Richard J. Barker

MODEL WEIGHTS AND THE FOUNDATIONS OF MULTIMODEL INFERENCE

Statistical thinking in wildlife biology and ecology has been profoundly influenced by the introduction of AIC (Akaikes information criterion) as a tool for model selection and as a basis for model averaging. In this paper, we advocate the Bayesian paradigm as a broader framework for multimodel inference, one in which model averaging and model selection are naturally linked, and in which the performance of AIC-based tools is naturally evaluated. Prior model weights implicitly associated with the use of AIC are seen to highly favor complex models: in some cases, all but the most highly parameterized models in the model set are virtually ignored a priori. We suggest the usefulness of the weighted BIC (Bayesian information criterion) as a computationally simple alternative to AIC, based on explicit selection of prior model probabilities rather than acceptance of default priors associated with AIC. We note, however, that both procedures are only approximate to the use of exact Bayes factors. We discuss and illustrate technical difficulties associated with Bayes factors, and suggest approaches to avoiding these difficulties in the context of model selection for a logistic regression. Our example highlights the predisposition of AIC weighting to favor complex models and suggests a need for caution in using the BIC for computing approximate posterior model weights.

JOINT MODELING OF LIVE-RECAPTURE, TAG-RESIGHT, AND TAG-RECOVERY DATA

We propose a model for a mark-recapture experiment with resightings obtained from marked animals any time between capture periods and throughout the geographic range of the animals. The likelihood is described for random movement of animals in and out of the capture site, and closed-form maximum likelihood estimators are reported. We show that the estimator is equivalent to that suggested by Jolly (1965, Biometrika 52, 239) but is only valid under the random emigration assumption. The model provides a common framework for most of the widely used mark-recapture models including live-recapture, tag-recovery, and tag-resight models and allows simultaneous analysis of data obtained in all three ways.

Chapter 3 Modeling Individual Animal Histories with Multistate Capture–Recapture Models

Summary Many fields of science begin with a phase of exploration and description, followed by investigations of the processes that account for observed patterns. The science of ecology is no exception, and recent decades have seen a focus on understanding key processes underlying the dynamics of ecological systems. In population ecology, emphasis has shifted from the state variable of population size to the demographic processes responsible for changes in this state variable: birth, death, immigration, and emigration. In evolutionary ecology, some of these same demographic processes, rates of birth and death, are also the determinants of fitness. In animal population ecology, the estimation of state variables and their associated vital rates is especially problematic because of the difficulties in sampling such populations and detecting individual animals. Indeed, early capture–recapture models were developed for the purpose of estimating population size, given the reality that all animals are not caught or detected at any sampling occasion. More recently, capture–recapture models for open populations were developed to draw inferences about survival in the face of these same sampling problems. The focus of this paper is on multi‐state mark–recapture models (MSMR), which first appeared in the 1970s but have undergone substantial development in the last 15 years. These models were developed to deal explicitly with biological variation, in that animals in different “states” (classes defined by location, physiology, behavior, reproductive status, etc.) may have different probabilities of survival and detection. Animal transitions between states are also stochastic and themselves of interest. These general models have proven to be extremely useful and provide a way of thinking about a remarkably wide range of important ecological processes. These methods are now at a stage of refinement and sophistication where they can readily be used by biologists to tackle a wide range of important issues in ecology. In this paper, we draw together information on the state of the art in multistate mark–recapture methods, explaining the models and illustrating their use. We provide a modeling philosophy and a series of general principles on how to conduct analyses. We cover key issues and features, and we anticipate the ways in which we expect the models to develop in the years ahead. In particular: – MSMR can now be used in a straightforward fashion by population biologists, thanks to the development of sound goodness‐of‐fit procedures, reliable parameter identifiability diagnostics, and robust user‐friendly computer software.Constrained models and model selection procedures can be used in the ANOVA‐like philosophy commonly used over the last 15 years for survival models, to answer a variety of biological questions. We develop as an example a treatment of meadow vole Microtus pennsylvanicus data. – As in survival models, random effects should be an integral part of this philosophy. Some simple approaches to random effects are illustrated. – States can be defined in a very general way, for example, by combining several criteria, such as sites and reproductive states, and can include nonobservable states. We develop as an example a multisite recruitment model of roseate terns Sterna dougallii . – MSMR models appear as a natural framework for combining different sources of information, viewed as different events that can be organized into mutually exclusive alternatives. – With the available developments, MSMR models are becoming a standard tool in population biology, as shown by a rapid growth of their use in the literature. In particular, given the ease with which a variety of constrained models can now be developed, MSMR models appear as less data hungry than was often feared. – MSMR models make it also possible to unify a large array of methodology, and, as such, are both a step towards further unification in a “mother of all” model, and a sound basis for further generalizations. – Future developments concern a variety of generalizations such as the reverse time approach and population size estimation. “Multievent” models, accounting for uncertainty in state determination, and integrated state–space models are key generalizations already in full development.

