Testing environmental and genetic effects in the presence of spatial autocorrelation

Spatial autocorrelation is a well‐recognized concern for observational data in general, and more specifically for spatial data in ecology. Generalized linear mixed models (GLMMs) with spatially autocorrelated random effects are a potential general framework for handling these spatial correlations. However, as the result of statistical and practical issues, such GLMMs have been fitted through the undocumented use of procedures based on penalized quasi‐likelihood approximations (PQL), and under restrictive models of spatial correlation. Alternatively, they are often neglected in favor of simpler but more questionable approaches. In this work we aim to provide practical and validated means of inference under spatial GLMMs, that overcome these limitations. For this purpose, a new software is developed to fit spatial GLMMs. We use it to assess the performance of likelihood ratio tests for fixed effects under spatial autocorrelation, based on Laplace or PQL approximations of the likelihood. Expectedly, the Laplace approximation performs generally slightly better, although a variant of PQL was better in the binary case. We show that a previous implementation of PQL methods in the R language, glmmPQL, is not appropriate for such applications. Finally, we illustrate the efficiency of a bootstrap procedure for correcting the small sample bias of the tests, which applies also to non‐spatial models.     

A spatial scan statistic for multiple clusters

Spatial scan statistics are commonly used for geographical disease surveillance and cluster detection. While there are multiple clusters coexisting in the study area, they become difficult to detect because of clusters’ shadowing effect to each other. The recently proposed sequential method showed its better power for detecting the second weaker cluster, but did not improve the ability of detecting the first stronger cluster which is more important than the second one. We propose a new extension of the spatial scan statistic which could be used to detect multiple clusters. Through constructing two or more clusters in the alternative hypothesis, our proposed method accounts for other coexisting clusters in the detecting and evaluating process. The performance of the proposed method is compared to the sequential method through an intensive simulation study, in which our proposed method shows better power in terms of both rejecting the null hypothesis and accurately detecting the coexisting clusters. In the real study of hand-foot-mouth disease data in Pingdu city, a true cluster town is successfully detected by our proposed method, which cannot be evaluated to be statistically significant by the standard method due to another cluster’s shadowing effect.     

Variable selection in Bayesian generalized linear‐mixed models: An illustration using candidate gene case‐control association studies

The problem of variable selection in the generalized linear‐mixed models (GLMMs) is pervasive in statistical practice. For the purpose of variable selection, many methodologies for determining the best subset of explanatory variables currently exist according to the model complexity and differences between applications. In this paper, we develop a “higher posterior probability model with bootstrap” (HPMB) approach to select explanatory variables without fitting all possible GLMMs involving a small or moderate number of explanatory variables. Furthermore, to save computational load, we propose an efficient approximation approach with Laplace's method and Taylor's expansion to approximate intractable integrals in GLMMs. Simulation studies and an application of HapMap data provide evidence that this selection approach is computationally feasible and reliable for exploring true candidate genes and gene–gene associations, after adjusting for complex structures among clusters.     

On estimation and prediction for spatial generalized linear mixed models

We use spatial generalized linear mixed models (GLMM) to model non-Gaussian spatial variables that are observed at sampling locations in a continuous area. In many applications, prediction of random effects in a spatial GLMM is of great practical interest. We show that the minimum mean-squared error (MMSE) prediction can be done in a linear fashion in spatial GLMMs analogous to linear kriging. We develop a Monte Carlo version of the EM gradient algorithm for maximum likelihood estimation of model parameters. A by-product of this approach is that it also produces the MMSE estimates for the realized random effects at the sampled sites. This method is illustrated through a simulation study and is also applied to a real data set on plant root diseases to obtain a map of disease severity that can facilitate the practice of precision agriculture.     

