Hierarchical models of animal abundance and occurrence

Much of animal ecology is devoted to studies of abundance and occurrence of species, based on surveys of spatially referenced sample units. These surveys frequently yield sparse counts that are contaminated by imperfect detection, making direct inference about abundance or occurrence based on observational data infeasible. This article describes a flexible hierarchical modeling framework for estimation and inference about animal abundance and occurrence from survey data that are subject to imperfect detection. Within this framework, we specify models of abundance and detectability of animals at the level of the local populations defined by the sample units. Information at the level of the local population is aggregated by specifying models that describe variation in abundance and detection among sites. We describe likelihood-based and Bayesian methods for estimation and inference under the resulting hierarchical model. We provide two examples of the application of hierarchical models to animal survey data, the first based on removal counts of stream fish and the second based on avian quadrat counts. For both examples, we provide a Bayesian analysis of the models using the software WinBUGS.     

Imputation Approaches for Animal Movement Modeling

The analysis of telemetry data is common in animal ecological studies. While the collection of telemetry data for individual animals has improved dramatically, the methods to properly account for inherent uncertainties (e.g., measurement error, dependence, barriers to movement) have lagged behind. Still, many new statistical approaches have been developed to infer unknown quantities affecting animal movement or predict movement based on telemetry data. Hierarchical statistical models are useful to account for some of the aforementioned uncertainties, as well as provide population-level inference, but they often come with an increased computational burden. For certain types of statistical models, it is straightforward to provide inference if the latent true animal trajectory is known, but challenging otherwise. In these cases, approaches related to multiple imputation have been employed to account for the uncertainty associated with our knowledge of the latent trajectory. Despite the increasing use of imputation approaches for modeling animal movement, the general sensitivity and accuracy of these methods have not been explored in detail. We provide an introduction to animal movement modeling and describe how imputation approaches may be helpful for certain types of models. We also assess the performance of imputation approaches in two simulation studies. Our simulation studies suggests that inference for model parameters directly related to the location of an individual may be more accurate than inference for parameters associated with higher-order processes such as velocity or acceleration. Finally, we apply these methods to analyze a telemetry data set involving northern fur seals (Callorhinus ursinus) in the Bering Sea. Supplementary materials accompanying this paper appear online.     

Marking springtails (Folsomia candida) with rubidium

Marking springtails is a basic tool to evaluate their fundamental ecological phenomena. Rb marking is based on the fact that enriched rubidium in an organism can be tracked trough the experiment. Our goal was to improve the rubidium-marking technique in Folsomia candida (Willem) for both microcosm and field experiments. We investigated four methodological problems of this technique, in particular, we determined the required Rb concentration in the diet to reach marking level, measured the period when labeling could be detected under two different feeding conditions, and we estimated the effects of Rb on springtails' growth. Because marked and unmarked animals are always mixed in the course of recapture we also measured the levels of contamination between labeled springtails and those in the control groups. For introducing rubidium, we fed animals with Rb-treated Baker's yeast. Rubidium-chloride labeling persisted in springtails for 27 days during which the Rb-levels in marked animals remained distinguishable from those in unmarked ones. Rb-elimination rate depended highly on the feeding conditions, with Rb-elimination being faster when food was in excess. The fitted exponential model to Rb-elimination suggested that Rb-labeling may be used for 46 and 103 days for experiments with and without food respectively. We found no effect on Collembola growth at low Rb-levels (1.2 μg Rb/g dry yeast) but at higher concentration growth was reduced. We found that contamination occurred when springtails were stored together in glycerin, however the unmarked sample with the highest Rb content was still just 4.8% of the lowest marked sample. These results provide a basis for mark-release-recapture and other studies using Rb marking on springtails.     

Generalized Linear Latent Variable Models for Multivariate Count and Biomass Data in Ecology

In this paper we consider generalized linear latent variable models that can handle overdispersed counts and continuous but non-negative data. Such data are common in ecological studies when modelling multivariate abundances or biomass. By extending the standard generalized linear modelling framework to include latent variables, we can account for any covariation between species not accounted for by the predictors, notably species interactions and correlations driven by missing covariates. We show how estimation and inference for the considered models can be performed efficiently using the Laplace approximation method and use simulations to study the finite-sample properties of the resulting estimates. In the overdispersed count data case, the Laplace-approximated estimates perform similarly to the estimates based on variational approximation method, which is another method that provides a closed form approximation of the likelihood. In the biomass data case, we show that ignoring the correlation between taxa affects the regression estimates unfavourably. To illustrate how our methods can be used in unconstrained ordination and in making inference on environmental variables, we apply them to two ecological datasets: abundances of bacterial species in three arctic locations in Europe and abundances of coral reef species in Indonesia.Supplementary materials accompanying this paper appear on-line.     

