Dynamic occupancy models for analyzing species' range dynamics across large geographic scales

Large‐scale biodiversity data are needed to predict species' responses to global change and to address basic questions in macroecology. While such data are increasingly becoming available, their analysis is challenging because of the typically large heterogeneity in spatial sampling intensity and the need to account for observation processes. Two further challenges are accounting for spatial effects that are not explained by covariates, and drawing inference on dynamics at these large spatial scales. We developed dynamic occupancy models to analyze large‐scale atlas data. In addition to occupancy, these models estimate local colonization and persistence probabilities. We accounted for spatial autocorrelation using conditional autoregressive models and autologistic models. We fitted the models to detection/nondetection data collected on a quarter‐degree grid across southern Africa during two atlas projects, using the hadeda ibis (Bostrychia hagedash) as an example. The model accurately reproduced the range expansion between the first (SABAP1: 1987–1992) and second (SABAP2: 2007–2012) Southern African Bird Atlas Project into the drier parts of interior South Africa. Grid cells occupied during SABAP1 generally remained occupied, but colonization of unoccupied grid cells was strongly dependent on the number of occupied grid cells in the neighborhood. The detection probability strongly varied across space due to variation in effort, observer identity, seasonality, and unexplained spatial effects. We present a flexible hierarchical approach for analyzing grid‐based atlas data using dynamical occupancy models. Our model is similar to a species' distribution model obtained using generalized additive models but has a number of advantages. Our model accounts for the heterogeneous sampling process, spatial correlation, and perhaps most importantly, allows us to examine dynamic aspects of species ranges.     

Safe space in the woods: Mechanistic spatial models for predicting risks of human–bear conflicts in India

Human–wildlife interactions can have negative consequences when they involve large carnivores. Spatial risk modelling could serve as a useful management approach for predicting and pre-emptively mitigating negative interactions. We present a mechanistic modelling framework and examine interactions between humans and sloth bears (Melursus ursinus) in a multi-use forest landscape of central India. We first assessed patterns and determinants of bear distribution across the landscape using indirect sign surveys. At the same spatial scale, we then estimated spatial probabilities of bear attacks on people using information from 675 interviews with local residents, incorporating estimates of distribution probabilities from the previous step. We found the average occupancy probability across 128 grid-cells to be 0.77 (SE = 0.03). Bear occupancy was influenced by terrain ruggedness, forest composition and configuration, vegetation productivity and size of human settlements. The average probability of a bear attack in any given grid-cell was 0.61 (SE = 0.03), mostly determined by bear occurrence patterns, forest cover, terrain ruggedness, and size of human settlements. Using spatial information on people's dependence on forest resources, we identified locations with the highest risk of bear attacks. Our study demonstrates that human attacks by bears—generally believed to be random or incidental—in fact showed deterministic patterns. Our framework can be applied to other scenarios involving human–wildlife conflicts. Based on our findings, we propose that a proactive co-management approach which involves collaboration between wildlife managers and local residents could help better manage human–bear conflicts in central India and elsewhere across the species' range.     

Occupancy modelling as a new approach to assess supranational trends using opportunistic data: a pilot study for the damselfly Calopteryx splendens

There is limited information available on changes in biodiversity at the European scale, because there is a lack of data from standardised monitoring for most species groups. However, a great number of observations made without a standardised field protocol is available in many countries for many species. Such opportunistic data offer an alternative source of information, but unfortunately such data suffer from non-standardised observation effort and geographical bias. Here we describe a new approach to compiling supranational trends using opportunistic data which adjusts for these two major imperfections. The non-standardised observation effort is dealt with by occupancy modelling, and the unequal geographical distribution of sites by a weighting procedure. The damselfly Calopteryx splendens was chosen as our test species. The data were collected from five countries (Ireland, Great Britain, the Netherlands, Belgium and France), covering the period 1990–2008. We used occupancy models to estimate the annual number of occupied 1 × 1 km sites per country. Occupancy models use presence-absence data, account for imperfect detection of species, and thereby correct for between-year variability in observation effort. The occupancy models were run per country in a Bayesian mode of inference using JAGS. The occupancy estimates per country were then aggregated to assess the supranational trend in the number of occupied 1 × 1 km². To adjust for the unequal geographical distribution of surveyed sites, we weighted the countries according to the number of sites surveyed and the range of the species per country. The distribution of C. splendens has increased significantly in the combined five countries. Our trial demonstrated that a supranational trend in distribution can be derived from opportunistic data, while adjusting for observation effort and geographical bias. This opens new perspectives for international monitoring of biodiversity.     