Statistical Modeling of Observational Data With Spatial Dependencies

Abstract I provide a brief introduction to the concept of spatial autocorrelation and its incorporation into regression-type models. Spatial autocorrelation occurs when the response variable is correlated with itself at other locations in the region of interest. The autocorrelation usually takes a specific form where observations close in space are more correlated than those farther apart, and the rate of decay of the correlation is a function of the distance separating 2 locations. I present 2 commonly used models: 1) geostatistical modeling in which data are collected at points in the study region and 2) conditional autoregression (lattice) models in which data are aggregated over small nonoverlapping sub-areas of the study region. I also describe incorporation of explanatory covariates, such as habitat or physico-chemical attributes. I emphasize frequentist methods, but I briefly describe Bayesian approaches. I also provide some advantages, such as obtaining correct standard errors for estimators, and disadvantages, such as requirements for larger sample sizes, of incorporating spatial autocorrelation into the modeling effort. This information can aid researchers in designing and analyzing models of the relationships between species distributions and habitat. As a result, more informative models can be developed which further aid in management of wildlife.     

Effects of multiple founder populations on spatial genetic structure of reintroduced American martens

Reintroductions and translocations are increasingly used to repatriate or increase probabilities of persistence for animal and plant species. Genetic and demographic characteristics of founding individuals and suitability of habitat at release sites are commonly believed to affect the success of these conservation programs. Genetic divergence among multiple source populations of American martens (Martes americana) and well documented introduction histories permitted analyses of post‐introduction dispersion from release sites and development of genetic clusters in the Upper Peninsula (UP) of Michigan <50 years following release. Location and size of spatial genetic clusters and measures of individual‐based autocorrelation were inferred using 11 microsatellite loci. We identified three genetic clusters in geographic proximity to original release locations. Estimated distances of effective gene flow based on spatial autocorrelation varied greatly among genetic clusters (30–90 km). Spatial contiguity of genetic clusters has been largely maintained with evidence for admixture primarily in localized regions, suggesting recent contact or locally retarded rates of gene flow. Data provide guidance for future studies of the effects of permeabilities of different land‐cover and land‐use features to dispersal and of other biotic and environmental factors that may contribute to the colonization process and development of spatial genetic associations.     

Selection of the Maximum Spatial Cluster Size of the Spatial Scan Statistic by Using the Maximum Clustering Set-Proportion Statistic

Spatial scan statistics are widely used in various fields. The performance of these statistics is influenced by parameters, such as maximum spatial cluster size, and can be improved by parameter selection using performance measures. Current performance measures are based on the presence of clusters and are thus inapplicable to data sets without known clusters. In this work, we propose a novel overall performance measure called maximum clustering set–proportion (MCS-P), which is based on the likelihood of the union of detected clusters and the applied dataset. MCS-P was compared with existing performance measures in a simulation study to select the maximum spatial cluster size. Results of other performance measures, such as sensitivity and misclassification, suggest that the spatial scan statistic achieves accurate results in most scenarios with the maximum spatial cluster sizes selected using MCS-P. Given that previously known clusters are not required in the proposed strategy, selection of the optimal maximum cluster size with MCS-P can improve the performance of the scan statistic in applications without identified clusters.     

Structured additive regression for categorical space-time data: a mixed model approach

Motivated by a space-time study on forest health with damage state of trees as the response, we propose a general class of structured additive regression models for categorical responses, allowing for a flexible semiparametric predictor. Nonlinear effects of continuous covariates, time trends, and interactions between continuous covariates are modeled by penalized splines. Spatial effects can be estimated based on Markov random fields, Gaussian random fields, or two-dimensional penalized splines. We present our approach from a Bayesian perspective, with inference based on a categorical linear mixed model representation. The resulting empirical Bayes method is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to inverse smoothing parameters, are estimated using (approximate) restricted maximum likelihood. In simulation studies we investigate the performance of different choices for the spatial effect, compare the empirical Bayes approach to competing methodology, and study the bias of mixed model estimates. As an application we analyze data from the forest health survey.     