Age Specificity in Conditional Ring-Recovery Models

We consider the case of age-specific ring-recovery data obtained only from recovered individual birds and modelled by conditioning a multinomial distribution on the recovery. These models may be appealing when the information about the numbers of marked individuals is missing, but they have previously been analyzed by ignoring a large set of nuisance parameters, the recovery probabilities. We investigate the consequences of this conditioning by relating the age-time specific structure of recovery probabilities to the estimation of survival.     

Nonparametric estimation of ROC curves based on Bayesian models when the true disease state is unknown

We develop a Bayesian methodology for nonparametric estimation of ROC curves used for evaluation of the accuracy of a diagnostic procedure. We consider the situation where there is no perfect reference test, that is, no “gold standard”. The method is based on a multinomial model for the joint distribution of test-positive and test-negative observations. We use a Bayesian approach which assures the natural monotonicity property of the resulting ROC curve estimate. MCMC methods are used to compute the posterior estimates of the sensitivities and specificities that provide the basis for inference concerning the accuracy of the diagnostic procedure. Because there is no gold standard, identifiability requires that the data come from at least two populations with different prevalences. No assumption is needed concerning the shape of the distributions of test values of the diseased and non diseased in these populations. We discuss an application to an analysis of ELISA scores in the diagnostic testing of paratuberculosis (Johne’s Disease) for several herds of dairy cows and compare the results to those obtained from some previously proposed methods.     

Inference on Partially Observed Quasi-stationary Markov Chains With Applications to Multistate Population Models

If the full capture histories of captured individuals are available, inferences on multistate open population models may be conducted using the well known Arnason–Schwarz model. However, data of this detail is not always available. It is well known that inference on the transition probabilities of a Markov chain may be conducted using aggregate data and we extend this approach to aggregate data on multistate open population models. We show that for parameters to be identifiable we need to augment the aggregate data and we achieve this by batch marking a cohort of individuals according to their initial state, so that the batch marking augments the aggregate data. Model performance is examined by conducting several simulation studies and the model is applied to a real data set where full capture histories are available so it may be compared with the Arnason–Schwarz estimates. This article has supplementary material online.     

Comparison of Models Analyzing a Small Number of Observed Meningitis Cases in Navrongo,Ghana

The “meningitis belt” is a region in sub-Saharan Africa where annual outbreaks of meningitis occur, with epidemics observed cyclically. While we know that meningitis is heavily dependent on seasonal trends, the exact pathways for contracting the disease are not fully understood and warrant further investigation. Most previous approaches have used large sample inference to assess impacts of weather on meningitis rates. However, in the case of rare events, the validity of such assumptions is uncertain. This work examines the meningitis trends in the context of rare events, with the specific objective of quantifying the underlying seasonal patterns in meningitis rates. We compare three main classes of models: the Poisson generalized linear model, the Poisson generalized additive model, and a Bayesian hazard model extended to accommodate count data and a changing at-risk population. We compare the accuracy and robustness of the models through the bias, RMSE, and standard deviation of the estimators, and also provide a detailed case study of meningitis patterns for data collected in Navrongo, Ghana.Supplementary materials accompanying this paper appear online.     

Spatial stochastic volatility for lattice data

Spatial heteroscedasticity may arise jointly with spatial autocorrelation in lattice data collected from agricultural trials and environmental studies. This leads to spatial clustering not only in the level but also in the variation of the data, the latter of which may be very important, for example, in constructing prediction intervals. This article introduces a spatial stochastic volatility (SSV) component into the widely used conditional autoregressive (CAR) model to capture the spatial clustering in heteroscedasticity. The SSV component is a mean zero, conditionally independent Gaussian process given a latent spatial process of the variances. The logarithm of the latent variance process is specified by an intrinsic Gaussian Markov random field. The SSV model relaxes the traditional homoscedasticity assumption for spatial heterogeneity and brings greater flexibility to the popular spatial statistical models. The Bayesian method is used for inference. The full conditional distribution of the heteroscedasticity components can be shown to be log-concave, which facilitates an adaptive rejection sampling algorithm. Application to the well-known wheat yield data illustrates that incorporating spatial stochastic volatility may reveal the spatial heteroscedasticity hidden from existing analyses.     

