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.

Performance of a Bayesian state-space model of semelparous species for stock-recruitment data subject to measurement error

Measurement errors in spawner abundance create problems for fish stock assessment scientists. To deal with measurement error, we develop a Bayesian state-space model for stock-recruitment data that contain measurement error in spawner abundance, process error in recruitment, and time series bias. Through extensive simulations across numerous scenarios, we compare the statistical performance of the Bayesian state-space model with that of standard regression for a traditional stock-recruitment model that only considers process error. Performance varies depending on the information content in data, as determined by stock productivity, types of harvest situations, and amount of measurement error. Overall, in terms of estimating optimal spawner abundance S_MSY, the Ricker density-dependence parameter β, and optimal harvest rate h_MSY, the Bayesian state-space model works best for informative data from low and variable harvest rate situations for high-productivity salmon stocks. The traditional stock-recruitment model (TSR) may be used for estimating α and h_MSY for low-productivity stocks from variable and high harvest rate situations. However, TSR can severely overestimate S_MSY when spawner abundance is measured with large error in low and variable harvest rate situations. We also found that there is substantial merit in using h_MSY (or benchmarks derived from it) instead of S_MSY as a management target.     

Bayesian nonparametric inference for the overlap of daily animal activity patterns

The study of the interaction among species is an active area of research in Ecology. In particular, it is of interest to evaluate the overlap of their ecological niches. Temporal activity is one of the niche’s axes most commonly used to explore ecological segregation among animal species, and many contributions focus on the overlap of this variable. Once the information of the temporal activity is obtained in the wild, the data is treated as a random sample. There exist different methods to estimate the overlap. Specifically, in the case of two species, one possibility is to estimate the density of the temporal activity of each species and then evaluate the overlap between these density functions. This leads naturally to the analysis of circular data. Most of the procedures currently in use impose some rather restrictive assumptions on the probabilistic models used to describe the phenomena, and only provide approximate measures of the uncertainty involved in the process. In this article, we propose a Bayesian nonparametric approach which incorporates a well-defined noninformative prior. We take advantage of the data structure to define such a prior in terms of the predictive distribution. To the best of our knowledge, this is a novel approach. Our procedure is compared with a well-known method using simulated data, and applied to the analysis of real camera-trap data concerning two mammalian species from the El Triunfo biosphere reserve (Chiapas, Mexico).     

Univariate Bayesian nonparametric binary regression with application in environmental management

In environmental management, we often have to deal with binary response variables whose outcome dictates the course of action. This paper introduces a nonparametric Bayesian binary regression model with a single predictor variable that is more flexible than the commonly used logistic or probit models. Due to the Bayesian feature, the model can be easily used to combine observed data with our knowledge of the subject to produce site-specific results. By using three examples, this paper shows the potential application of the model in the environmental management, and its advantages in terms of flexibility in model specification, robustness to outliers, and realistic interpretation of data.     

Bayesian areal wombling via adjacency modeling

Recently, public health professionals and other geostatistical researchers have shown increasing interest in boundary analysis, the detection or testing of zones or boundaries that reveal sharp changes in the values of spatially oriented variables. For areal data (i.e., data which consist only of sums or averages over geopolitical regions), Lu and Carlin (Geogr Anal 37: 265–285, 2005) suggested a fully model-based framework for areal wombling using Bayesian hierarchical models with posterior summaries computed using Markov chain Monte Carlo (MCMC) methods, and showed the approach to have advantages over existing non-stochastic alternatives. In this paper, we develop Bayesian areal boundary analysis methods that estimate the spatial neighborhood structure using the value of the process in each region and other variables that indicate how similar two regions are. Boundaries may then be determined by the posterior distribution of either this estimated neighborhood structure or the regional mean response differences themselves. Our methods do require several assumptions (including an appropriate prior distribution, a normal spatial random effect distribution, and a Bernoulli distribution for a set of spatial weights), but also deliver more in terms of full posterior inference for the boundary segments (e.g., direct probability statements regarding the probability that a particular border segment is part of the boundary). We illustrate three different remedies for the computing difficulties encountered in implementing our method. We use simulation to compare among existing purely algorithmic approaches, the Lu and Carlin (2005) method, and our new adjacency modeling methods. We also illustrate more practical modeling issues (e.g., covariate selection) in the context of a breast cancer late detection data set collected at the county level in the state of Minnesota.     

Hierarchical Bayesian strategy for modeling correlated compositional data with observed zero counts

This article proposes a hierarchical multivariate conditional autoregressive model applied to a compositional response vector. We particularly focus on situations when the composition is discrete occurring when observations are based on small multinomial counts. We address drawbacks that exist in current modeling approaches for such data. Our hierarchical model will be demonstrated with data used to help manage a commercial sockeye salmon fishery in the Fraser River of British Columbia.     