Estimating occupancy dynamics for large‐scale monitoring networks: amphibian breeding occupancy across protected areas in the northeast United States

Regional monitoring strategies frequently employ a nested sampling design where a finite set of study areas from throughout a region are selected and intensive sampling occurs within a subset of sites within the individual study areas. This sampling protocol naturally lends itself to a hierarchical analysis to account for dependence among subsamples. Implementing such an analysis using a classic likelihood framework is computationally challenging when accounting for detection errors in species occurrence models. Bayesian methods offer an alternative approach for fitting models that readily allows for spatial structure to be incorporated. We demonstrate a general approach for estimating occupancy when data come from a nested sampling design. We analyzed data from a regional monitoring program of wood frogs (Lithobates sylvaticus) and spotted salamanders (Ambystoma maculatum) in vernal pools using static and dynamic occupancy models. We analyzed observations from 2004 to 2013 that were collected within 14 protected areas located throughout the northeast United States. We use the data set to estimate trends in occupancy at both the regional and individual protected area levels. We show that occupancy at the regional level was relatively stable for both species. However, substantial variation occurred among study areas, with some populations declining and some increasing for both species. In addition, When the hierarchical study design is not accounted for, one would conclude stronger support for latitudinal gradient in trends than when using our approach that accounts for the nested design. In contrast to the model that does not account for nesting, the nested model did not include an effect of latitude in the 95% credible interval. These results shed light on the range‐level population status of these pond‐breeding amphibians, and our approach provides a framework that can be used to examine drivers of local and regional occurrence dynamics.     

Twenty‐five years of change in southern African passerine diversity: nonclimatic factors of change

We analysed more than 25 years of change in passerine bird distribution in South Africa, Swaziland and Lesotho, to show that species distributions can be influenced by processes that are at least in part independent of the local strength and direction of climate change: land use and ecological succession. We used occupancy models that separate species' detection from species' occupancy probability, fitted to citizen science data from both phases of the Southern African Bird Atlas Project (1987–1996 and 2007–2013). Temporal trends in species' occupancy probability were interpreted in terms of local extinction/colonization, and temporal trends in detection probability were interpreted in terms of change in abundance. We found for the first time at this scale that, as predicted in the context of bush encroachment, closed‐savannah specialists increased where open‐savannah specialists decreased. In addition, the trend in the abundance of species a priori thought to be favoured by agricultural conversion was negatively correlated with human population density, which is in line with hypotheses explaining the decline in farmland birds in the Northern Hemisphere. In addition to climate, vegetation cover and the intensity and time since agricultural conversion constitute important predictors of biodiversity changes in the region. Their inclusion will improve the reliability of predictive models of species distribution.     

When mechanism matters: Bayesian forecasting using models of ecological diffusion

Ecological diffusion is a theory that can be used to understand and forecast spatio‐temporal processes such as dispersal, invasion, and the spread of disease. Hierarchical Bayesian modelling provides a framework to make statistical inference and probabilistic forecasts, using mechanistic ecological models. To illustrate, we show how hierarchical Bayesian models of ecological diffusion can be implemented for large data sets that are distributed densely across space and time. The hierarchical Bayesian approach is used to understand and forecast the growth and geographic spread in the prevalence of chronic wasting disease in white‐tailed deer (Odocoileus virginianus). We compare statistical inference and forecasts from our hierarchical Bayesian model to phenomenological regression‐based methods that are commonly used to analyse spatial occurrence data. The mechanistic statistical model based on ecological diffusion led to important ecological insights, obviated a commonly ignored type of collinearity, and was the most accurate method for forecasting.     

