A robust-design formulation of the incidence function model of metapopulation dynamics applied to two species of rails

The incidence function model (IFM) uses area and connectivity to predict metapopulation dynamics. However, false absences and missing data can lead to underestimates of the number of sites contributing to connectivity, resulting in overestimates of dispersal ability and turnovers (extinctions plus colonizations). We extend estimation methods for the IFM by using a hierarchical Bayesian model to account both for false absences due to imperfect detection and for missing data due to sites not surveyed in some years. We compare parameter estimates, measures of metapopulation dynamics, and forecasts using stochastic patch occupancy models (SPOMs) among three IFM models: (1) a Bayesian formulation assuming no false absences and omitting site-year combinations with missing data; (2) a hierarchical Bayesian formulation assuming no false absences but incorporating missing data; and (3) a hierarchical Bayesian formulation allowing for imperfect detection and incorporating missing data. We fit the models to multiyear data sets of occupancy for two bird species that differ in body size and presumed dispersal ability but inhabit the same network of sites: the small Black Rail (Laterallus jamaicensis) and the medium-sized Virginia Rail (Rallus limicola). Incorporating missing data affected colonization parameters and led to lower estimates of dispersal ability for the Black Rail. Detection rates were high for the Black Rail in most years but moderate for the Virginia Rail. Incorporating imperfect detection resulted in higher occupancy and lower turnover rates for both species, with largest effects for the Virginia Rail. Forecasts using SPOMs were sensitive to both missing data and false absences; persistence in models assuming no false absences was more optimistic than from robust models. Our results suggest that incorporating false absences and missing data into the IFM can improve (1) estimates of dispersal ability and the effect of connectivity on colonization, (2) the scaling of extinction risk with patch area, and (3) forecasts of occupancy and turnover rates.     

Multimethod,multistate Bayesian hierarchical modeling approach for use in regional monitoring of wolves

In many cases, the first step in large‐carnivore management is to obtain objective, reliable, and cost‐effective estimates of population parameters through procedures that are reproducible over time. However, monitoring predators over large areas is difficult, and the data have a high level of uncertainty. We devised a practical multimethod and multistate modeling approach based on Bayesian hierarchical‐site‐occupancy models that combined multiple survey methods to estimate different population states for use in monitoring large predators at a regional scale. We used wolves (Canis lupus) as our model species and generated reliable estimates of the number of sites with wolf reproduction (presence of pups). We used 2 wolf data sets from Spain (Western Galicia in 2013 and Asturias in 2004) to test the approach. Based on howling surveys, the naïve estimation (i.e., estimate based only on observations) of the number of sites with reproduction was 9 and 25 sites in Western Galicia and Asturias, respectively. Our model showed 33.4 (SD 9.6) and 34.4 (3.9) sites with wolf reproduction, respectively. The number of occupied sites with wolf reproduction was 0.67 (SD 0.19) and 0.76 (0.11), respectively. This approach can be used to design more cost‐effective monitoring programs (i.e., to define the sampling effort needed per site). Our approach should inspire well‐coordinated surveys across multiple administrative borders and populations and lead to improved decision making for management of large carnivores on a landscape level. The use of this Bayesian framework provides a simple way to visualize the degree of uncertainty around population‐parameter estimates and thus provides managers and stakeholders an intuitive approach to interpreting monitoring results. Our approach can be widely applied to large spatial scales in wildlife monitoring where detection probabilities differ between population states and where several methods are being used to estimate different population parameters.     

Lifting a veil on diversity: a Bayesian approach to fitting relative-abundance models.

Bayesian methods incorporate prior knowledge into a statistical analysis. This prior knowledge is usually restricted to assumptions regarding the form of probability distributions of the parameters of interest, leaving their values to be determined mainly through the data. Here we show how a Bayesian approach can be applied to the problem of drawing inference regarding species abundance distributions and comparing diversity indices between sites. The classic log series and the lognormal models of relative- abundance distribution are apparently quite different in form. The first is a sampling distribution while the other is a model of abundance of the underlying population. Bayesian methods help unite these two models in a common framework. Markov chain Monte Carlo simulation can be used to fit both distributions as small hierarchical models with shared common assumptions. Sampling error can be assumed to follow a Poisson distribution. Species not found in a sample, but suspected to be present in the region or community of interest, can be given zero abundance. This not only simplifies the process of model fitting, but also provides a convenient way of calculating confidence intervals for diversity indices. The method is especially useful when a comparison of species diversity between sites with different sample sizes is the key motivation behind the research. We illustrate the potential of the approach using data on fruit-feeding butterflies in southern Mexico. We conclude that, once all assumptions have been made transparent, a single data set may provide support for the belief that diversity is negatively affected by anthropogenic forest disturbance. Bayesian methods help to apply theory regarding the distribution of abundance in ecological communities to applied conservation.     

