Matthew R. Schofield

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).

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.

A unified capture-recapture framework

This article develops a hierarchical framework for capture-recapture data that separates the capture process from the demographic processes of interest, such as birth and survival. This allows users to parameterize in terms of meaningful demographic parameters. The framework is very flexible with many of the current capture-recapture models included as special cases. The hierarchical nature of the model allows natural expression of relationships, both between parameters and between parameters and the realization of random variables, such as population size. Previously, many of these relationships, such as density dependence have been unable to be explored using capture-recapture data. Density dependence, where survival and birth rates depend on the population size, is an interesting special case. We fit a density-dependent model to male Gonodontis bidentata data and report evidence of negative density dependence in percapita birth rates and weak evidence of negative density dependence in survival.

A Model-Based Approach to Climate Reconstruction Using Tree-Ring Data

Quantifying long-term historical climate is fundamental to understanding recent climate change. Most instrumentally recorded climate data are only available for the past 200 years, so proxy observations from natural archives are often considered. We describe a model-based approach to reconstructing climate defined in terms of raw tree-ring measurement data that simultaneously accounts for nonclimatic and climatic variability. In this approach, we specify a joint model for the tree-ring data and climate variable that we fit using Bayesian inference. We consider a range of prior densities and compare the modeling approach to current methodology using an example case of Scots pine from Torneträsk, Sweden, to reconstruct growing season temperature. We describe how current approaches translate into particular model assumptions. We explore how changes to various components in the model-based approach affect the resulting reconstruction. We show that minor changes in model specification can have little effect on model fit but lead to large changes in the predictions. In particular, the periods of relatively warmer and cooler temperatures are robust between models, but the magnitude of the resulting temperatures is highly model dependent. Such sensitivity may not be apparent with traditional approaches because the underlying statistical model is often hidden or poorly described. Supplementary materials for this article are available online.

Comparing rates of springtail predation by web‐building spiders using Bayesian inference

A major goal of gut‐content analysis is to quantify predation rates by predators in the field, which could provide insights into the mechanisms behind ecosystem structure and function, as well as quantification of ecosystem services provided. However, percentage‐positive results from molecular assays are strongly influenced by factors other than predation rate, and thus can only be reliably used to quantify predation rates under very restrictive conditions. Here, we develop two statistical approaches, one using a parametric bootstrap and the other in terms of Bayesian inference, to build upon previous techniques that use DNA decay rates to rank predators by their rate of prey consumption, by allowing a statistical assessment of confidence in the inferred ranking. To demonstrate the utility of this technique in evaluating ecological data, we test web‐building spiders for predation on a primary prey item, springtails. Using these approaches we found that an orb‐weaving spider consumes springtail prey at a higher rate than a syntopic sheet‐weaving spider, despite occupying microhabitats where springtails are less frequently encountered. We suggest that spider‐web architecture (orb web vs. sheet web) is a primary determinant of prey‐consumption rates within this assemblage of predators, which demonstrates the potential influence of predator foraging behaviour on trophic web structure. We also discuss how additional assumptions can be incorporated into the same analysis to allow broader application of the technique beyond the specific example presented. We believe that such modelling techniques can greatly advance the field of molecular gut‐content analysis.

MC(MC)MC: exploring Monte Carlo integration within MCMC for mark–recapture models with individual covariates

Summary Estimating abundance from mark–recapture data is challenging when capture probabilities vary among individuals. Initial solutions to this problem were based on fitting conditional likelihoods and estimating abundance as a derived parameter. More recently, Bayesian methods using full likelihoods have been implemented via reversible jump Markov chain Monte Carlo sampling (RJMCMC) or data augmentation (DA). The latter approach is easily implemented in available software and has been applied to fit models that allow for heterogeneity in both open and closed populations. However, both RJMCMC and DA may be inefficient when modelling large populations. We describe an alternative approach using Monte Carlo (MC) integration to approximate the posterior density within a Markov chain Monte Carlo (MCMC) sampling scheme. We show how this Monte Carlo within MCMC (MCWM) approach may be used to fit a simple, closed population model including a single individual covariate and present results from a simulation study comparing RJMCMC, DA and MCWM. We found that MCWM can provide accurate inference about population size and can be more efficient than both RJMCMC and DA. The efficiency of MCWM can also be improved by using advanced MC methods like antithetic sampling. Finally, we apply MCWM to estimate the abundance of meadow voles (Microtus pennsylvanicus) at the Patuxent Wildlife Research Center in 1982 allowing for capture probabilities to vary as a function body mass.

Modeling individual specific fish length from capture-recapture data using the von Bertalanffy growth curve.

We use Bayesian methods to explore fitting the von Bertalanffy length model to tag-recapture data. We consider two popular parameterizations of the von Bertalanffy model. The first models the data relative to age at first capture; the second models in terms of length at first capture. Using data from a rainbow trout Oncorhynchus mykiss study we explore the relationship between the assumptions and resulting inference using posterior predictive checking, cross validation and a simulation study. We find that untestable hierarchical assumptions placed on the nuisance parameters in each model can influence the resulting inference about parameters of interest. Researchers should carefully consider these assumptions when modeling growth from tag-recapture data.

A Further Step Toward the Mother-of-All-Models: Flexibility and Functionality in the Modeling of Capture–Recapture Data

The idea behind the mother-of-all-models is to have the likelihoods for commonly used capture–recapture models factorized into conditional likelihoods that can be called and combined on request to give a user specified model. Barker and White (2004) mapped out a conceptual plan for the mother-of-all-models that included the robust design model and joint recapture, live re-sighting models. However they were unable to obtain a factorization that could easily include the multi-state model. Including any missing data directly into the model using data augmentation allows us to write the model in terms of the complete data likelihood (CDL). The CDL is a more natural representation of the model that factors into separate components that can be combined to give many different capture–recapture models, including the multi-state model. Overcoming the obstacles in the factorization brings the mother-of-all-models one step closer with the development of software the next step.

Closed-population capture–recapture modeling of samples drawn one at a time

Motivated by field sampling of DNA fragments, we describe a general model for capture-recapture modeling of samples drawn one at a time in continuous-time. Our model is based on Poisson sampling where the sampling time may be unobserved. We show that previously described models correspond to partial likelihoods from our Poisson model and their use may be justified through arguments concerning S- and Bayes-ancillarity of discarded information. We demonstrate a further link to continuous-time capture-recapture models and explain observations that have been made about this class of models in terms of partial ancillarity. We illustrate application of our models using data from the European badger (Meles meles) in which genotyping of DNA fragments was subject to error.