Integrating multiple lines of evidence into historical biogeography hypothesis testing: a Bison bison case study

One of the grand goals of historical biogeography is to understand how and why species'' population sizes and distributions change over time. Multiple types of data drawn from disparate fields, combined into a single modelling framework, are necessary to document changes in a species''s demography and distribution, and to determine the drivers responsible for change. Yet truly integrated approaches are challenging and rarely performed. Here, we discuss a modelling framework that integrates spatio-temporal fossil data, ancient DNA, palaeoclimatological reconstructions, bioclimatic envelope modelling and coalescence models in order to statistically test alternative hypotheses of demographic and potential distributional changes for the iconic American bison (Bison bison). Using different assumptions about the evolution of the bioclimatic niche, we generate hypothetical distributional and demographic histories of the species. We then test these demographic models by comparing the genetic signature predicted by serial coalescence against sequence data derived from subfossils and modern populations. Our results supported demographic models that include both climate and human-associated drivers of population declines. This synthetic approach, integrating palaeoclimatology, bioclimatic envelopes, serial coalescence, spatio-temporal fossil data and heterochronous DNA sequences, improves understanding of species'' historical biogeography by allowing consideration of both abiotic and biotic interactions at the population level.     

Accounting for preferential sampling in species distribution models

Species distribution models (SDMs) are now being widely used in ecology for management and conservation purposes across terrestrial, freshwater, and marine realms. The increasing interest in SDMs has drawn the attention of ecologists to spatial models and, in particular, to geostatistical models, which are used to associate observations of species occurrence or abundance with environmental covariates in a finite number of locations in order to predict where (and how much of) a species is likely to be present in unsampled locations. Standard geostatistical methodology assumes that the choice of sampling locations is independent of the values of the variable of interest. However, in natural environments, due to practical limitations related to time and financial constraints, this theoretical assumption is often violated. In fact, data commonly derive from opportunistic sampling (e.g., whale or bird watching), in which observers tend to look for a specific species in areas where they expect to find it. These are examples of what is referred to as preferential sampling, which can lead to biased predictions of the distribution of the species. The aim of this study is to discuss a SDM that addresses this problem and that it is more computationally efficient than existing MCMC methods. From a statistical point of view, we interpret the data as a marked point pattern, where the sampling locations form a point pattern and the measurements taken in those locations (i.e., species abundance or occurrence) are the associated marks. Inference and prediction of species distribution is performed using a Bayesian approach, and integrated nested Laplace approximation (INLA) methodology and software are used for model fitting to minimize the computational burden. We show that abundance is highly overestimated at low abundance locations when preferential sampling effects not accounted for, in both a simulated example and a practical application using fishery data. This highlights that ecologists should be aware of the potential bias resulting from preferential sampling and account for it in a model when a survey is based on non‐randomized and/or non‐systematic sampling.     

Predictor species: Improving assessments of rare species occurrence by modeling environmental co‐responses

Designing an effective conservation strategy requires understanding where rare species are located. Because rare species can be difficult to find, ecologists often identify other species called conservation surrogates that can help inform the distribution of rare species. Species distribution models typically rely on environmental data when predicting the occurrence of species, neglecting the effect of species' co‐occurrences and biotic interactions. Here, we present a new approach that uses Bayesian networks to improve predictions by modeling environmental co‐responses among species. For species from a European peat bog community, our approach consistently performs better than single‐species models and better than conventional multi‐species approaches that include the presence of nontarget species as additional independent variables in regression models. Our approach performs particularly well with rare species and when calibration data are limited. Furthermore, we identify a group of “predictor species” that are relatively common, insensitive to the presence of other species, and can be used to improve occurrence predictions of rare species. Predictor species are distinct from other categories of conservation surrogates such as umbrella or indicator species, which motivates focused data collection of predictor species to enhance conservation practices.     

Evaluating Multi-Level Models to Test Occupancy State Responses of Plethodontid Salamanders

Plethodontid salamanders are diverse and widely distributed taxa and play critical roles in ecosystem processes. Due to salamander use of structurally complex habitats, and because only a portion of a population is available for sampling, evaluation of sampling designs and estimators is critical to provide strong inference about Plethodontid ecology and responses to conservation and management activities. We conducted a simulation study to evaluate the effectiveness of multi-scale and hierarchical single-scale occupancy models in the context of a Before-After Control-Impact (BACI) experimental design with multiple levels of sampling. Also, we fit the hierarchical single-scale model to empirical data collected for Oregon slender and Ensatina salamanders across two years on 66 forest stands in the Cascade Range, Oregon, USA. All models were fit within a Bayesian framework. Estimator precision in both models improved with increasing numbers of primary and secondary sampling units, underscoring the potential gains accrued when adding secondary sampling units. Both models showed evidence of estimator bias at low detection probabilities and low sample sizes; this problem was particularly acute for the multi-scale model. Our results suggested that sufficient sample sizes at both the primary and secondary sampling levels could ameliorate this issue. Empirical data indicated Oregon slender salamander occupancy was associated strongly with the amount of coarse woody debris (posterior mean = 0.74; SD = 0.24); Ensatina occupancy was not associated with amount of coarse woody debris (posterior mean = -0.01; SD = 0.29). Our simulation results indicate that either model is suitable for use in an experimental study of Plethodontid salamanders provided that sample sizes are sufficiently large. However, hierarchical single-scale and multi-scale models describe different processes and estimate different parameters. As a result, we recommend careful consideration of study questions and objectives prior to sampling data and fitting models.     