Multistate Survival Models and Their Extensions in Program MARK

Abstract Program MARK provides >100 models for the estimation of population parameters from mark–encounter data. The multistate model of Brownie et al. (1993) and Hestbeck et al. (1991) allows animals to move between states with a probability of transition. The simplest multistate model is an extension of the Cormack–Jolly–Seber (CJS) live recapture model. Parameters estimated are state-specific survival rates and encounter probabilities and transition probabilities between states. The multistate model provides a valuable framework to evaluate important ecological questions. For example, estimation of state-specific survival and transition probabilities between the biological states of breeders and nonbreeders allows estimation of the cost of reproduction. Transitions between physical states, such as spatial areas, provide estimates needed for meta-population models. The basic multistate model uses only live recaptures, but 3 extensions are included in MARK. A multistate model with live and dead encounters is available, although the dead encounters are not state specific. Robust-design multistate models are also included in MARK, with both open and closed robust designs. These models assume that animals move between states only between primary sessions of the robust design. For the closed robust design, we can specify 12 different data types for the modeling of encounter probabilities during the primary session, including 6 versions of the closed model likelihood incorporating population size (N) directly in the likelihood, and 6 versions of the Huggins model in which N is estimated as a derived parameter outside the likelihood. One assumption that is generally necessary to estimate state-specific survival rates in the multistate model is that transitions take place immediately before encounter occasions. Otherwise, survival rates over the interval between encounter occasions are a mix of survival rates over multiple states. Advantages of using MARK to estimate the parameters of the various multistate models include flexibility of model specification to include group, time, and individual covariates, estimation of variance components, model averaging of parameter estimates, and Bayesian parameter estimation using Markov chain Monte Carlo procedures on the logit scale.

Modeling the Relationship Between Fecal Pellet Indices and Deer Density

Abstract Although fecal pellet counts have been widely used to index changes in deer abundance in forests, few studies have modeled the relationship between the indices and deer density. We examined the relationships between 3 fecal pellet indices (total pellets, pellet groups, and pellet frequency) and the density of deer (primarily red deer [Cervus elaphus scoticus]) in 20 enclosures in the North and South islands of New Zealand. In each enclosure we estimated the 3 indices on 30 randomly located 150-m transects, with each transect having 30 circular plots of 3.14 m2. We developed 4 candidate models (1 linear and 3 nonlinear) to describe the relationship between the indices and deer density. We used a Bayesian analysis to account for uncertainty in the estimates of deer abundance and to facilitate fitting models that included random transect effects. The 4 models explained the relationship between the 3 indices and deer density similarly well. The slopes of the linear relationships between the 3 indices and deer density were positive. Our results suggest that fecal pellet counts may be useful indices of deer abundance.

Incorporating Genotype Uncertainty into Mark–Recapture-Type Models For Estimating Abundance Using DNA Samples

Sampling DNA noninvasively has advantages for identifying animals for uses such as mark-recapture modeling that require unique identification of animals in samples. Although it is possible to generate large amounts of data from noninvasive sources of DNA, a challenge is overcoming genotyping errors that can lead to incorrect identification of individuals. A major source of error is allelic dropout, which is failure of DNA amplification at one or more loci. This has the effect of heterozygous individuals being scored as homozygotes at those loci as only one allele is detected. If errors go undetected and the genotypes are naively used in mark-recapture models, significant overestimates of population size can occur. To avoid this it is common to reject low-quality samples but this may lead to the elimination of large amounts of data. It is preferable to retain these low-quality samples as they still contain usable information in the form of partial genotypes. Rather than trying to minimize error or discarding error-prone samples we model dropout in our analysis. We describe a method based on data augmentation that allows us to model data from samples that include uncertain genotypes. Application is illustrated using data from the European badger (Meles meles).