Sensitivity of ecological models to their climate drivers: statistical ensembles for forcing.

Global and regional numerical models for terrestrial ecosystem dynamics require fine spatial resolution and temporally complete historical climate fields as input variables. However, because climate observations are unevenly spaced and have incomplete records, such fields need to be estimated. In addition, uncertainty in these fields associated with their estimation are rarely assessed. Ecological models are usually driven with a geostatistical model's mean estimate (kriging) of these fields without accounting for this uncertainty, much less evaluating such errors in terms of their propagation in ecological simulations. We introduce a Bayesian statistical framework to model climate observations to create spatially uniform and temporally complete fields, taking into account correlation in time and space, spatial heterogeneity, lack of normality, and uncertainty about all these factors. A key benefit of the Bayesian model is that it generates uncertainty measures for the generated fields. To demonstrate this method, we reconstruct historical monthly precipitation fields (a driver for ecological models) on a fine resolution grid for a climatically heterogeneous region in the western United States. The main goal of this work is to evaluate the sensitivity of ecological models to the uncertainty associated with prediction of their climate drivers. To assess their numerical sensitivity to predicted input variables, we generate a set of ecological model simulations run using an ensemble of different versions of the reconstructed fields. We construct such an ensemble by sampling from the posterior predictive distribution of the climate field. We demonstrate that the estimated prediction error of the climate field can be very high. We evaluate the importance of such errors in ecological model experiments using an ensemble of historical precipitation time series in simulations of grassland biogeochemical dynamics with an ecological numerical model, Century. We show how uncertainty in predicted precipitation fields is propagated into ecological model results and that this propagation had different modes. Depending on output variable, the response of model dynamics to uncertainty in inputs ranged from uncertainty in outputs that matched that of inputs to those that were muted or that were biased, as well as uncertainty that was persistent in time after input errors dropped.     

Building statistical models to analyze species distributions.

Models of the geographic distributions of species have wide application in ecology. But the nonspatial, single-level, regression models that ecologists have often employed do not deal with problems of irregular sampling intensity or spatial dependence, and do not adequately quantify uncertainty. We show here how to build statistical models that can handle these features of spatial prediction and provide richer, more powerful inference about species niche relations, distributions, and the effects of human disturbance. We begin with a familiar generalized linear model and build in additional features, including spatial random effects and hierarchical levels. Since these models are fully specified statistical models, we show that it is possible to add complexity without sacrificing interpretability. This step-by-step approach, together with attached code that implements a simple, spatially explicit, regression model, is structured to facilitate self-teaching. All models are developed in a Bayesian framework. We assess the performance of the models by using them to predict the distributions of two plant species (Proteaceae) from South Africa's Cape Floristic Region. We demonstrate that making distribution models spatially explicit can be essential for accurately characterizing the environmental response of species, predicting their probability of occurrence, and assessing uncertainty in the model results. Adding hierarchical levels to the models has further advantages in allowing human transformation of the landscape to be taken into account, as well as additional features of the sampling process.     

Hierarchical Bayesian space-time models

Space-time data are ubiquitous in the environmental sciences. Often, as is the case with atmo- spheric and oceanographic processes, these data contain many different scales of spatial and temporal variability. Such data are often non-stationary in space and time and may involve many observation/prediction locations. These factors can limit the effectiveness of traditional space- time statistical models and methods. In this article, we propose the use of hierarchical space-time models to achieve more flexible models and methods for the analysis of environmental data distributed in space and time. The first stage of the hierarchical model specifies a measurement- error process for the observational data in terms of some 'state' process. The second stage allows for site-specific time series models for this state variable. This stage includes large-scale (e.g. seasonal) variability plus a space-time dynamic process for the anomalies'. Much of our interest is with this anomaly proc ess. In the third stage, the parameters of these time series models, which are distributed in space, are themselves given a joint distribution with spatial dependence (Markov random fields). The Bayesian formulation is completed in the last two stages by speci- fying priors on parameters. We implement the model in a Markov chain Monte Carlo framework and apply it to an atmospheric data set of monthly maximum temperature.     

