Effects of residual smoothing on the posterior of the fixed effects in disease-mapping models

Disease-mapping models for areal data often have fixed effects to measure the effect of spatially varying covariates and random effects with a conditionally autoregressive (CAR) prior to account for spatial clustering. In such spatial regressions, the objective may be to estimate the fixed effects while accounting for the spatial correlation. But adding the CAR random effects can cause large changes in the posterior mean and variance of fixed effects compared to the nonspatial regression model. This article explores the impact of adding spatial random effects on fixed effect estimates and posterior variance. Diagnostics are proposed to measure posterior variance inflation from collinearity between the fixed effect covariates and the CAR random effects and to measure each region's influence on the change in the fixed effect's estimates by adding the CAR random effects. A new model that alleviates the collinearity between the fixed effect covariates and the CAR random effects is developed and extensions of these methods to point-referenced data models are discussed.     

The Impact of Spatial Scales and Spatial Smoothing on the Outcome of Bayesian Spatial Model

Discretization of a geographical region is quite common in spatial analysis. There have been few studies into the impact of different geographical scales on the outcome of spatial models for different spatial patterns. This study aims to investigate the impact of spatial scales and spatial smoothing on the outcomes of modelling spatial point-based data. Given a spatial point-based dataset (such as occurrence of a disease), we study the geographical variation of residual disease risk using regular grid cells. The individual disease risk is modelled using a logistic model with the inclusion of spatially unstructured and/or spatially structured random effects. Three spatial smoothness priors for the spatially structured component are employed in modelling, namely an intrinsic Gaussian Markov random field, a second-order random walk on a lattice, and a Gaussian field with Matérn correlation function. We investigate how changes in grid cell size affect model outcomes under different spatial structures and different smoothness priors for the spatial component. A realistic example (the Humberside data) is analyzed and a simulation study is described. Bayesian computation is carried out using an integrated nested Laplace approximation. The results suggest that the performance and predictive capacity of the spatial models improve as the grid cell size decreases for certain spatial structures. It also appears that different spatial smoothness priors should be applied for different patterns of point data.     

Spatial modelling of multinomial data with latent structure: an application to geographical mapping of human gene and haplotype frequencies

We develop hierarchical models for spatial multinomial data with missing categories, to analyse a database of HLA-A and -B gene and haplotype frequencies from Papua New Guinea, with a highly variable number of samples per spatial unit. The spatial structure of the multinomial data is incorporated by adopting conditional autoregressive (CAR) priors for the random effects, reflecting extra-multinomial variation. Different spatial structures are investigated, and covariate effects are evaluated using a novel model selection criterion. Tables and maps reveal strong spatial association and the importance of altitude, a covariate anticipated to be significant in explaining genetic variation. Our approach can be used in identifying associations with environmental factors, linguistic or epidemiological patterns and hence potential causes of genetic diversity (population movements, natural selection, stochastic effects).     

Joint Spatial Modeling of Recurrent Infection and Growth with Processes under Intermittent Observation

Summary In this article, we present a new statistical methodology for longitudinal studies in forestry, where trees are subject to recurrent infection, and the hazard of infection depends on tree growth over time. Understanding the nature of this dependence has important implications for reforestation and breeding programs. Challenges arise for statistical analysis in this setting with sampling schemes leading to panel data, exhibiting dynamic spatial variability, and incomplete covariate histories for hazard regression. In addition, data are collected at a large number of locations, which poses computational difficulties for spatiotemporal modeling. A joint model for infection and growth is developed wherein a mixed nonhomogeneous Poisson process, governing recurring infection, is linked with a spatially dynamic nonlinear model representing the underlying height growth trajectories. These trajectories are based on the von Bertalanffy growth model and a spatially varying parameterization is employed. Spatial variability in growth parameters is modeled through a multivariate spatial process derived through kernel convolution. Inference is conducted in a Bayesian framework with implementation based on hybrid Monte Carlo. Our methodology is applied for analysis in an 11‐year study of recurrent weevil infestation of white spruce in British Columbia.     

On the precision of the conditionally autoregressive prior in spatial models

Bayesian analyses of spatial data often use a conditionally autoregressive (CAR) prior, which can be written as the kernel of an improper density that depends on a precision parameter tau that is typically unknown. To include tau in the Bayesian analysis, the kernel must be multiplied by tau(k) for some k. This article rigorously derives k = (n - I)/2 for the L2 norm CAR prior (also called a Gaussian Markov random field model) and k = n - I for the L1 norm CAR prior, where n is the number of regions and I the number of "islands" (disconnected groups of regions) in the spatial map. Since I = 1 for a spatial structure defining a connected graph, this supports Knorr-Held's (2002, in Highly Structured Stochastic Systems, 260-264) suggestion that k = (n - 1)/2 in the L2 norm case, instead of the more common k = n/2. We illustrate the practical significance of our results using a periodontal example.     