Mark-recapture with occasion and individual effects: Abundance estimation through Bayesian model selection in a fixed dimensional parameter space

We present a Bayesian mark-recapture method for explicitly communicating uncertainty about the size of a closed population where capture probabilities vary across both individuals and sampling occasions. Heterogeneity is modeled hierarchically using a continuous logistic-Normal model to specify the capture probabilities for both individuals that are captured on at least one occasion and individuals that are never captured and so remain undetected. Inference about how many undetected individuals to include in the model is accomplished through a Bayesian model selection procedure using MCMC, applied to a product space of possible models for different numbers of undetected individuals. Setting the estimation problem in a fixed dimensional parameter space enables the model selection procedure to be performed using the freely available WinBUGS software. The outcome of inference is a full “posterior” probability distribution for the population size parameter. We demonstrate this method through an example involving real mark-recapture data.     

Estimating survival from joint analysis of resighting and radiotelemetry capture-recapture data for wild animals

Biologists often use more than one marking technique in wildlife studies. For each of the mark types, it is common to conduct a separate analysis of the recapture data to estimate parameters of interest, such as survival rates. Two data types that can be used in estimating survival rates are resighting and radiotelemetry data. The Cormack-Jolly-Seber model is commonly used to analyze the resighting data, while the Kaplan-Meier product limit estimator, modified for staggered entry of animals, is used to analyze the radi otelemetry data. In a study where some animals receive two types of tags and others receive just one tag type, the separate Cormack-Jolly-Seber and Kaplan-Meier analyses do not exploit all of the information in the combined data sets. In this article, we propose a model and likelihood for the combined analysis of resighting and radi otelemetry data. In comparison with the separate analyses, this richer model provides more information about the biology and sampling processes. For example, the richer model permits assessment of assumptions required by the separate analyses and allows estimation of additional parameters. We apply the model to annual resighting and monthly telemetry data from a population of snail kites in Florida. The snail kite is a threatened species of bird in the United States, and our results on survival are very important. In this example, all birds are marked using leg bands and some of them receive radios.     

Bayesian Inference for Multistate ‘Step and Turn’ Animal Movement in Continuous Time

Mechanistic modelling of animal movement is often formulated in discrete time despite problems with scale invariance, such as handling irregularly timed observations. A natural solution is to formulate in continuous time, yet uptake of this has been slow. This lack of implementation is often excused by a difficulty in interpretation. Here we aim to bolster usage by developing a continuous-time model with interpretable parameters, similar to those of popular discrete-time models that use turning angles and step lengths. Movement is defined by a joint bearing and speed process, with parameters dependent on a continuous-time behavioural switching process, creating a flexible class of movement models. Methodology is presented for Markov chain Monte Carlo inference given irregular observations, involving augmenting observed locations with a reconstruction of the underlying movement process. This is applied to well-known GPS data from elk (Cervus elaphus), which have previously been modelled in discrete time. We demonstrate the interpretable nature of the continuous-time model, finding clear differences in behaviour over time and insights into short-term behaviour that could not have been obtained in discrete time.     

Bayesian Spatial Point Process Modeling of Line Transect Data

This paper develops a Bayesian approach for spatial inference on animal density from line transect survey data. We model the spatial distribution of animals within a geographical area of interest by an inhomogeneous Poisson process whose intensity function incorporates both covariate effects and spatial smoothing of residual variation. Independently thinning the animal locations according to their estimated detection probabilities results into another spatial Poisson process for the sightings (the observations). Prior distributions are elicited for all unknown model parameters. Due to the sparsity of data in the application we consider, eliciting sensible prior distributions is important in order to get meaningful estimation results. A reversible jump Markov Chain Monte Carlo (MCMC) algorithm for simulation of the posterior distribution is developed. We present results for simulated data and a real data set of minke whale pods from Antarctic waters. The main advantages of our method compared to design-based analyses are that it can use data arising from sources other than specifically designed surveys and its ability to link covariate effects to variation of animal density. The Bayesian paradigm provides a coherent framework for quantifying uncertainty in estimation results.     

Life-history and ecological correlates of population change in Dutch breeding birds

Predicting relative extinction risks of animals has become a major challenge in conservation biology. Identifying life-history and ecological traits related to the decline of species helps understand what causes population decreases and sets priorities for conservation action. Here, we use Dutch breeding bird data to correlate species characteristics with national population changes. We modelled population changes between 1990 and 2005 of all 170 breeding bird species using 25 life-history, ecological and behavioural traits as explanatory variables. We used multiple regression and multi-model inference to account for intercorrelated variables, to assess the relative importance of traits that best explain interspecific differences in population trend, and to identify the environmental changes most likely responsible. We found that more breeding birds have increased than decreased in number. The most parsimonious models suggest that ground-nesting and late arrival at the breeding grounds in migratory birds are most strongly correlated with decline. Increasing populations are mainly found among herbivores, sedentary and short-distance migrants, herb- and shrub-nesting birds and large species with a small European range. Declines in ground-nesting and late arriving migrant birds suggest that agricultural intensification, eutrophication and climate change are most likely responsible for changes in Dutch breeding bird diversity. We illustrate that management strategies should primarily focus on the traits and causes responsible for the population changes, in order to be effective and sustainable.     