On the latent state estimation of nonlinear population dynamics using Bayesian and non-Bayesian state-space models

Nonlinear state-space models have been increasingly applied to study population dynamics and data assimilation in environmental sciences. State-space models can account for process error and measurement error simultaneously to correct for the bias in the estimates of system state and model parameters. However, few studies have compared the performance of different nonlinear state-space models for reconstructing the state of population dynamics from noisy time series. This study compared the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and Bayesian nonlinear state-space models (BNSSM) through simulations. Synthetic population time series were generated using the theta logistic model with known parameters, and normally distributed process and measurement errors were introduced using the Monte Carlo simulations. At higher levels of nonlinearity, the UKF and BNSSM had lower root mean square error (RMSE) than the EKF. The BNSSM performed reliably across all levels of nonlinearity, whereas increased levels of nonlinearity resulted in higher RMSE of the EKF. The Metropolis–Hastings algorithm within the Gibbs algorithm was used to fit the theta logistic model to synthetic time series to estimate model parameters. The estimated posterior distribution of the parameter θ indicated that the 95% credible intervals included the true values of θ (=0.5 and 1.5), but did not include 1.0 and 0.0. Future studies need to incorporate the adaptive Metropolis algorithm to estimate unknown model parameters for broad applications of Bayesian nonlinear state-space models in ecological studies.     

The Use of Bayesian Model Averaging to Better Represent Uncertainty in Ecological Models

Abstract:   In conservation biology, uncertainty about the choice of a statistical model is rarely considered. Model-selection uncertainty occurs whenever one model is chosen over plausible alternative models to represent understanding about a process and to make predictions about future observations. The standard approach to representing prediction uncertainty involves the calculation of prediction (or confidence) intervals that incorporate uncertainty about parameter estimates contingent on the choice of a "best" model chosen to represent truth. However, this approach to prediction based on statistical models tends to ignore model-selection uncertainty, resulting in overconfident predictions. Bayesian model averaging (BMA) has been promoted in a range of disciplines as a simple means of incorporating model-selection uncertainty into statistical inference and prediction. Bayesian model averaging also provides a formal framework for incorporating prior knowledge about the process being modeled. We provide an example of the application of BMA in modeling and predicting the spatial distribution of an arboreal marsupial in the Eden region of southeastern Australia. Other approaches to estimating prediction uncertainty are discussed.     

Estimating parameters of hidden Markov models based on marked individuals: use of robust design data

Development and use of multistate mark-recapture models, which provide estimates of parameters of Markov processes in the face of imperfect detection, have become common over the last 20 years. Recently, estimating parameters of hidden Markov models, where the state of an individual can be uncertain even when it is detected, has received attention. Previous work has shown that ignoring state uncertainty biases estimates of survival and state transition probabilities, thereby reducing the power to detect effects. Efforts to adjust for state uncertainty have included special cases and a general framework for a single sample per period of interest. We provide a flexible framework for adjusting for state uncertainty in multistate models, while utilizing multiple sampling occasions per period of interest to increase precision and remove parameter redundancy. These models also produce direct estimates of state structure for each primary period, even for the case where there is just one sampling occasion. We apply our model to expected-value data, and to data from a study of Florida manatees, to provide examples of the improvement in precision due to secondary capture occasions. We have also implemented these models in program MARK. This general framework could also be used by practitioners to consider constrained models of particular interest, or to model the relationship between within-primary-period parameters (e.g., state structure) and between-primary-period parameters (e.g., state transition probabilities).     

