Space-time Bayesian small area disease risk models: development and evaluation with a focus on cluster detection

This paper extends the spatial local-likelihood model and the spatial mixture model to the space-time (ST) domain. For comparison, a standard random effect space-time (SREST) model is examined to allow evaluation of each model’s ability in relation to cluster detection. To pursue this evaluation, we use the ST counterparts of spatial cluster detection diagnostics. The proposed criteria are based on posterior estimates (e.g., misclassification rate) and some are based on post-hoc analysis of posterior samples (e.g., exceedance probability). In addition, we examine more conventional model fit criteria including mean square error (MSE). We illustrate the methodology with a real ST dataset, Georgia throat cancer mortality data for the years 1994–2005, and a simulated dataset where different levels and shapes of clusters are embedded. Overall, it is found that conventional SREST models fair well in ST cluster detection and in goodness-of-fit, while for extreme risk detection the local likelihood ST model does best.     

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.     

Isomorphic chain graphs for modeling spatial dependence in ecological data

Graphical models (alternatively, Bayesian belief networks, path analysis models) are increasingly used for modeling complex ecological systems (e.g., Lee, In: Ferson S, Burgman M(eds) Quantative methods for conservation biology. Springer, Berlin Heilin Heideslperk New York, pp.127–147, 2000; Borsuk et al., J Water Res Plann Manage 129:271–282, 2003). Their implementation in this context leverages their utility in modeling interrelationships in multivariate systems, and in a Bayesian implementation, their intuitive appeal of yielding easily interpretable posterior probability estimates. However, methods for incorporating correlational structure to account for observations collected through time and/or space—features of most ecological data—have not been widely studied; Haas et al. (AI Appl 8:15–27, 1994) is one exception. In this paper, an “isomorphic” chain graph (ICG) model is introduced to account for correlation between samples by linking site-specific Bayes network models. Several results show that the ICG preserves many of the Markov properties (conditional and marginal dependencies) of the site-specific models. The ICG model is compared with a model that does not account for spatial correlation. Data from several stream networks in the Willamette River valley, Oregon (USA) are used. Significant correlation between sites within the same stream network is shown with an ICG model.     

Accounting for rate instability and spatial patterns in the boundary analysis of cancer mortality maps

Boundary analysis of cancer maps may highlight areas where causative exposures change through geographic space, the presence of local populations with distinct cancer incidences, or the impact of different cancer control methods. Too often, such analysis ignores the spatial pattern of incidence or mortality rates and overlooks the fact that rates computed from sparsely populated geographic entities can be very unreliable. This paper proposes a new methodology that accounts for the uncertainty and spatial correlation of rate data in the detection of significant edges between adjacent entities or polygons. Poisson kriging is first used to estimate the risk value and the associated standard error within each polygon, accounting for the population size and the risk semivariogram computed from raw rates. The boundary statistic is then defined as half the absolute difference between kriged risks. Its reference distribution, under the null hypothesis of no boundary, is derived through the generation of multiple realizations of the spatial distribution of cancer risk values. This paper presents three types of neutral models generated using methods of increasing complexity: the common random shuffle of estimated risk values, a spatial re-ordering of these risks, or p-field simulation that accounts for the population size within each polygon. The approach is illustrated using age-adjusted pancreatic cancer mortality rates for white females in 295 US counties of the Northeast (1970–1994). Simulation studies demonstrate that Poisson kriging yields more accurate estimates of the cancer risk and how its value changes between polygons (i.e., boundary statistic), relatively to the use of raw rates or local empirical Bayes smoother. When used in conjunction with spatial neutral models generated by p-field simulation, the boundary analysis based on Poisson kriging estimates minimizes the proportion of type I errors (i.e., edges wrongly declared significant) while the frequency of these errors is predicted well by the p-value of the statistical test.