Spatial Autocorrelation of Amino Acid Replacement Rates in the Vasopressin Receptor Family

Evolutionary rates of sites can be independent of one another or correlated in some fashion. Significant spatial autocorrelation was observed for site amino acid replacement rates in vasopressin receptor family proteins (VPRs). Spatial autocorrelation of rates is the propensity of residues to lie near other residues of similar rate in the folded protein structure. Optimal correlation occurred at a distance suggesting that residues in contact had correlated rates. As another way to study the same phenomenon, VPR was partitioned into >40 × 10 Å³ contiguous spatial clusters for amino acid replacement rate estimation. Partitioning was done without preconception of functional regions of the protein and with a random partition control. Cluster rates exhibited an overdispersed distribution suggesting that rates were not randomly distributed in the spatial partitions. In tests, cluster partitioning improved maximum likelihood and Bayesian likelihood models for VPR evolution. Spatial clusters with outlier rates, or lineage-specific clusters differing in rate, proved to contain VPR features likely to be under selection. Thus the spatial autocorrelation observed is probably not just a statistical finding, but likely has an evolutionary basis in protein function.     

Spatial organization in bacterial chemotaxis

Spatial organization of signalling is not an exclusive property of eukaryotic cells. Despite the fact that bacterial signalling pathways are generally simpler than those in eukaryotes, there are several well‐documented examples of higher‐order intracellular signalling structures in bacteria. One of the most prominent and best‐characterized structures is formed by proteins that control bacterial chemotaxis. Signals in chemotaxis are processed by ordered arrays, or clusters, of receptors and associated proteins, which amplify and integrate chemotactic stimuli in a highly cooperative manner. Receptor clusters further serve to scaffold protein interactions, enhancing the efficiency and specificity of the pathway reactions and preventing the formation of signalling gradients through the cell body. Moreover, clustering can also ensure spatial separation of multiple chemotaxis systems in one bacterium. Assembly of receptor clusters appears to be a stochastic process, but bacteria evolved mechanisms to ensure optimal cluster distribution along the cell body for partitioning to daughter cells at division.     

Comparison of regression methods for spatially‐autocorrelated count data on regularly‐ and irregularly‐spaced locations

It has long been known that insufficient consideration of spatial autocorrelation leads to unreliable hypothesis‐tests and inaccurate parameter estimates. Yet, ecologists are confronted with a confusing array of methods to account for spatial autocorrelation. Although Beale et al. (2010) provided guidance for continuous data on regular grids, researchers still need advice for other types of data in more flexible spatial contexts. In this paper, we extend Beale et al. (2010)‘s work to count data on both regularly‐ and irregularly‐spaced plots, the latter being commonly encountered in ecological studies. Through a simulation‐based approach, we assessed the accuracy and the type I errors of two frequentist and two Bayesian ready‐to‐use methods in the family of generalized mixed models, with distance‐based or neighbourhood‐based correlated random effects. In addition, we tested whether the methods are robust to spatial non‐stationarity, and over‐ and under‐dispersion – both typical features of species distribution count data which violate standard regression assumptions. In the simplest of our simulated datasets, the two frequentist methods gave inflated type I errors, while the two Bayesian methods provided satisfying results. When facing real‐world complexities, the distance‐based Bayesian method (MCMC with Langevin–Hastings updates) performed best of all. We hope that, in the light of our results, ecological researchers will feel more comfortable including spatial autocorrelation in their analyses of count data.     

A spatial scan statistic for survival data

Spatial scan statistics with Bernoulli and Poisson models are commonly used for geographical disease surveillance and cluster detection. These models, suitable for count data, were not designed for data with continuous outcomes. We propose a spatial scan statistic based on an exponential model to handle either uncensored or censored continuous survival data. The power and sensitivity of the developed model are investigated through intensive simulations. The method performs well for different survival distribution functions including the exponential, gamma, and log-normal distributions. We also present a method to adjust the analysis for covariates. The cluster detection method is illustrated using survival data for men diagnosed with prostate cancer in Connecticut from 1984 to 1995.     