Partitioning of \alpha and \beta diversity using hierarchical Bayesian modeling of species distribution and abundance

Diversity partitioning is becoming widely used to decompose the total number of species recorded in an area or region \((\gamma )\) into the average number of species within samples \((\alpha )\) and the average difference in species composition \((\beta )\) among samples. Single-value metrics of \(\alpha \) and \(\beta \) diversity are popular because they may be applied at multiple scales and because of their ease in computation and interpretation. Studies thus far, however, have emphasized observed diversity components or comparisons to randomized, null distributions. In addition, prediction of \(\alpha \) and \(\beta \) components using environmental or spatial variables has been limited to more extensive data sets because multiple samples are required to estimate single \(\alpha \) and \(\beta \) components. Lastly, observed diversity components do not incorporate variation in detection probabilities among species or samples. In this study, we used hierarchical Bayesian models of species abundances to provide predictions of \(\alpha \) and \(\beta \) components in species richness and composition using environmental and spatial variables. We illustrate our approach using butterfly data collected from 26 grassland remnants to predict spatially nested patterns of \(\alpha \) and \(\beta \) based on the predicted counts of butterflies. Diversity partitioning using a Bayesian hierarchical model incorporated variation in detection probabilities by butterfly species and habitat patches, and provided prediction intervals for \(\alpha \) and \(\beta \) components using environmental and spatial variables.     

Ecological boundary detection using Bayesian areal wombling

The study of ecological boundaries and their dynamics is of fundamental importance to much of ecology, biogeography, and evolution. Over the past two decades, boundary analysis (of which wombling is a subfield) has received considerable research attention, resulting in multiple approaches for the quantification of ecological boundaries. Nonetheless, few methods have been developed that can simultaneously (1) analyze spatially homogenized data sets (i.e., areal data in the form of polygons rather than point-reference data); (2) account for spatial structure in these data and uncertainty associated with them; and (3) objectively assign probabilities to boundaries once detected. Here we describe the application of a Bayesian hierarchical framework for boundary detection developed in public health, which addresses these issues but which has seen limited application in ecology. As examples, we analyze simulated spread data and the historic pattern of spread of an invasive species, the hemlock woolly adelgid (Adelges tsugae), using county-level summaries of the year of first reported infestation and several covariates potentially important to influencing the observed spread dynamics. Bayesian areal wombling is a promising approach for analyzing ecological boundaries and dynamics related to changes in the distributions of native and invasive species.     

Using structural equation modeling to investigate relationships among ecological variables

Structural equation modeling is an advanced multivariate statistical process with which a researcher can construct theoretical concepts, test their measurement reliability, hypothesize and test a theory about their relationships, take into account measurement errors, and consider both direct and indirect effects of variables on one another. Latent variables are theoretical concepts that unite phenomena under a single term, e.g., ecosystem health, environmental condition, and pollution (Bollen, 1989). Latent variables are not measured directly but can be expressed in terms of one or more directly measurable variables called indicators. For some researchers, defining, constructing, and examining the validity of latent variables may be the end task of itself. For others, testing hypothesized relationships of latent variables may be of interest. We analyzed the correlation matrix of eleven environmental variables from the U.S. Environmental Protection Agency's (USEPA) Environmental Monitoring and Assessment Program for Estuaries (EMAP-E) using methods of structural equation modeling. We hypothesized and tested a conceptual model to characterize the interdependencies between four latent variables-sediment contamination, natural variability, biodiversity, and growth potential. In particular, we were interested in measuring the direct, indirect, and total effects of sediment contamination and natural variability on biodiversity and growth potential. The model fit the data well and accounted for 81% of the variability in biodiversity and 69% of the variability in growth potential. It revealed a positive total effect of natural variability on growth potential that otherwise would have been judged negative had we not considered indirect effects. That is, natural variability had a negative direct effect on growth potential of magnitude –0.3251 and a positive indirect effect mediated through biodiversity of magnitude 0.4509, yielding a net positive total effect of 0.1258. Natural variability had a positive direct effect on biodiversity of magnitude 0.5347 and a negative indirect effect mediated through growth potential of magnitude –0.1105 yielding a positive total effects of magnitude 0.4242. Sediment contamination had a negative direct effect on biodiversity of magnitude –0.1956 and a negative indirect effect on growth potential via biodiversity of magnitude –0.067. Biodiversity had a positive effect on growth potential of magnitude 0.8432, and growth potential had a positive effect on biodiversity of magnitude 0.3398. The correlation between biodiversity and growth potential was estimated at 0.7658 and that between sediment contamination and natural variability at –0.3769.     