Prediction of potential areas of species distributions based on presence-only data

We introduce a methodology to infer zones of high potential for the habitat of a species, useful for management of biodiversity, conservation, biogeography, ecology, or sustainable use. Inference is based on a set of sites where the presence of the species has been reported. Each site is associated with covariate values, measured on discrete scales. We compute the predictive probability that the species is present at each node of a regular grid. Possible spatial bias for sites of presence is accounted for. Since the resulting posterior distribution does not have a closed form, a Markov chain Monte Carlo (MCMC) algorithm is implemented. However, we also describe an approximation to the posterior distribution, which avoids MCMC. Relevant features of the approach are that specific notions of data acquisition such as sampling intensity and detectability are accounted for, and that available a priori information regarding areas of distribution of the species is incorporated in a clear-cut way. These concepts, arising in the presence-only context, are not addressed in alternative methods. We also consider an uncertainty map, which measures the variability for the predictive probability at each node on the grid. A simulation study is carried out to test and compare our approach with other standard methods. Two case studies are also presented.     

Use of Spatial Capture‐Recapture Modeling and DNA Data to Estimate Densities of Elusive Animals

Abstract: Assessment of abundance, survival, recruitment rates, and density (i.e., population assessment) is especially challenging for elusive species most in need of protection (e.g., rare carnivores). Individual identification methods, such as DNA sampling, provide ways of studying such species efficiently and noninvasively. Additionally, statistical methods that correct for undetected animals and account for locations where animals are captured are available to efficiently estimate density and other demographic parameters. We collected hair samples of European wildcat (Felis silvestris) from cheek‐rub lure sticks, extracted DNA from the samples, and identified each animals’ genotype. To estimate the density of wildcats, we used Bayesian inference in a spatial capture‐recapture model. We used WinBUGS to fit a model that accounted for differences in detection probability among individuals and seasons and between two lure arrays. We detected 21 individual wildcats (including possible hybrids) 47 times. Wildcat density was estimated at 0.29/km² (SE 0.06), and 95% of the activity of wildcats was estimated to occur within 1.83 km from their home‐range center. Lures located systematically were associated with a greater number of detections than lures placed in a cell on the basis of expert opinion. Detection probability of individual cats was greatest in late March. Our model is a generalized linear mixed model; hence, it can be easily extended, for instance, to incorporate trap‐ and individual‐level covariates. We believe that the combined use of noninvasive sampling techniques and spatial capture‐recapture models will improve population assessments, especially for rare and elusive animals.     

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.     

Some Interpolation Estimators in Environmental Risk Assessment for Spatially Misaligned Health Data

Ecological regression studies are widely used in geographical epidemiology to assess the relationships between health hazard and putative risk factors. Very often, health data are measured at an aggregate level because of confidentiality restrictions, while putative risk factors are measured on a different grid, i.e., independent (exposure) variable and response (counts) variable are spatially misaligned. To perform a regression of risk on exposure, one needs to realign the spatial support of the variables. Bayesian hierarchical models constitute a natural approach to the problem because of their ability to model the exposure field and the relationship between exposure and relative risk at different levels of the hierarchy, taking proper account of the variability induced by the covariate estimation. In the current paper, we propose two fully Bayesian solutions to the problem. The first one is based on the kernel-smoothing technique, while the second one is built on the tessellation of the study region. We illustrate our methods by assessing the relationship between exposure to uranium in drinkable waters and cancer incidence, in South Carolina (USA).     

Multinomial mixture model with heterogeneous classification probabilities

Royle and Link (Ecology 86(9):2505?C2512, 2005) proposed an analytical method that allowed estimation of multinomial distribution parameters and classification probabilities from categorical data measured with error. While useful, we demonstrate algebraically and by simulations that this method yields biased multinomial parameter estimates when the probabilities of correct category classifications vary among sampling units. We address this shortcoming by treating these probabilities as logit-normal random variables within a Bayesian framework. We use Markov chain Monte Carlo to compute Bayes estimates from a simulated sample from the posterior distribution. Based on simulations, this elaborated Royle-Link model yields nearly unbiased estimates of multinomial and correct classification probability estimates when classification probabilities are allowed to vary according to the normal distribution on the logit scale or according to the Beta distribution. The method is illustrated using categorical submersed aquatic vegetation data.     