Effect of spatial coupling of amino acid residues on the possibility of mutational substitution in globular proteins

Amino acid residues can be divided into similar groups by frequencies of interreplacements in the evolutionary pathway and by trends to spatial contacts at the tertiary structures of globular proteins. Each residue was compared to the cluster of spatial surrounding--the totality of residues spacially drawn together. 5210 clusters in 32 unhomologous proteins with established tertiary structure and 6447 clusters formed only by variables amino acid residues were analysed. Spatial contacts among residues were studied depending on the secondary structure and the amount of residues in a cluster. It was assumed that functionally admissible mutations may be defined, first of all, by the degree of neighboring of amino acid residues in the spatial surrounding.     

Dealing with spatial autocorrelation when learning predictive clustering trees

Spatial autocorrelation is the correlation among data values which is strictly due to the relative spatial proximity of the objects that the data refer to. Inappropriate treatment of data with spatial dependencies, where spatial autocorrelation is ignored, can obfuscate important insights. In this paper, we propose a data mining method that explicitly considers spatial autocorrelation in the values of the response (target) variable when learning predictive clustering models. The method is based on the concept of predictive clustering trees (PCTs), according to which hierarchies of clusters of similar data are identified and a predictive model is associated to each cluster. In particular, our approach is able to learn predictive models for both a continuous response (regression task) and a discrete response (classification task). We evaluate our approach on several real world problems of spatial regression and spatial classification. The consideration of the autocorrelation in the models improves predictions that are consistently clustered in space and that clusters try to preserve the spatial arrangement of the data, at the same time providing a multi-level insight into the spatial autocorrelation phenomenon. The evaluation of SCLUS in several ecological domains (e.g. predicting outcrossing rates within a conventional field due to the surrounding genetically modified fields, as well as predicting pollen dispersal rates from two lines of plants) confirms its capability of building spatial aware models which capture the spatial distribution of the target variable. In general, the maps obtained by using SCLUS do not require further post-smoothing of the results if we want to use them in practice.     

Spatially explicit analyses of anopheline mosquitoes indoor resting density: implications for malaria control

Background

The question of sampling and spatial aggregation of malaria vectors is central to vector control efforts and estimates of transmission. Spatial patterns of anopheline populations are complex because mosquitoes'' habitats and behaviors are strongly heterogeneous. Analyses of spatially referenced counts provide a powerful approach to delineate complex distribution patterns, and contributions of these methods in the study and control of malaria vectors must be carefully evaluated.

Methodology/Principal Findings

We used correlograms, directional variograms, Local Indicators of Spatial Association (LISA) and the Spatial Analysis by Distance IndicEs (SADIE) to examine spatial patterns of Indoor Resting Densities (IRD) in two dominant malaria vectors sampled with a 5×5 km grid over a 2500 km² area in the forest domain of Cameroon. SADIE analyses revealed that the distribution of Anopheles gambiae was different from regular or random, whereas there was no evidence of spatial pattern in Anopheles funestus (Ia = 1.644, Pa<0.05 and Ia = 1.464, Pa>0.05, respectively). Correlograms and variograms showed significant spatial autocorrelations at small distance lags, and indicated the presence of large clusters of similar values of abundance in An. gambiae while An. funestus was characterized by smaller clusters. The examination of spatial patterns at a finer spatial scale with SADIE and LISA identified several patches of higher than average IRD (hot spots) and clusters of lower than average IRD (cold spots) for the two species. Significant changes occurred in the overall spatial pattern, spatial trends and clusters when IRDs were aggregated at the house level rather than the locality level. All spatial analyses unveiled scale-dependent patterns that could not be identified by traditional aggregation indices.

Conclusions/Significance

Our study illustrates the importance of spatial analyses in unraveling the complex spatial patterns of malaria vectors, and highlights the potential contributions of these methods in malaria control.     