Bayesian analysis of mass spectrometry proteomic data using wavelet-based functional mixed models

Summary .   In this article, we apply the recently developed Bayesian wavelet-based functional mixed model methodology to analyze MALDI-TOF mass spectrometry proteomic data. By modeling mass spectra as functions, this approach avoids reliance on peak detection methods. The flexibility of this framework in modeling nonparametric fixed and random effect functions enables it to model the effects of multiple factors simultaneously, allowing one to perform inference on multiple factors of interest using the same model fit, while adjusting for clinical or experimental covariates that may affect both the intensities and locations of peaks in the spectra. For example, this provides a straightforward way to account for systematic block and batch effects that characterize these data. From the model output, we identify spectral regions that are differentially expressed across experimental conditions, in a way that takes both statistical and clinical significance into account and controls the Bayesian false discovery rate to a prespecified level. We apply this method to two cancer studies.     

When parks work: Effect of anthropogenic disturbance on occupancy of tropical forest mammals

Protected areas (PAs) in the tropics are vulnerable to human encroachment, and, despite formal protection, they do not fully mitigate anthropogenic threats to habitats and biodiversity. However, attempts to quantify the effectiveness of PAs and to understand the status and changes of wildlife populations in relation to protection efficiency remain limited. Here, we used camera‐trapping data collected over 8 consecutive years (2009–2016) to investigate the yearly occurrences of medium‐to‐large mammals within the Udzungwa Mountains National Park (Tanzania), an area of outstanding importance for biological endemism and conservation. Specifically, we evaluated the effects of habitat and proxies of human disturbance, namely illegal hunting with snares and firewood collection (a practice that was banned in 2011 in the park), on species' occurrence probabilities. Our results showed variability in species' responses to disturbance: The only species that showed a negative effect of the number of snares found on occurrence probability was the Harvey's duiker, a relatively widespread forest antelope. Similarly, we found a moderate positive effect of the firewood collection ban on only the suni, another common antelope, and a negative effect on a large opportunistic rodent, the giant‐pouched rat. Importantly, we found evidence of temporal stability in occurrence probability for all species over the 8‐year study period. Our findings suggest that well‐managed PAs can sustain mammal populations in tropical forests. However, variability among species in their responses to anthropogenic disturbance necessitates consideration in the design of conservation action plans for multiple taxa.     

Incorporating spatial autocorrelation into species distribution models alters forecasts of climate‐mediated range shifts

Species distribution models (SDMs) are widely used to forecast changes in the spatial distributions of species and communities in response to climate change. However, spatial autocorrelation (SA) is rarely accounted for in these models, despite its ubiquity in broad‐scale ecological data. While spatial autocorrelation in model residuals is known to result in biased parameter estimates and the inflation of type I errors, the influence of unmodeled SA on species' range forecasts is poorly understood. Here we quantify how accounting for SA in SDMs influences the magnitude of range shift forecasts produced by SDMs for multiple climate change scenarios. SDMs were fitted to simulated data with a known autocorrelation structure, and to field observations of three mangrove communities from northern Australia displaying strong spatial autocorrelation. Three modeling approaches were implemented: environment‐only models (most frequently applied in species' range forecasts), and two approaches that incorporate SA; autologistic models and residuals autocovariate (RAC) models. Differences in forecasts among modeling approaches and climate scenarios were quantified. While all model predictions at the current time closely matched that of the actual current distribution of the mangrove communities, under the climate change scenarios environment‐only models forecast substantially greater range shifts than models incorporating SA. Furthermore, the magnitude of these differences intensified with increasing increments of climate change across the scenarios. When models do not account for SA, forecasts of species' range shifts indicate more extreme impacts of climate change, compared to models that explicitly account for SA. Therefore, where biological or population processes induce substantial autocorrelation in the distribution of organisms, and this is not modeled, model predictions will be inaccurate. These results have global importance for conservation efforts as inaccurate forecasts lead to ineffective prioritization of conservation activities and potentially to avoidable species extinctions.     