Toward accurate and precise estimates of lion density

Reliable estimates of animal density are fundamental to understanding ecological processes and population dynamics. Furthermore, their accuracy is vital to conservation because wildlife authorities rely on estimates to make decisions. However, it is notoriously difficult to accurately estimate density for wide‐ranging carnivores that occur at low densities. In recent years, significant progress has been made in density estimation of Asian carnivores, but the methods have not been widely adapted to African carnivores, such as lions (Panthera leo). Although abundance indices for lions may produce poor inferences, they continue to be used to estimate density and inform management and policy. We used sighting data from a 3‐month survey and adapted a Bayesian spatially explicit capture‐recapture (SECR) model to estimate spatial lion density in the Maasai Mara National Reserve and surrounding conservancies in Kenya. Our unstructured spatial capture‐recapture sampling design incorporated search effort to explicitly estimate detection probability and density on a fine spatial scale, making our approach robust in the context of varying detection probabilities. Overall posterior mean lion density was estimated to be 17.08 (posterior SD 1.310) lions >1 year old/100 km², and the sex ratio was estimated at 2.2 females to 1 male. Our modeling framework and narrow posterior SD demonstrate that SECR methods can produce statistically rigorous and precise estimates of population parameters, and we argue that they should be favored over less reliable abundance indices. Furthermore, our approach is flexible enough to incorporate different data types, which enables robust population estimates over relatively short survey periods in a variety of systems. Trend analyses are essential to guide conservation decisions but are frequently based on surveys of differing reliability. We therefore call for a unified framework to assess lion numbers in key populations to improve management and policy decisions.     

Modeling and spatial prediction of pre-settlement patterns of forest distribution using witness tree data

At the time of European settlement, land surveys were conducted progressively westward throughout the United States. Outside of the original 13 colonies, surveys generally followed the Public Land Survey system in which trees, called witness trees, were regularly recorded at 1 mi by 1 mi grid intersections. This unintentional sampling provides insight into the composition and structure of pre-European settlement forests, which is used as baseline data to assess forest change following settlement. In this paper, a model for the Public Land Surveys of east central Alabama is developed. Assuming that the locations of trees of each species are realized from independent Poisson processes whose respective log intensities are linear functions of environmental covariates (i.e., elevation, landform, and physiographic province), the species observed at the survey grid intersections are independently sampled from a generalized logistic regression model. If all 68 species found in the survey were included, the model would be highly over-parameterized, so only the distribution of the most common taxon, pines, will be considered at this time. To assess the impact of environmental factors not included in the model, a hidden Gaussian random field shall be added as a random effect. A Markov Chain Monte Carlo algorithm is developed for Bayesian inference on model parameters, and for Bayes posterior prediction of the spatial distribution of pines in east central Alabama. Received: June 2004 / Revised: November 2004     

Bayesian spatial prediction

This paper presents a complete Bayesian methodology for analyzing spatial data, one which employs proper priors and features diagnostic methods in the Bayesian spatial setting. The spatial covariance structure is modeled using a rich class of covariance functions for Gaussian random fields. A general class of priors for trend, scale, and structural covariance parameters is considered. In particular, we obtain analytic results that allow easy computation of the predictive distribution for an arbitrary prior on the parameters of the covariance function using importance sampling. The computations, as well as model diagnostics and sensitivity analysis, are illustrated with a set of precipitation data.     

Sampling design trade-offs in occupancy studies with imperfect detection: examples and software.

Researchers have used occupancy, or probability of occupancy, as a response or state variable in a variety of studies (e.g., habitat modeling), and occupancy is increasingly favored by numerous state, federal, and international agencies engaged in monitoring programs. Recent advances in estimation methods have emphasized that reliable inferences can be made from these types of studies if detection and occupancy probabilities are simultaneously estimated. The need for temporal replication at sampled sites to estimate detection probability creates a trade-off between spatial replication (number of sample sites distributed within the area of interest/inference) and temporal replication (number of repeated surveys at each site). Here, we discuss a suite of questions commonly encountered during the design phase of occupancy studies, and we describe software (program GENPRES) developed to allow investigators to easily explore design trade-offs focused on particularities of their study system and sampling limitations. We illustrate the utility of program GENPRES using an amphibian example from Greater Yellowstone National Park, U.S.A.     