Estimating metabolic rate for butadiene at steady state using a Bayesian physiologically-based pharmacokinetic model

In a study of 133 volunteer subjects, demographic, physiologic and pharmacokinetic data through exposure to 1,3-Butadiene (BD) were collected in order to estimate the percentage of BD concentration metabolized at steady state, and to determine whether this percentage varies across gender, racial, and age groups. During the 20 min of continuous exposure to 2 parts per million (ppm) of BD, five measurements of exhaled concentration were made on each subject. In the following 40 min washout period, another five measurements were collected. A Bayesian hierarchical compartmental physiologically-based pharmacokinetic model (PKPB) was used. Using prior information on the model parameters, Markov Chain Monte Carlo (MCMC) simulation was conducted to obtain posterior distributions. The overall estimate of the mean percent of BD metabolized at steady state was 12.7% (95% credible interval: 7.7–17.8%). There was no significant difference in gender with males having a mean of 13.5%, and females 12.3%. Among the racial groups, Hispanic (13.9%), White (13.0%), Asian (12.1%), and Black (10.9%), the significant difference came from the difference between Black and Hispanic with a 95% credible interval from −5.63 to −0.30%. Those older than 30 years had a mean of 12.2% versus 12.9% for the younger group; although this was not a statistically significant difference. Given a constant inhalation input of 2 ppm, at steady state, the overall mean exhaled concentration was estimated to be 1.75ppm (95% credible interval: 1.64–1.84). An equivalent parameter, first-order metabolic rate constant, was also estimated and found to be consistent with the percent of BD metabolized at steady state across gender, race, and age strata.     

Disentangling trophic relationships in a High Arctic tundra ecosystem through food web modeling

Determining the manner in which food webs will respond to environmental changes is difficult because the relative importance of top-down vs. bottom-up forces in controlling ecosystems is still debated. This is especially true in the Arctic tundra where, despite relatively simple food webs, it is still unclear which forces dominate in this ecosystem. Our primary goal was to assess the extent to which a tundra food web was dominated by plant-herbivore or predator-prey interactions. Based on a 17-year (1993-2009) study of terrestrial wildlife on Bylot Island, Nunavut, Canada, we developed trophic mass balance models to address this question. Snow Geese were the dominant herbivores in this ecosystem, followed by two sympatric lemming species (brown and collared lemmings). Arctic foxes, weasels, and several species of birds of prey were the dominant predators. Results of our trophic models encompassing 19 functional groups showed that <10% of the annual primary production was consumed by herbivores in most years despite the presence of a large Snow Goose colony, but that 20-100% of the annual herbivore production was consumed by predators. The impact of herbivores on vegetation has also weakened over time, probably due to an increase in primary production. The impact of predators was highest on lemmings, intermediate on passerines, and lowest on geese and shorebirds, but it varied with lemming abundance. Predation of collared lemmings exceeded production in most years and may explain why this species remained at low density. In contrast, the predation rate on brown lemmings varied with prey density and may have contributed to the high-amplitude, periodic fluctuations in the abundance of this species. Our analysis provided little evidence that herbivores are limited by primary production on Bylot Island. In contrast, we measured strong predator-prey interactions, which supports the hypothesis that this food web is primarily controlled by top-down forces. The presence of allochthonous resources subsidizing top predators and the absence of large herbivores may partly explain the predominant role of predation in this low-productivity ecosystem.     

Ensemble Bayesian model averaging using Markov Chain Monte Carlo sampling

Bayesian model averaging (BMA) has recently been proposed as a statistical method to calibrate forecast ensembles from numerical weather models. Successful implementation of BMA however, requires accurate estimates of the weights and variances of the individual competing models in the ensemble. In their seminal paper (Raftery et al. Mon Weather Rev 133:1155–1174, 2005) has recommended the Expectation–Maximization (EM) algorithm for BMA model training, even though global convergence of this algorithm cannot be guaranteed. In this paper, we compare the performance of the EM algorithm and the recently developed DiffeRential Evolution Adaptive Metropolis (DREAM) Markov Chain Monte Carlo (MCMC) algorithm for estimating the BMA weights and variances. Simulation experiments using 48-hour ensemble data of surface temperature and multi-model streamflow forecasts show that both methods produce similar results, and that their performance is unaffected by the length of the training data set. However, MCMC simulation with DREAM is capable of efficiently handling a wide variety of BMA predictive distributions, and provides useful information about the uncertainty associated with the estimated BMA weights and variances.     