How to deal with ground truthing affected by human‐induced habitat change?: Identifying high‐quality habitats for the Critically Endangered Red Siskin

Species distribution models (SDM) can be valuable for identifying key habitats for conservation management of threatened taxa, but anthropogenic habitat change can undermine SDM accuracy. We used data for the Red Siskin (Spinus cucullatus), a critically endangered bird and ground truthing to examine anthropogenic habitat change as a source of SDM inaccuracy. We aimed to estimate: (1) the Red Siskin's historic distribution in Venezuela; (2) the portion of this historic distribution lost to vegetation degradation; and (3) the location of key habitats or areas with both, a high probability of historic occurrence and a low probability of vegetation degradation. We ground‐truthed 191 locations and used expert opinion as well as landscape characteristics to classify species' habitat suitability as excellent, good, acceptable, or poor. We fit a Random Forest model (RF) and Enhanced Vegetation Index (EVI) time series to evaluate the accuracy and precision of the expert categorization of habitat suitability. We estimated the probability of historic occurrence by fitting a MaxLike model using 88 presence records (1960–2013) and data on forest cover and aridity index. Of the entire study area, 23% (20,696 km²) had a historic probability of Red Siskin occurrence over 0.743. Furthermore, 85% of ground‐truthed locations had substantial reductions in mean EVI, resulting in key habitats totaling just 976 km², in small blocks in the western and central regions. Decline in Area of Occupancy over 15 years was between 40% and 95%, corresponding to an extinction risk category between Vulnerable and Critically Endangered. Relating key habitats with other landscape features revealed significant risks and opportunities for proposed conservation interventions, including the fact that ongoing vegetation degradation could limit the establishment of reintroduced populations in eastern areas, while the conservation of remaining key habitats on private lands could be improved with biodiversity‐friendly agri‐ and silviculture programs.     

Bayesian Inference of Reticulate Phylogenies under the Multispecies Network Coalescent

The multispecies coalescent (MSC) is a statistical framework that models how gene genealogies grow within the branches of a species tree. The field of computational phylogenetics has witnessed an explosion in the development of methods for species tree inference under MSC, owing mainly to the accumulating evidence of incomplete lineage sorting in phylogenomic analyses. However, the evolutionary history of a set of genomes, or species, could be reticulate due to the occurrence of evolutionary processes such as hybridization or horizontal gene transfer. We report on a novel method for Bayesian inference of genome and species phylogenies under the multispecies network coalescent (MSNC). This framework models gene evolution within the branches of a phylogenetic network, thus incorporating reticulate evolutionary processes, such as hybridization, in addition to incomplete lineage sorting. As phylogenetic networks with different numbers of reticulation events correspond to points of different dimensions in the space of models, we devise a reversible-jump Markov chain Monte Carlo (RJMCMC) technique for sampling the posterior distribution of phylogenetic networks under MSNC. We implemented the methods in the publicly available, open-source software package PhyloNet and studied their performance on simulated and biological data. The work extends the reach of Bayesian inference to phylogenetic networks and enables new evolutionary analyses that account for reticulation.     