Zero‐inflated spatio‐temporal models for disease mapping

In this paper, our aim is to analyze geographical and temporal variability of disease incidence when spatio‐temporal count data have excess zeros. To that end, we consider random effects in zero‐inflated Poisson models to investigate geographical and temporal patterns of disease incidence. Spatio‐temporal models that employ conditionally autoregressive smoothing across the spatial dimension and B‐spline smoothing over the temporal dimension are proposed. The analysis of these complex models is computationally difficult from the frequentist perspective. On the other hand, the advent of the Markov chain Monte Carlo algorithm has made the Bayesian analysis of complex models computationally convenient. Recently developed data cloning method provides a frequentist approach to mixed models that is also computationally convenient. We propose to use data cloning, which yields to maximum likelihood estimation, to conduct frequentist analysis of zero‐inflated spatio‐temporal modeling of disease incidence. One of the advantages of the data cloning approach is that the prediction and corresponding standard errors (or prediction intervals) of smoothing disease incidence over space and time is easily obtained. We illustrate our approach using a real dataset of monthly children asthma visits to hospital in the province of Manitoba, Canada, during the period April 2006 to March 2010. Performance of our approach is also evaluated through a simulation study.     

Diagnosis of random-effect model misspecification in generalized linear mixed models for binary response

Summary .  Generalized linear mixed models (GLMMs) are widely used in the analysis of clustered data. However, the validity of likelihood-based inference in such analyses can be greatly affected by the assumed model for the random effects. We propose a diagnostic method for random-effect model misspecification in GLMMs for clustered binary response. We provide a theoretical justification of the proposed method and investigate its finite sample performance via simulation. The proposed method is applied to data from a longitudinal respiratory infection study.     

Simple temporal models for ecological systems with complex spatial patterns

Spatial patterns are ubiquitous in nature. Because these patterns modify the temporal dynamics and stability properties of population densities at a range of spatial scales, their effects must be incorporated in temporal ecological models that do not represent space explicitly. We demonstrate a connection between a simple parameterization of spatial effects and the geometry of clusters in an individual‐based predator–prey model that is both nonlinear and stochastic. Specifically we show that clusters exhibit a power‐law scaling of perimeter to area with an exponent close to unity. In systems with a high degree of patchiness, similar power‐law scalings can provide a basis for applying simple temporal models that assume well‐mixed conditions.     

Spatial Pattern Analysis of Plant-Parasitic Nematodes

Spatial patterns of Meloidogyne incognita, Tylenchorhynchus claytoni, Helicotylenchus dihystera, and Criconemella ornata were analyzed using Hill''s two-term local quadrat variance method (TTLQV), spectral analysis, and spatial correlation. Data were collected according to a systematic grid sampling plan from seven tobacco fields in North Carolina. Different estimates of nematode cluster size were obtained through TTLQV and spectral analysis. No relationship was observed between either estimate and nematode species, time of sampling (spring vs. fall), or mean density. Cluster size estimates obtained from spectral analysis depended on sampling block size. For each species examined, spatial correlations among nematode population densities were greater within plant rows than across rows, indicating that clusters were ellipsoidal with long axes oriented along plant rows. Analysis of mean square errors indicated that significant gains in sampling efficiency resulted from orienting the long axis of sampling blocks across plant rows. Spatial correlation was greater in the fall than in spring and was greater among 1 × 1-m quadrats than among 3 × 3-m quadrats.     

Hypercycles versus parasites in the origin of life: model dependence in spatial hypercycle systems

Spatial hypercycle systems can be modelled by means of cellular automata or partial differential equations. In either model, two dimensional spirals or clusters can be formed. Different models give rise to slightly different spatial structures, but the response to parasites is fundamentally different: In cellular automata the hypercycle is resistant to parasites that are fatal in a partial differential equations model. In three dimensions scroll rings correspond to the two dimensional spirals. Numerical simulations on a partial differential equations model indicate that the scroll rings are unstable: The contract by a power law and disappear. Therefore, in three dimensions clusters seem to be the best candidate for the hypercycle resistant to parasites.