Predicting arsenic concentration in groundwater of Bangladesh using Bayesian geostatistical model

The pattern of the spatial variation in arsenic concentration in groundwater of Bangladesh is usually needed for the planning of safe drinking water. Often a model-based prediction is required for this purpose. In this paper, we fit a Bayesian hierarchical geostatistical model by utilizing data from the project, ‘Groundwater studies of arsenic concentration in Bangladesh’ conducted by the British Geological Survey and the Department of Public Health Engineering of Bangladesh. We also develop a predictive model for arsenic concentration at different levels of well-depth using the same approach. The resulting predictive model has been cross-validated by appropriate statistical tools. Finally, we obtained reliable spatially continuous predictive maps and predictive probability maps showing the areas with high probability of arsenic concentration for different levels of well-depth. Results indicate that our model fits the data well and captures a substantial amount of spatial variation. Moreover, well-depth is found to have a significant contribution in explaining the observed variation in arsenic concentration. The predictive maps that have been produced are observed to be different for various levels of well-depths and are expected to be helpful to the policy makers in preparing proper regional planning for safe drinking water.     

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 nonparametric procedure for analyzing repeated measures of spatially correlated data

Many agricultural, biological, and environmental studies involve detecting temporal changes of a response variable, based on data observed at sampling sites in a spatial region and repeatedly over several time points. That is, data are repeated measures over time and are potentially correlated across space. The traditional repeated-measures analysis allows for time dependence but assumes that the observations at different sampling sites are mutually independent, which may not be suitable for field data that are correlated across space. In this paper, a nonparametric large-sample inference procedure is developed to assess the time effects while accounting for the spatial dependence using a block bootstrap. For illustration, the methodology is applied to describe the population changes of root-lesion nematodes over time in a production field in Wisconsin.     

Examination of conflicts and improved strategies for the management of an endangered Eucalypt species using Bayesian networks

Bayesian decision support tools are becoming increasingly popular as a modelling framework that can analyse complex problems, resolve controversies, and support future decision-making in an adaptive management framework. This paper introduces a model designed to assist the management of an endangered Eucalypt species, the Swamp Gum (Eucalyptus camphora). This tree species is found in the Yellingbo Nature Conservation Reserve (YNCR), an isolated patch of forest in the Yarra Valley (Victoria, Australia), where E. camphora has become increasingly threatened by dieback. In order to maintain and rehabilitate existing trees and encourage regeneration, management strategies and action plans have concentrated on restoring the hydrological regime, which has been altered due to agricultural activities within the catchment. However, research suggests that nutrient enrichment from surrounding horticulture and livestock is having a greater impact on the health of the trees. A Bayesian network model has been developed for E. camphora and used to explore the differences between these two hypotheses. Model outputs suggest that the influencing factors of E. camphora condition are (a) spatially specific and (b) differ according to the group conducting the study in the YNCR. Given the poor quality of data and knowledge available, further research is required to identify the causal factors of dieback. The model offers a framework to guide future integrative and iterative monitoring and research in the YNCR.     

Spatial analyses of groundwater level differences using geostatistical modeling

The purpose of this study was to determine and evaluate the spatial changes in the depletion of groundwater level differences by using geostatistical methods based on data from 58 groundwater wells during the period from April 1999 to April 2008 in the study area. Geostatistical methods have been used widely as a convenient tool to make decision on the management of groundwater levels. To evaluate the spatial changes in the level of the groundwater, geographic information system is used for the application of universal kriging method with cross-validation leading to the estimation of groundwater levels. The resulting prediction mappings identify the locations of groundwater level fluctuations of the study area. The average range of variogram (spherical model) for the spatial analysis is about 9,200 m. Results of universal kriging for groundwater level differences drops were underestimated by 15 %. Cross-validation errors are within an acceptable level. The maps show that this area of high decrease of groundwater level is located at the southwest. Kriging model helps also to detect sensitively risk prone areas for groundwater withdrawing. Such areas must be protected with an effective management procedure for future groundwater exploitations.     

Bayesian estimation of species relative abundances and habitat preferences using opportunistic data

We develop a new statistical procedure to monitor relative species abundances and their respective preferences for different habitat types, using opportunistic data. Following Giraud et al. (Biometrics 72(2):649–658, 2015), we combine the opportunistic data with some standardized data in order to correct the bias inherent to the opportunistic data collection. Species observations are modeled by Poisson distributions whose parameters quantify species abundances and habitat preferences, and are estimated using Bayesian computations. Our main contributions are (i) to tackle the bias induced by habitat selection behaviors, (ii) to handle data where the habitat type associated to each observation is unknown, (iii) to estimate probabilities of selection of habitat for the species. As an illustration, we estimate common bird species habitat preferences and abundances in the region of Aquitaine (France).