Estimating breeding season abundance of golden-cheeked warblers in Texas,USA

Population abundance estimates using predictive models are important for describing habitat use and responses to population-level impacts, evaluating conservation status of a species, and for establishing monitoring programs. The golden-cheeked warbler (Setophaga chrysoparia) is a neotropical migratory bird that was listed as federally endangered in 1990 because of threats related to loss and fragmentation of its woodland habitat. Since listing, abundance estimates for the species have mainly relied on localized population studies on public lands and qualitative-based methods. Our goal was to estimate breeding population size of male warblers using a predictive model based on metrics for patches of woodland habitat throughout the species' breeding range. We first conducted occupancy surveys to determine range-wide distribution. We then conducted standard point-count surveys on a subset of the initial sampling locations to estimate density of males. Mean observed patch-specific density was 0.23 males/ha (95% CI = 0.197–0.252, n = 301). We modeled the relationship between patch-specific density of males and woodland patch characteristics (size and landscape composition) and predicted patch occupancy. The probability of patch occupancy, derived from a model that used patch size and landscape composition as predictor variables while addressing effects of spatial relatedness, best predicted patch-specific density. We predicted patch-specific densities as a function of occupancy probability and estimated abundance of male warblers across 63,616 woodland patches accounting for 1.678 million ha of potential warbler habitat. Using a Monte Carlo simulation, our approach yielded a range-wide male warbler population estimate of 263,339 (95% CI: 223,927–302,620). Our results provide the first abundance estimate using habitat and count data from a sampling design focused on range-wide inference. Managers can use the resulting model as a tool to support conservation planning and guide recovery efforts. © 2012 The Wildlife Society.     

Precision gain versus effort with joint models using detection/non‐detection and banding data

Capture–recapture techniques provide valuable information, but are often more cost‐prohibitive at large spatial and temporal scales than less‐intensive sampling techniques. Model development combining multiple data sources to leverage data source strengths and for improved parameter precision has increased, but with limited discussion on precision gain versus effort. We present a general framework for evaluating trade‐offs between precision gained and costs associated with acquiring multiple data sources, useful for designing future or new phases of current studies.We illustrated how Bayesian hierarchical joint models using detection/non‐detection and banding data can improve abundance, survival, and recruitment inference, and quantified data source costs in a northern Arizona, USA, western bluebird (Sialia mexicana) population. We used an 8‐year detection/non‐detection (distributed across the landscape) and banding (subset of locations within landscape) data set to estimate parameters. We constructed separate models using detection/non‐detection and banding data, and a joint model using both data types to evaluate parameter precision gain relative to effort.Joint model parameter estimates were more precise than single data model estimates, but parameter precision varied (apparent survival > abundance > recruitment). Banding provided greater apparent survival precision than detection/non‐detection data. Therefore, little precision was gained when detection/non‐detection data were added to banding data. Additional costs were minimal; however, additional spatial coverage and ability to estimate abundance and recruitment improved inference. Conversely, more precision was gained when adding banding to detection/non‐detection data at higher cost. Spatial coverage was identical, yet survival and abundance estimates were more precise. Justification of increased costs associated with additional data types depends on project objectives.We illustrate a general framework for evaluating precision gain relative to effort, applicable to joint data models with any data type combination. This framework evaluates costs and benefits from and effort levels between multiple data types, thus improving population monitoring designs.     

Predicting fundamental and realized distributions based on thermal niche: A case study of a freshwater turtle

Species distribution models (SDM) have been broadly used in ecology to address theoretical and practical problems. Currently, there are two main approaches to generate SDMs: (i) correlative, which is based on species occurrences and environmental predictor layers and (ii) process-based models, which are constructed based on species' functional traits and physiological tolerances. The distributions estimated by each approach are based on different components of species niche. Predictions of correlative models approach species realized niches, while predictions of process-based are more akin to species fundamental niche. Here, we integrated the predictions of fundamental and realized distributions of the freshwater turtle Trachemys dorbigni. Fundamental distribution was estimated using data of T. dorbigni's egg incubation temperature, and realized distribution was estimated using species occurrence records. Both types of distributions were estimated using the same regression approaches (logistic regression and support vector machines), both considering macroclimatic and microclimatic temperatures. The realized distribution of T. dorbigni was generally nested in its fundamental distribution reinforcing theoretical assumptions that the species' realized niche is a subset of its fundamental niche. Both modelling algorithms produced similar results but microtemperature generated better results than macrotemperature for the incubation model. Finally, our results reinforce the conclusion that species realized distributions are constrained by other factors other than just thermal tolerances.     

Hierarchical models for combining ecological and case-control data

The ecological study design suffers from a broad range of biases that result from the loss of information regarding the joint distribution of individual-level outcomes, exposures, and confounders. The consequent nonidentifiability of individual-level models cannot be overcome without additional information; we combine ecological data with a sample of individual-level case-control data. The focus of this article is hierarchical models to account for between-group heterogeneity. Estimation and inference pose serious computational challenges. We present a Bayesian implementation based on a data augmentation scheme where the unobserved data are treated as auxiliary variables. The methods are illustrated with a dataset of county-specific infant mortality data from the state of North Carolina.