Assessing conservation status with extensive but low-resolution data: Application of frequentist and Bayesian models to endangered Athabasca River rainbow trout

Use of extensive but low-resolution abundance data is common in the assessment of species at-risk status based on quantitative decline criteria under International Union for Conservation of Nature (IUCN) and national endangered species legislation. Such data can be problematic for 3 reasons. First, statistical power to reject the null hypothesis of no change is often low because of small sample size and high sampling uncertainty leading to a high frequency of type II errors. Second, range-wide assessments composed of multiple site-specific observations do not effectively weight site-specific trends into global trends. Third, uncertainty in site-specific temporal trends and relative abundance are not propagated at the appropriate spatial scale. A common result is the propensity to underestimate the magnitude of declines and therefore fail to identify the appropriate at-risk status for a species. We used 3 statistical approaches, from simple to more complex, to estimate temporal decline rates for a designatable unit (DU) of rainbow trout in the Athabasca River watershed in western Canada. This DU is considered a native species for purposes of listing because of its genetic composition characterized as >0.95 indigenous origin in the face of continuing introgressive hybridization with introduced populations in the watershed. Analysis of abundance trends from 57 time series with a fixed-effects model identified 33 sites with negative trends, but only 2 were statistically significant. By contrast, a hierarchical linear mixed model weighted by site-specific abundance provided a DU-wide decline estimate of 16.4% per year and a 3-generation decline of 93.2%. A hierarchical Bayesian mixed model yielded a similar 3-generation decline trend of 91.3% and the posterior distribution showed that the estimate had a >99% probability of exceeding thresholds for an endangered listing. We conclude that the Bayesian approach was the most useful because it provided a probabilistic statement of threshold exceedance in support of an at-risk status recommendation.     

Effects of uncertainty and variability on population declines and IUCN Red List classifications

The International Union for Conservation of Nature (IUCN) Red List Categories and Criteria is a quantitative framework for classifying species according to extinction risk. Population models may be used to estimate extinction risk or population declines. Uncertainty and variability arise in threat classifications through measurement and process error in empirical data and uncertainty in the models used to estimate extinction risk and population declines. Furthermore, species traits are known to affect extinction risk. We investigated the effects of measurement and process error, model type, population growth rate, and age at first reproduction on the reliability of risk classifications based on projected population declines on IUCN Red List classifications. We used an age‐structured population model to simulate true population trajectories with different growth rates, reproductive ages and levels of variation, and subjected them to measurement error. We evaluated the ability of scalar and matrix models parameterized with these simulated time series to accurately capture the IUCN Red List classification generated with true population declines. Under all levels of measurement error tested and low process error, classifications were reasonably accurate; scalar and matrix models yielded roughly the same rate of misclassifications, but the distribution of errors differed; matrix models led to greater overestimation of extinction risk than underestimations; process error tended to contribute to misclassifications to a greater extent than measurement error; and more misclassifications occurred for fast, rather than slow, life histories. These results indicate that classifications of highly threatened taxa (i.e., taxa with low growth rates) under criterion A are more likely to be reliable than for less threatened taxa when assessed with population models. Greater scrutiny needs to be placed on data used to parameterize population models for species with high growth rates, particularly when available evidence indicates a potential transition to higher risk categories.     

A hierarchical model for spatial capture-recapture data

Estimating density is a fundamental objective of many animal population studies. Application of methods for estimating population size from ostensibly closed populations is widespread, but ineffective for estimating absolute density because most populations are subject to short-term movements or so-called temporary emigration. This phenomenon invalidates the resulting estimates because the effective sample area is unknown. A number of methods involving the adjustment of estimates based on heuristic considerations are in widespread use. In this paper, a hierarchical model of spatially indexed capture-recapture data is proposed for sampling based on area searches of spatial sample units subject to uniform sampling intensity. The hierarchical model contains explicit models for the distribution of individuals and their movements, in addition to an observation model that is conditional on the location of individuals during sampling. Bayesian analysis of the hierarchical model is achieved by the use of data augmentation, which allows for a straightforward implementation in the freely available software WinBUGS. We present results of a simulation study that was carried out to evaluate the operating characteristics of the Bayesian estimator under variable densities and movement patterns of individuals. An application of the model is presented for survey data on the flat-tailed horned lizard (Phrynosoma mcallii) in Arizona, USA.     