Bayesian LASSO for quantitative trait loci mapping

The mapping of quantitative trait loci (QTL) is to identify molecular markers or genomic loci that influence the variation of complex traits. The problem is complicated by the facts that QTL data usually contain a large number of markers across the entire genome and most of them have little or no effect on the phenotype. In this article, we propose several Bayesian hierarchical models for mapping multiple QTL that simultaneously fit and estimate all possible genetic effects associated with all markers. The proposed models use prior distributions for the genetic effects that are scale mixtures of normal distributions with mean zero and variances distributed to give each effect a high probability of being near zero. We consider two types of priors for the variances, exponential and scaled inverse-chi(2) distributions, which result in a Bayesian version of the popular least absolute shrinkage and selection operator (LASSO) model and the well-known Student's t model, respectively. Unlike most applications where fixed values are preset for hyperparameters in the priors, we treat all hyperparameters as unknowns and estimate them along with other parameters. Markov chain Monte Carlo (MCMC) algorithms are developed to simulate the parameters from the posteriors. The methods are illustrated using well-known barley data.     

Random effects selection in linear mixed models

We address the important practical problem of how to select the random effects component in a linear mixed model. A hierarchical Bayesian model is used to identify any random effect with zero variance. The proposed approach reparameterizes the mixed model so that functions of the covariance parameters of the random effects distribution are incorporated as regression coefficients on standard normal latent variables. We allow random effects to effectively drop out of the model by choosing mixture priors with point mass at zero for the random effects variances. Due to the reparameterization, the model enjoys a conditionally linear structure that facilitates the use of normal conjugate priors. We demonstrate that posterior computation can proceed via a simple and efficient Markov chain Monte Carlo algorithm. The methods are illustrated using simulated data and real data from a study relating prenatal exposure to polychlorinated biphenyls and psychomotor development of children.     

Bayesian analysis for generalized linear models with nonignorably missing covariates

We propose Bayesian methods for estimating parameters in generalized linear models (GLMs) with nonignorably missing covariate data. We show that when improper uniform priors are used for the regression coefficients, phi, of the multinomial selection model for the missing data mechanism, the resulting joint posterior will always be improper if (i) all missing covariates are discrete and an intercept is included in the selection model for the missing data mechanism, or (ii) at least one of the covariates is continuous and unbounded. This impropriety will result regardless of whether proper or improper priors are specified for the regression parameters, beta, of the GLM or the parameters, alpha, of the covariate distribution. To overcome this problem, we propose a novel class of proper priors for the regression coefficients, phi, in the selection model for the missing data mechanism. These priors are robust and computationally attractive in the sense that inferences about beta are not sensitive to the choice of the hyperparameters of the prior for phi and they facilitate a Gibbs sampling scheme that leads to accelerated convergence. In addition, we extend the model assessment criterion of Chen, Dey, and Ibrahim (2004a, Biometrika 91, 45-63), called the weighted L measure, to GLMs and missing data problems as well as extend the deviance information criterion (DIC) of Spiegelhalter et al. (2002, Journal of the Royal Statistical Society B 64, 583-639) for assessing whether the missing data mechanism is ignorable or nonignorable. A novel Markov chain Monte Carlo sampling algorithm is also developed for carrying out posterior computation. Several simulations are given to investigate the performance of the proposed Bayesian criteria as well as the sensitivity of the prior specification. Real datasets from a melanoma cancer clinical trial and a liver cancer study are presented to further illustrate the proposed methods.     

Testing for spatial correlation in nonstationary binary data, with application to aberrant crypt foci in colon carcinogenesis

In an experiment to understand colon carcinogenesis, all animals were exposed to a carcinogen, with half the animals also being exposed to radiation. Spatially, we measured the existence of what are referred to as aberrant crypt foci (ACF), namely, morphologically changed colonic crypts that are known to be precursors of colon cancer development. The biological question of interest is whether the locations of these ACFs are spatially correlated: if so, this indicates that damage to the colon due to carcinogens and radiation is localized. Statistically, the data take the form of binary outcomes (corresponding to the existence of an ACF) on a regular grid. We develop score-type methods based upon the Matern and conditionally autoregressive (CAR) correlation models to test for the spatial correlation in such data, while allowing for nonstationarity. Because of a technical peculiarity of the score-type test, we also develop robust versions of the method. The methods are compared to a generalization of Moran's test for continuous outcomes, and are shown via simulation to have the potential for increased power. When applied to our data, the methods indicate the existence of spatial correlation, and hence indicate localization of damage.     