SOURCES OF VARIATION IN WATERFOWL SURVIVAL RATES

Because of the need to manage hunted populations of waterfowl (Anatidae), biologists have studied many demographic traits of waterfowl by analyzing band recoveries. These analyses have produced the most extensive and best estimates of survival available for any group of birds. Using these data, we examined several factors that might explain variation among annual survival rates to explore large-scale patterns that might be useful in under- standing waterfowl population dynamics. We found that geography, body mass, and tribe (i.e. phylogeny) were important in explaining variation in average waterfowl survival rates. Received 22 April 1996, accepted 18 September 1996. ALTHOUGH MODERN METHODS for estimating survival rates have been available for 30 years (Cormack 1964, Jolly 1965, Seber 1965), these estimators have not been applied evenly among all groups of birds. Modern estimation proce- dures have been put to their fullest use in only one order, the Anseriformes. One reason for this emphasis is the need to understand the relationship between hunter harvest and

EFFICIENT ESTIMATION OF ABUNDANCE FOR PATCHILY DISTRIBUTED POPULATIONS VIA TWO-PHASE, ADAPTIVE SAMPLING

Many organisms are patchily distributed, with some patches occupied at high density, others at lower densities, and others not occupied. Estimation of overall abundance can be difficult and is inefficient via intensive approaches such as capture-mark-recapture (CMR) or distance sampling. We propose a two-phase sampling scheme and model in a Bayesian framework to estimate abundance for patchily distributed populations. In the first phase, occupancy is estimated by binomial detection samples taken on all selected sites, where selection may be of all sites available, or a random sample of sites. Detection can be by visual surveys, detection of sign, physical captures, or other approach. At the second phase, if a detection threshold is achieved, CMR or other intensive sampling is conducted via standard procedures (grids or webs) to estimate abundance. Detection and CMR data are then used in a joint likelihood to model probability of detection in the occupancy sample via an abundance-detection model. CMR modeling is used to estimate abundance for the abundance-detection relationship, which in turn is used to predict abundance at the remaining sites, where only detection data are collected. We present a full Bayesian modeling treatment of this problem, in which posterior inference on abundance and other parameters (detection, capture probability) is obtained under a variety of assumptions about spatial and individual sources of heterogeneity. We apply the approach to abundance estimation for two species of voles (Microtus spp.) in Montana, USA. We also use a simulation study to evaluate the frequentist properties of our procedure given known patterns in abundance and detection among sites as well as design criteria. For most population characteristics and designs considered, bias and mean-square error (MSE) were low, and coverage of true parameter values by Bayesian credibility intervals was near nominal. Our two-phase, adaptive approach allows efficient estimation of abundance of rare and patchily distributed species and is particularly appropriate when sampling in all patches is impossible, but a global estimate of abundance is required.

Flexible hierarchical mark-recapture modeling for open populations using WinBUGS

Hierarchical mark-recapture models offer three advantages over classical mark-recapture models: (i) they allow expression of complicated models in terms of simple components; (ii) they provide a convenient way of modeling missing data and latent variables in a way that allows expression of relationships involving latent variables in the model; (iii) they provide a convenient way of introducing parsimony into models involving many nuisance parameters. Expressing models using the complete data likelihood we show how many of the standard mark-recapture models for open populations can be readily fitted using the software WinBUGS. We include examples that illustrate fitting the Cormack–Jolly–Seber model, multi-state and multi-event models, models including auxiliary data, and models including density dependence.

Modeling tag loss in New Zealand fur seal pups

Mark-recapture studies of pinnipeds commonly use double-tagging to reduce bias of parameter estimates and to allow estimation of tag retention rates. However, most tag retention estimates assume independence of tag loss. Here we were able to identify when individual New Zealand fur seal (Arctocephalus forsteri) pups had lost both tags ; therefore, we tested the assumption of no association between the tag-loss rates of left and right tags. We also tested for differences in tag retention among three different types of plastic tag (Allflex® cattle, mini and button tags), between two attachment types (i.e., fixed or swivel), and whether retention varied among years and colonies sampled. We found strong evidence of within-individual tag loss association for most tags in most years, but little evidence that this varied among colonies. We found that ignoring within-individual association of tag loss led to a bias in estimated tag retention of 7.4-10.1%. Smaller rocks and greater crevice and ledge densities in colonies were associated with lower probabilities of tag retention. We suggest researchers should attempt to use permanent marks in combination with tags to assess unbiased estimates of tag retention.