Continuous-Time Proportional Hazards Regression for Ecological Monitoring Data

We consider a continuous-time proportional hazards model for the analysis of ecological monitoring data where subjects are monitored at discrete times and fixed sites across space. Since the exact time of event occurrence is not directly observed, we rely on dichotomous event indicators observed at monitoring times to make inference about the model parameters. We use autoregression on the response at neighboring sites from a previous time point to take into account spatial dependence. The interesting fact is utilized that the probability of observing an event at a monitoring time when the underlying hazards is proportional falls under the class of generalized linear models with binary responses and complementary log-log link functions. Thus, a maximum likelihood approach can be taken for inference and the computation can be carried out using standard statistical software packages. This approach has significant computational advantages over some of the existing methods that rely on Monte Carlo simulations. Simulation experiments are conducted and demonstrate that our method has sound finite-sample properties. A real dataset from an ecological study that monitored bark beetle colonization of red pines in Wisconsin is analyzed using the proposed models and inference. Supplementary materials that contain technical details are available online.     

Maximum likelihood inference for seed and pollen dispersal distributions

Seed and pollen dispersal are both important factors in the demography and population genetic structure of plant populations. How does one model and infer dispersal patterns? One approach is to map the locations of individuals in a population and use genetic information to suggest which parents generated which offspring. This article develops models and a maximum likelihood inference framework for data of this type. The procedure will be illustrated on data from a population of Chamaelirium luteum, an herbacious plant of the forest floor. This article shows how the proposed method avoids some of the problems found in the original analysis of these data. The approach also allows us to uncover some additional patterns in the data: differencesin the seed dispersal distributions between years.     

Reflected Stochastic Differential Equation Models for Constrained Animal Movement

Movement for many animal species is constrained in space by barriers such as rivers, shorelines, or impassable cliffs. We develop an approach for modeling animal movement constrained in space by considering a class of constrained stochastic processes, reflected stochastic differential equations. Our approach generalizes existing methods for modeling unconstrained animal movement. We present methods for simulation and inference based on augmenting the constrained movement path with a latent unconstrained path and illustrate this augmentation with a simulation example and an analysis of telemetry data from a Steller sea lion (Eumatopias jubatus) in southeast Alaska.     

A Capture–Recapture Model with Double-Marking,Live and Dead Encounters,and Heterogeneity of Reporting Due to Auxiliary Mark Loss

Capture–recapture (CR) models assume marked individuals remain at risk of capture, which may not be true if individuals lose their mark or emigrate definitively from the study area. Using a double-marking protocol, with a main and auxiliary mark, and both live encounters and dead recoveries at a large scale, partially frees CR models from this assumption. However, the auxiliary mark may fall off and its presence is often not mentioned when dead individuals are reported. We propose a new model to deal with heterogeneity of detection and uncertainty of the presence of an auxiliary mark in a multi-event framework. Our general model, based on a double-marking protocol, uses information from physical captures/recaptures, distant observations and main mark recoveries from dead animals. We applied our model to a 13-year data set of a harvested species, the Greater Snow Goose. We obtained seasonal survival estimates for adults of both sexes. Survival estimates differed between models where the presence of the auxiliary mark upon recovery was ignored versus those where the presence was accounted for. In the multi-event framework, seasonal survival estimates are no longer biased because the heterogeneity due to the presence of an auxiliary mark is accounted for in the estimation of recovery rates.     

D-optimal designs for mixed discrete and continuous outcomes analyzed using nonlinear models

Many dose-response experiments in toxicology and other biological sciences are designed to measure multiple outcomes. Unfortunately, most of these studies are powered or designed for a single response, and the inference on the under-powered endpoints is limited. As additional design challenges, the outcomes may have different regions and shapes of activity or have different response types. As a new application to the traditional D-optimality criterion, we have developed optimal designs for mixed discrete and continuous outcomes that are analyzed with nonlinear models. These designs use a numerical algorithm to choose the location of the dose groups and proportion of total sample size allocated to each group that minimize the generalized variance of a model-based covariance matrix that incorporates the correlation between outcomes. Using this methodology, we designed a dose-response experiment with binary, count, and continuous outcomes to evaluate neurotoxicity. In this example, the optimal designs placed dose groups at the predicted dose thresholds and throughout the active range. The designs were generally robust to different correlation structures. In addition, when the expected correlation was moderate or large, we observed a substantial gain in efficiency compared to optimal designs created for each outcome separately.