A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove

The spread of invasive species is a long studied subject that garners much interest in the ecological research community. Historically the phenomenon has been approached using a purely deterministic mathematical framework (usually involving differential equations of some form). These methods, while scientifically meaningful, are generally highly simplified and fail to account for uncertainty in the data and process, of which our knowledge could not possibly exist without error. We propose a hierarchical Bayesian model for population spread that accommodates data sources with errors, dependence structures between population dynamics parameters, and takes into account prior scientific understanding via non-linear relationships between model parameters and space-time response variables. We model the process (i.e., the bird population in this case) as a Poisson response with spatially varying diffusion coefficients as well as a logistic population growth term using a common reaction-diffusion equation that realistically mimics the ecological process. We focus the application on the ongoing invasion of the Eurasian Collared-Dove.     

Poor-data and data-poor species stock assessment using a Bayesian hierarchical approach

Appropriate inference for stocks or species with low-quality data (poor data) or limited data (data poor) is extremely important. Hierarchical Bayesian methods are especially applicable to small-area, small-sample-size estimation problems because they allow poor-data species to borrow strength from species with good-quality data. We used a hammerhead shark complex as an example to investigate the advantages of using hierarchical Bayesian models in assessing the status of poor-data and data-poor exploited species. The hammerhead shark complex (Sphyrna spp.) along the Atlantic and Gulf of Mexico coasts of the United States is composed of three species: the scalloped hammerhead (S. lewini), the great hammerhead (S. mokarran), and the smooth hammerhead (S. zygaena) sharks. The scalloped hammerhead comprises 70-80% of the catch and has catch and relative abundance data of good quality, whereas great and smooth hammerheads have relative abundance indices that are both limited and of low quality presumably because of low stock density and limited sampling. Four hierarchical Bayesian state-space surplus production models were developed to simulate variability in population growth rates, carrying capacity, and catchability of the species. The results from the hierarchical Bayesian models were considerably more robust than those of the nonhierarchical models. The hierarchical Bayesian approach represents an intermediate strategy between traditional models that assume different population parameters for each species and those that assume all species share identical parameters. Use of the hierarchical Bayesian approach is suggested for future hammerhead shark stock assessments and for modeling fish complexes with species-specific data, because the poor-data species can borrow strength from the species with good data, making the estimation more stable and robust.     

A Bayesian approach to retransformation bias in transformed regression

Ecological data analysis often involves fitting linear or nonlinear equations to data after transforming either the response variable, the right side of the equation, or both, so that the standard suite of regression assumptions are more closely met. However, inference is usually done in the natural metric and it is well known that retransforming back to the original metric provides a biased estimator for the mean of the response variable. For the normal linear model, fit under a log-transformation, correction factors are available to reduce this bias, but these factors may not be generally applicable to all model forms or other transformations. We demonstrate that this problem is handled in a straightforward manner using a Bayesian approach, which is general for linear and nonlinear models and other transformations and model error structures. The Bayesian framework provides a predictive distribution for the response variable so that inference can be made at the mean, or over the entire distribution to incorporate the predictive uncertainty.     

Hierarchical Bayesian Spatial Models for Multispecies Conservation Planning and Monitoring

Abstract: Biologists who develop and apply habitat models are often familiar with the statistical challenges posed by their data's spatial structure but are unsure of whether the use of complex spatial models will increase the utility of model results in planning. We compared the relative performance of nonspatial and hierarchical Bayesian spatial models for three vertebrate and invertebrate taxa of conservation concern (Church's sideband snails [Monadenia churchi], red tree voles [Arborimus longicaudus], and Pacific fishers [Martes pennanti pacifica]) that provide examples of a range of distributional extents and dispersal abilities. We used presence–absence data derived from regional monitoring programs to develop models with both landscape and site‐level environmental covariates. We used Markov chain Monte Carlo algorithms and a conditional autoregressive or intrinsic conditional autoregressive model framework to fit spatial models. The fit of Bayesian spatial models was between 35 and 55% better than the fit of nonspatial analogue models. Bayesian spatial models outperformed analogous models developed with maximum entropy (Maxent) methods. Although the best spatial and nonspatial models included similar environmental variables, spatial models provided estimates of residual spatial effects that suggested how ecological processes might structure distribution patterns. Spatial models built from presence–absence data improved fit most for localized endemic species with ranges constrained by poorly known biogeographic factors and for widely distributed species suspected to be strongly affected by unmeasured environmental variables or population processes. By treating spatial effects as a variable of interest rather than a nuisance, hierarchical Bayesian spatial models, especially when they are based on a common broad‐scale spatial lattice (here the national Forest Inventory and Analysis grid of 24 km² hexagons), can increase the relevance of habitat models to multispecies conservation planning.     