Locally adaptive function estimation for binary regression models

In this paper we present a nonparametric Bayesian approach for fitting unsmooth or highly oscillating functions in regression models with binary responses. The approach extends previous work by Lang et al. for Gaussian responses. Nonlinear functions are modelled by first or second order random walk priors with locally varying variances or smoothing parameters. Estimation is fully Bayesian and uses latent utility representations of binary regression models for efficient block sampling from the full conditionals of nonlinear functions.     

Assessing the spatial variability of density dependence in waterfowl populations

Population models commonly assume that the demographic parameters are spatially invariant, but there is considerable evidence that population growth rate (r) and the strength of density dependence (β) can vary over a species' range. To address this issue we developed a spatially explicit Gompertz population model based on the spatially varying coefficients approach to assess the spatial variation in population drivers. The model was fit to spatially stratified time series population estimates of the mallard Anas platyrhynchos in western North America. We included precipitation during the previous year and spring maximum temperature in the current year as environmental factors in the density dependent population model. Because density dependent models can give biased estimates for time series of abundance data, we fit a naïve model without informative priors and a model where we constrained the mean and variance of r to biologically realistic values that were derived via a comparative demography approach. In the naïve model, r and β were not separately identifiable and their values were overestimated, leading to unrealistic population growth. The naïve model also implied spatial variation in population r and the return time to equilibrium [1?(– β)] across the survey area. In contrast, in the informative model, r and the return time to equilibrium did not vary markedly among populations and were generally equal across populations. The effects of the climatic factors were similar across models. Population growth rates in the Prairie‐pothole region were positively correlated with precipitation, while in Alaska rates were positively correlated with spring temperature. Although it has been argued in the past that adding ecological realism could help avoid the pitfalls associated with density dependent models, our results demonstrate that imposing constraints on the population parameters is still the best course of action.     

Functional specialization of seven mouse visual cortical areas

To establish the mouse as a genetically tractable model for high-order visual processing, we characterized fine-scale retinotopic organization of visual cortex and determined functional specialization of layer 2/3 neuronal populations in seven retinotopically identified areas. Each area contains a distinct visuotopic representation and encodes a unique combination of spatiotemporal features. Areas LM, AL, RL, and AM prefer up to three times faster temporal frequencies and significantly lower spatial frequencies than V1, while V1 and PM prefer high spatial and low temporal frequencies. LI prefers both high spatial and temporal frequencies. All extrastriate areas except LI increase orientation selectivity compared to V1, and three areas are significantly more direction selective (AL, RL, and AM). Specific combinations of spatiotemporal representations further distinguish areas. These results reveal that mouse higher visual areas are functionally distinct, and separate groups of areas may be specialized for motion-related versus pattern-related computations, perhaps forming pathways analogous to dorsal and ventral streams in other species.     

Penalized local polynomial regression for spatial data

When performing spatial regression analysis in environmental data applications, spatial heterogeneity in the regression coefficients is often observed. Spatially varying coefficient models, including geographically weighted regression and spline models, are standard tools for quantifying such heterogeneity. In this paper, we propose a spatially varying coefficient model that represents the spatially varying parameters as a mixture of local polynomials at selected locations. The local polynomial parameters have attractive interpretations, indicating various types of spatial heterogeneity. Instead of estimating the spatially varying regression coefficients directly, we develop a penalized least squares regression procedure for the local polynomial parameter estimation, which both shrinks the parameter estimation and penalizes the differences among parameters that are associated with neighboring locations. We develop confidence intervals for the varying regression coefficients and prediction intervals for the response. We apply the proposed method to characterize the spatially varying association between particulate matter concentrations ( ${\text{PM}}_{2.5}$) and pollutant gases related to the secondary aerosol formulation in China. The identified regression coefficients show distinct spatial patterns for nitrogen dioxide, sulfur dioxide, and carbon monoxide during different seasons.     

geophylobuilder 1.0: an arcgis extension for creating 'geophylogenies'

Evolution is inherently a spatiotemporal process; however, despite this, phylogenetic and geographical data and models remain largely isolated from one another. Geographical information systems provide a ready-made spatial modelling, analysis and dissemination environment within which phylogenetic models can be explicitly linked with their associated spatial data and subsequently integrated with other georeferenced data sets describing the biotic and abiotic environment. geophylobuilder 1.0 is an extension for the arcgis geographical information system that builds a 'geophylogenetic' data model from a phylogenetic tree and associated geographical data. Geophylogenetic database objects can subsequently be queried, spatially analysed and visualized in both 2D and 3D within a geographical information systems.     