A Bayesian approach to line-transect analysis for estimating abundance

Line-transect analysis is a widely used method of estimating plant and animal density and abundance. A Bayesian approach to a basic line-transect analysis is developed for a half-normal detection function. We extend the model of Karunamuni and Quinn [Karunamuni, R.J., Quinn II, T.J., 1995. Bayesian estimation of animal abundance for line-transect sampling. Biometrics 51, 1325–1337] by including a binomial likelihood function for the number of objects detected. The method computes a joint posterior distribution on the effective strip width and the density of objects in the sampled area. Analytical and computational methods for binned and unbinned perpendicular distance data are provided. Existing information about effective strip width and density can be brought into the analysis via prior distributions. The Bayesian approach is compared to a standard line-transect analysis using both real and simulated data. Results of the Bayesian and non-Bayesian analyses are similar when there are no prior data on effective strip width or density, but the Bayesian approach performs better when such data are available from previous or related studies. Practical methods for including prior data on effective strip width and density are suggested. A numerical example shows how the Bayesian approach can provide valid estimates when the sample size is too small for the standard approach to work reliably. The proposed Bayesian approach can form the basis for developing more advanced analyses.     

Bayesian Methods in Conservation Biology

Abstract: Bayesian statistical inference provides an alternate way to analyze data that is likely to be more appropriate to conservation biology problems than traditional statistical methods. I contrast Bayesian techniques with traditional hypothesis-testing techniques using examples applicable to conservation. I use a trend analysis of two hypothetical populations to illustrate how easy it is to understand Bayesian results, which are given in terms of probability. Bayesian trend analysis indicated that the two populations had very different chances of declining at biologically important rates. For example, the probability that the first population was declining faster than 5% per year was 0.00, compared to a probability of 0.86 for the second population. The Bayesian results appropriately identified which population was of greater conservation concern. The Bayesian results contrast with those obtained with traditional hypothesis testing. Hypothesis testing indicated that the first population, which the Bayesian analysis indicated had no chance of declining at > 5% per year, was declining significantly because it was declining at a slow rate and the abundance estimates were precise. Despite the high probability that the second population was experiencing a serious decline, hypothesis testing failed to reject the null hypothesis of no decline because the abundance estimates were imprecise. Finally, I extended the trend analysis to illustrate Bayesian decision theory, which allows for choice between more than two decisions and allows explicit specification of the consequences of various errors. The Bayesian results again differed from the traditional results: the decision analysis led to the conclusion that the first population was declining slowly and the second population was declining rapidly.     

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.     

An interregional ecological approach for modelling sustainability in a globalizing world—Reviewing existing approaches and emerging directions

In recent decades international trade has become a major source of supplying the need and wants of billions of people around the world. Virtually everyone now consumes resource commodities and manufactured products imported from ‘elsewhere’. In effect, globalization and trade enable consuming populations to support themselves on the output of distant ecosystems half a world away. However, while economic integration implies greater ‘connectivity’ within the global village, the spatial separation of material production (including resource extraction) from consumption eliminates some of the signals i.e., the negative feedbacks coming from supporting eco-systems from reaching those who depend on these ecosystems for their sustainability. At present, despite increasing global connectedness, most environmental studies and models apply to a single spatial scale: local, national or global; analysing diverse pressures on human well-being and ecosystems integrity. This paper argues that both economic globalization and global ecological change should force us to add an interregional scale for quantifying and modelling sustainability. Such an approach recognizes that, in a globalizing world, the sustainability of any given region increasingly depends, directly and indirectly, on the sustainability of many other regions. The following pages describe the interregional approach and illustrate some existing and emerging methods for quantifying, analysing and modelling interregional linkages. It then identifies some of what is still missing, and discusses some of the implications in a changing world.     

