging. Priors on the parameters stabilize the estimates and produce realistic credible intervals for population size for overparameterized models, in contrast to likelihood‐based methods. To demonstrate the approach we analyze the standard Snowshoe hare and Cottontail rabbit data sets from ecology, a reliability testing data set.     

Estimating migratory connectivity of birds when re‐encounter probabilities are heterogeneous

Understanding the biology and conducting effective conservation of migratory species requires an understanding of migratory connectivity – the geographic linkages of populations between stages of the annual cycle. Unfortunately, for most species, we are lacking such information. The North American Bird Banding Laboratory (BBL) houses an extensive database of marking, recaptures and recoveries, and such data could provide migratory connectivity information for many species. To date, however, few species have been analyzed for migratory connectivity largely because heterogeneous re‐encounter probabilities make interpretation problematic. We accounted for regional variation in re‐encounter probabilities by borrowing information across species and by using effort covariates on recapture and recovery probabilities in a multistate capture–recapture and recovery model. The effort covariates were derived from recaptures and recoveries of species within the same regions. We estimated the migratory connectivity for three tern species breeding in North America and over‐wintering in the tropics, common (Sterna hirundo), roseate (Sterna dougallii), and Caspian terns (Hydroprogne caspia). For western breeding terns, model‐derived estimates of migratory connectivity differed considerably from those derived directly from the proportions of re‐encounters. Conversely, for eastern breeding terns, estimates were merely refined by the inclusion of re‐encounter probabilities. In general, eastern breeding terns were strongly connected to eastern South America, and western breeding terns were strongly linked to the more western parts of the nonbreeding range under both models. Through simulation, we found this approach is likely useful for many species in the BBL database, although precision improved with higher re‐encounter probabilities and stronger migratory connectivity. We describe an approach to deal with the inherent biases in BBL banding and re‐encounter data to demonstrate that this large dataset is a valuable source of information about the migratory connectivity of the birds of North America.     

Homogeneity/Heterogeneity Hypotheses for Standardized Mortality Ratios Based on Minimum Power‐divergence Estimators

This paper analyzes the power divergence estimators when homogeneity/heterogeneity hypotheses among standardized mortality ratios (SMRs) are taken into account. A Monte Carlo study shows that when the standard mortality rate is not external, that is it is estimated from the sample data, these estimators have a good performance even for small sample sets and in particular the minimum chi‐square estimators have a better behavior compared to the classical maximum likelihood estimators. In order to make decisions under homogeneity/heterogeneity hypotheses of SMRs we propose some test‐statistics which consider the minimum power divergence estimators. Through a numerical example focused on SMRs of melanoma mortality ratios in different regions of the US, a homogeneity/heterogeneity study is illustrated.     

Latent multinomial models for extended batch-mark data

Batch marking is common and useful for many capture–recapture studies where individual marks cannot be applied due to various constraints such as timing, cost, or marking difficulty. When batch marks are used, observed data are not individual capture histories but a set of counts including the numbers of individuals first marked, marked individuals that are recaptured, and individuals captured but released without being marked (applicable to some studies) on each capture occasion. Fitting traditional capture–recapture models to such data requires one to identify all possible sets of capture–recapture histories that may lead to the observed data, which is computationally infeasible even for a small number of capture occasions. In this paper, we propose a latent multinomial model to deal with such data, where the observed vector of counts is a non-invertible linear transformation of a latent vector that follows a multinomial distribution depending on model parameters. The latent multinomial model can be fitted efficiently through a saddlepoint approximation based maximum likelihood approach. The model framework is very flexible and can be applied to data collected with different study designs. Simulation studies indicate that reliable estimation results are obtained for all parameters of the proposed model. We apply the model to analysis of golden mantella data collected using batch marks in Central Madagascar.     

Estimating the abundance of rare and elusive carnivores from photographic-sampling data when the population size is very small

Conservation and management agencies require accurate and precise estimates of abundance when considering the status of a species and the need for directed actions. Due to the proliferation of remote sampling cameras, there has been an increase in capture–recapture studies that estimate the abundance of rare and/or elusive species using closed capture–recapture estimators (C–R). However, data from these studies often do not meet necessary statistical assumptions. Common attributes of these data are (1) infrequent detections, (2) a small number of individuals detected, (3) long survey durations, and (4) variability in detection among individuals. We believe there is a need for guidance when analyzing this type of sparse data. We highlight statistical limitations of closed C–R estimators when data are sparse and suggest an alternative approach over the conventional use of the Jackknife estimator. Our approach aims to maximize the probability individuals are detected at least once over the entire sampling period, thus making the modeling of variability in the detection process irrelevant, estimating abundance accurately and precisely. We use simulations to demonstrate when using the unconditional-likelihood M ₀ (constant detection probability) closed C–R estimator with profile-likelihood confidence intervals provides reliable results even when detection varies by individual. If each individual in the population is detected on average of at least 2.5 times, abundance estimates are accurate and precise. When studies sample the same species at multiple areas or at the same area over time, we suggest sharing detection information across datasets to increase precision when estimating abundance. The approach suggested here should be useful for monitoring small populations of species that are difficult to detect.     

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.     

Mark‐resight abundance and survival estimation of Indo‐Pacific bottlenose dolphins,Tursiops aduncus,in the Swatch‐of‐No‐Ground,Bangladesh

A mark‐resight analysis under Pollock's robust design was applied to Indo‐Pacific bottlenose dolphins Tursiops aduncus in the Swatch‐of‐No‐Ground (SoNG) submarine canyon, Bangladesh, during the winter seasons of 2005–2009. Information from sightings of photo‐identified individuals (1,144) and unmarked individuals generated abundance estimates of 1,701 (95% confidence interval [CI]= 1,533–1,888), 1,927 (95% CI = 1,851–2,006), 2,150 (95% CI = 1,906–2,425), and 2,239 (95% CI = 1,985–2,524) individuals for seasons 1–4, respectively. This makes the population among the largest assessed of the species. Overall apparent survival was estimated as 0.958 (95% CI = 0.802–0.992). Interseasonal probabilities of transitioning to an unobservable state were estimated as 0.045, 0.363, and 0.300 for years 1–2, 2–3, and 3–4, respectively, and the overall probability of remaining in an unobservable state was 0.688. These probabilities, together with an apparent increase in abundance during the study period, indicate that the identified dolphins are part of a larger superpopulation moving throughout a more extensive geographic area. Of the photo‐identified dolphins, 28.2% exhibited injuries related to entanglements with fishing gear. This implies a strong potential for fatal interactions that could jeopardize the conservation status of the population, which otherwise appears favorable.     

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.     

Complete tag loss in capture–recapture studies affects abundance estimates: An elephant seal case study

1. In capture–recapture studies, recycled individuals occur when individuals lose all of their tags and are recaptured as though they were new individuals. Typically, the effect of these recycled individuals is assumed negligible.
2. Through a simulation‐based study of double‐tagging experiments, we examined the effect of recycled individuals on parameter estimates in the Jolly–Seber model with tag loss (Cowen & Schwarz, 2006). We validated the simulation framework using long‐term census data of elephant seals.
3. Including recycled individuals did not affect estimates of capture, survival, and tag‐retention probabilities. However, with low tag‐retention rates, high capture rates, and high survival rates, recycled individuals produced overestimates of population size. For the elephant seal case study, we found population size estimates to be between 8% and 53% larger when recycled individuals were ignored.
4. Ignoring the effects of recycled individuals can cause large biases in population size estimates. These results are particularly noticeable in longer studies.