A predictive model of avian natal dispersal distance provides prior information for investigating response to landscape change

1. Informative Bayesian priors can improve the precision of estimates in ecological studies or estimate parameters for which little or no information is available. While Bayesian analyses are becoming more popular in ecology, the use of strongly informative priors remains rare, perhaps because examples of informative priors are not readily available in the published literature. 2. Dispersal distance is an important ecological parameter, but is difficult to measure and estimates are scarce. General models that provide informative prior estimates of dispersal distances will therefore be valuable. 3. Using a world-wide data set on birds, we develop a predictive model of median natal dispersal distance that includes body mass, wingspan, sex and feeding guild. This model predicts median dispersal distance well when using the fitted data and an independent test data set, explaining up to 53% of the variation. 4. Using this model, we predict a priori estimates of median dispersal distance for 57 woodland-dependent bird species in northern Victoria, Australia. These estimates are then used to investigate the relationship between dispersal ability and vulnerability to landscape-scale changes in habitat cover and fragmentation. 5. We find evidence that woodland bird species with poor predicted dispersal ability are more vulnerable to habitat fragmentation than those species with longer predicted dispersal distances, thus improving the understanding of this important phenomenon. 6. The value of constructing informative priors from existing information is also demonstrated. When used as informative priors for four example species, predicted dispersal distances reduced the 95% credible intervals of posterior estimates of dispersal distance by 8-19%. Further, should we have wished to collect information on avian dispersal distances and relate it to species' responses to habitat loss and fragmentation, data from 221 individuals across 57 species would have been required to obtain estimates with the same precision as those provided by the general model.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

Estimating local interaction from spatiotemporal forest data, and Monte Carlo bias correction

We point out a general problem in fitting continuous time spatially explicit models to a temporal sequence of spatial data observed at discrete times. To illustrate the problem, we examined the continuous time Markov model for forest gap dynamics. A forest is assumed to be apportioned into discrete cells (or sites) arranged in a regular square lattice. Each site is characterized as either a gap or a non-gap site according to the vegetation height of trees. The model incorporates the influence of neighboring sites on transition rate: transition rate from a non-gap to a gap site increases linearly with the number of neighbors that are currently in the gap state, and vice versa. We fitted the model to the spatiotemporal data of canopy height observed at the permanent plot in Barro Colorado Island (BCI). When we used the approximate maximum likelihood method to estimate the parameters of the model, the estimated transition rates included a large bias-in particular, the strength of interaction between nearby sites was underestimated. This bias originated from the assumption that each transition between two observation times is independent. The interaction between sites at local scale creates a long chain of transitions within a single census interval, which violates the independence of each transition. We show that a computer-intensive method, called Monte Carlo bias correction (MCBC), is very effective in removing the bias included in the estimate. The global and local gap densities measuring spatial aggregation of gap sites were computed from simulated and real gap dynamics to assess the model. When the approximate likelihood estimates were applied to the model, the predicted local gap density was clearly lower than the observed one. The use of MCBC estimates, suggesting a strong interaction between sites, improved this discrepancy.     

Modeling Trends from North American Breeding Bird Survey Data: A Spatially Explicit Approach

Population trends, defined as interval-specific proportional changes in population size, are often used to help identify species of conservation interest. Efficient modeling of such trends depends on the consideration of the correlation of population changes with key spatial and environmental covariates. This can provide insights into causal mechanisms and allow spatially explicit summaries at scales that are of interest to management agencies. We expand the hierarchical modeling framework used in the North American Breeding Bird Survey (BBS) by developing a spatially explicit model of temporal trend using a conditional autoregressive (CAR) model. By adopting a formal spatial model for abundance, we produce spatially explicit abundance and trend estimates. Analyses based on large-scale geographic strata such as Bird Conservation Regions (BCR) can suffer from basic imbalances in spatial sampling. Our approach addresses this issue by providing an explicit weighting based on the fundamental sample allocation unit of the BBS. We applied the spatial model to three species from the BBS. Species have been chosen based upon their well-known population change patterns, which allows us to evaluate the quality of our model and the biological meaning of our estimates. We also compare our results with the ones obtained for BCRs using a nonspatial hierarchical model (Sauer and Link 2011). Globally, estimates for mean trends are consistent between the two approaches but spatial estimates provide much more precise trend estimates in regions on the edges of species ranges that were poorly estimated in non-spatial analyses. Incorporating a spatial component in the analysis not only allows us to obtain relevant and biologically meaningful estimates for population trends, but also enables us to provide a flexible framework in order to obtain trend estimates for any area.     

Smoothing for spatiotemporal models and its application to modeling muskrat-mink interaction

For a set of spatially dependent dynamical models, we propose a method for estimating parameters that control temporal dynamics by spatial smoothing. The new approach is particularly relevant for analyzing spatially distributed panels of short time series. The asymptotic results show that spatial smoothing will improve the estimation in the presence of nugget effect, even when the sample size in each location is large. The proposed methodology is used to analyze the annual mink and muskrat data collected in a period of 25 years in 81 Canadian locations. Based on the proposed method, we are able to model the temporal dynamics which reflects the food chain interaction of the two species.