Combining state and transition models with dynamic Bayesian networks

Bashari et al. (2009) propose combining state and transition models (STMs) with Bayesian networks for decision support tools where the focus is on modelling the system dynamics. There is already an extension of Bayesian networks - so-called dynamic Bayesian networks (DBNs) - for explicitly modelling systems that change over time, that has also been applied in ecological modelling. In this paper we propose a combination of STMs and DBNs that overcome some of the limitations of Bashari et al.’s approach including providing an explicit representation of the next state, while retaining its advantages, such an the explicit representation of transitions. We then show that the new model can be applied iteratively to predict into the future consistently with different time frames. We use Bashari et al.’s rangeland management problem as an illustrative case study. We present a comparative complexity analysis of the different approaches, based on the structure inherent in the problem being modelled. This analysis showed that any models that explicitly represent all the transitions only remain tractable when there are natural constraints in the domain. Thus we recommend modellers should analyse these aspects of their problem before deciding whether to use the framework.     

Estimating age from recapture data: integrating incremental growth measures with ancillary data to infer age-at-length

Estimating the age of individuals in wild populations can be of fundamental importance for answering ecological questions, modeling population demographics, and managing exploited or threatened species. Significant effort has been devoted to determining age through the use of growth annuli, secondary physical characteristics related to age, and growth models. Many species, however, either do not exhibit physical characteristics useful for independent age validation or are too rare to justify sacrificing a large number of individuals to establish the relationship between size and age. Length-at-age models are well represented in the fisheries and other wildlife management literature. Many of these models overlook variation in growth rates of individuals and consider growth parameters as population parameters. More recent models have taken advantage of hierarchical structuring of parameters and Bayesian inference methods to allow for variation among individuals as functions of environmental covariates or individual-specific random effects. Here, we describe hierarchical models in which growth curves vary as individual-specific stochastic processes, and we show how these models can be fit using capture-recapture data for animals of unknown age along with data for animals of known age. We combine these independent data sources in a Bayesian analysis, distinguishing natural variation (among and within individuals) from measurement error. We illustrate using data for African dwarf crocodiles, comparing von Bertalanffy and logistic growth models. The analysis provides the means of predicting crocodile age, given a single measurement of head length. The von Bertalanffy was much better supported than the logistic growth model and predicted that dwarf crocodiles grow from 19.4 cm total length at birth to 32.9 cm in the first year and 45.3 cm by the end of their second year. Based on the minimum size of females observed with hatchlings, reproductive maturity was estimated to be at nine years. These size benchmarks are believed to represent thresholds for important demographic parameters; improved estimates of age, therefore, will increase the precision of population projection models. The modeling approach that we present can be applied to other species and offers significant advantages when multiple sources of data are available and traditional aging techniques are not practical.     

Accounting for matching uncertainty in two stage capture–recapture experiments using photographic measurements of natural marks

We propose a Bayesian hierarchical modeling approach for estimating the size of a closed population from data obtained by identifying individuals through photographs of natural markings. We assume that noisy measurements of a set of distinctive features are available for each individual present in a photographic catalogue. To estimate the population size from two catalogues obtained during two different sampling occasions, we embed the standard two-stage $M_t$ capture–recapture model for closed population into a multivariate normal data matching model that identifies the common individuals across the catalogues. In addition to estimating the population size while accounting for the matching process uncertainty, this hierarchical modelling approach allows to identify the common individuals by using the information provided by the capture–recapture model. This way, our model also represents a novel and reliable tool able to reduce the amount of effort researchers have to expend in matching individuals. We illustrate and motivate the proposed approach via a real data set of photo-identification of narwhals. Moreover, we compare our method with a set of possible alternative approaches by using both the empirical data set and a simulation study.