Bayesian Ecological Regression with Latent Factors: Atmospheric Pollutants Emissions and Mortality for Lung Cancer

In this work we propose a Bayesian ecological analysis in which a latent variable summarizes data on emissions of atmospheric pollutants. We specified a hierarchical Bayesian model with spatially structured and unstructured random terms with a nested latent factor model. This can be considered a combination of the convolution spatial model of Besag et al. (1991) and an ecological regression analysis in which a latent variable plays the role of the covariate. The unified approach allows to proper account for the uncertainty in the latent score estimation in the regression analysis. The Bayesian Latent Factor model is used to summarize the information on environmental pressure derived from three stressors: Carbon Monoxide, Nitrogen Oxides and Inhalable Particles. We found evidence of positive correlation between Lung cancer mortality and environmental pressure indicators, in males, Tuscany (Italy), 1995–1999. Environmental pressure seems to be restricted to fourteen municipalities (top 5% of the Latent Factor distribution). The model identified two areas with high point source emissions.     

Site‐Occupancy Distribution Modeling to Correct Population‐Trend Estimates Derived from Opportunistic Observations

Abstract: Species’ assessments must frequently be derived from opportunistic observations made by volunteers (i.e., citizen scientists). Interpretation of the resulting data to estimate population trends is plagued with problems, including teasing apart genuine population trends from variations in observation effort. We devised a way to correct for annual variation in effort when estimating trends in occupancy (species distribution) from faunal or floral databases of opportunistic observations. First, for all surveyed sites, detection histories (i.e., strings of detection–nondetection records) are generated. Within‐season replicate surveys provide information on the detectability of an occupied site. Detectability directly represents observation effort; hence, estimating detectablity means correcting for observation effort. Second, site‐occupancy models are applied directly to the detection‐history data set (i.e., without aggregation by site and year) to estimate detectability and species distribution (occupancy, i.e., the true proportion of sites where a species occurs). Site‐occupancy models also provide unbiased estimators of components of distributional change (i.e., colonization and extinction rates). We illustrate our method with data from a large citizen‐science project in Switzerland in which field ornithologists record opportunistic observations. We analyzed data collected on four species: the widespread Kingfisher (Alcedo atthis) and Sparrowhawk (Accipiter nisus) and the scarce Rock Thrush (Monticola saxatilis) and Wallcreeper (Tichodroma muraria). Our method requires that all observed species are recorded. Detectability was <1 and varied over the years. Simulations suggested some robustness, but we advocate recording complete species lists (checklists), rather than recording individual records of single species. The representation of observation effort with its effect on detectability provides a solution to the problem of differences in effort encountered when extracting trend information from haphazard observations. We expect our method is widely applicable for global biodiversity monitoring and modeling of species distributions.     

Neighborhood Level Spatial Analysis of the Relationship Between Alcohol Outlet Density and Criminal Violence

Misuse of alcohol is a significant public health problem, potentially resulting in unintentional injuries, motor vehicle crashes, drownings, and, perhaps of greatest concern, serious acts of violence, including assaults, rapes, suicides, and homicides. Although previous research establishes a link between alcohol consumption increased levels of violence, studies relating the density of alcohol outlets (e.g., restaurants, bars, liquor stores) and the likelihood of violent crime have been less common. In this paper we test for such a relationship at the small area level, using data from 79 neighborhoods in the city of Minneapolis, Minnesota. We adopt a fully Bayesian point of view using Markov chain Monte Carlo (MCMC) computational methods as available in the popular and freely available WinBUGS language. Our models control for important covariates (e.g., neighborhood racial heterogeneity, age heterogeneity) and also account for spatial association in unexplained variability using conditionally autoregressive (CAR) random effects. Our results indicate a significant positive relationship between alcohol outlet density and violent crime, while also permitting easy mapping of neighborhood-level predicted and residual values, the former useful for intervention in the most at-risk neighborhoods and the latter potentially useful in identifying covariates still missing from the fixed effects portion of the model.