A new method for dealing with residual spatial autocorrelation in species distribution models

Species distribution modelling (SDM) is a widely used tool and has many applications in ecology and conservation biology. Spatial autocorrelation (SAC), a pattern in which observations are related to one another by their geographic distance, is common in georeferenced ecological data. SAC in the residuals of SDMs violates the ‘independent errors’ assumption required to justify the use of statistical models in modelling species’ distributions. The autologistic modelling approach accounts for SAC by including an additional term (the autocovariate) representing the similarity between the value of the response variable at a location and neighbouring locations. However, autologistic models have been found to introduce bias in the estimation of parameters describing the influence of explanatory variables on habitat occupancy. To address this problem we developed an extension to the autologistic approach by calculating the autocovariate on SAC in residuals (the RAC approach). Performance of the new approach was tested on simulated data with a known spatial structure and on strongly autocorrelated mangrove species’ distribution data collected in northern Australia. The RAC approach was implemented as generalized linear models (GLMs) and boosted regression tree (BRT) models. We found that the BRT models with only environmental explanatory variables can account for some SAC, but applying the standard autologistic or RAC approaches further reduced SAC in model residuals and substantially improved model predictive performance. The RAC approach showed stronger inferential performance than the standard autologistic approach, as parameter estimates were more accurate and statistically significant variables were accurately identified. The new RAC approach presented here has the potential to account for spatial autocorrelation while maintaining strong predictive and inferential performance, and can be implemented across a range of modelling approaches.     

Linearized Forms of Individual-Level Models for Large-Scale Spatial Infectious Disease Systems

Individual-level models (ILMs) for infectious diseases, fitted in a Bayesian MCMC framework, are an intuitive and flexible class of models that can take into account population heterogeneity via various individual-level covariates. ILMs containing a geometric distance kernel to account for geographic heterogeneity provide a natural way to model the spatial spread of many diseases. However, in even only moderately large populations, the likelihood calculations required can be prohibitively time consuming. It is possible to speed up the computation via a technique which makes use a linearized distance kernel. Here we examine some methods of carrying out this linearization and compare the performances of these methods.     

Exponential distance statistics to detect the effects of population subdivision.

Statistical tests are needed to determine whether spatial structure has had a significant effect on the genetic differentiation of subpopulations. Here we introduce a new family of statistics based on a sum of an exponential function of the distances between individuals, which can be used with any genetic distance (e.g., nucleotide differences, number of nonshared alleles, or separation on a phylogenetic tree). The power of the tests to detect genetic differentiation in Wright-Fisher island models and stepping stone models was calculated for various sample sizes, rates of migration and mutation, and definitions of spatial neighborhoods. We found that our new test was in some cases more powerful than the Ks* statistic of Hudson et al. (Mol. Biol. Evol. 9, 138-151, 1992), but in all cases was slightly less powerful than both a traditional chi2 test without lumping of rare haplotypes and the S(nn) test of Hudson (Genetics 155, 2011-2014, 2000). However, when we applied our new tests to three data sets, we found in some cases highly significant results that were missed by the other tests.     

Landscape Habitat Suitability Index Software

ABSTRACT Habitat suitability index (HSI) models are traditionally used to evaluate habitat quality for wildlife at a local scale. Rarely have such models incorporated spatial relationships of habitat components. We introduce Landscape HSImodels, a new Microsoft Windows® (Microsoft, Redmond, WA)—based program that incorporates local habitat as well as landscape-scale attributes to evaluate habitats for 21 species of wildlife. Models for additional species can be constructed using the generic model option. At a landscape scale, attributes include edge effects, patch area, distance to resources, and habitat composition. A moving window approach is used to evaluate habitat composition and interspersion within areas typical of home ranges and territories or larger. The software and sample data are available free of charge from the United States Forest Service, Northern Research Station at http:www.nrs.fs.fed.ushsi .     

Performance of individual vs. group sampling for inferring dispersal under isolation‐by‐distance

Models of isolation‐by‐distance formalize the effects of genetic drift and gene flow in a spatial context where gene dispersal is spatially limited. These models have been used to show that, at an appropriate spatial scale, dispersal parameters can be inferred from the regression of genetic differentiation against geographic distance between sampling locations. This approach is compelling because it is relatively simple and robust and has rather low sampling requirements. In continuous populations, dispersal can be inferred from isolation‐by‐distance patterns using either individuals or groups as sampling units. Intrigued by empirical findings where individual samples seemed to provide more power, we used simulations to compare the performances of the two methods in a range of situations with different dispersal distributions. We found that sampling individuals provide more power in a range of dispersal conditions that is narrow but fits many realistic situations. These situations were characterized not only by the general steepness of isolation‐by‐distance but also by the intrinsic shape of the dispersal kernel. The performances of the two approaches are otherwise similar, suggesting that the choice of a sampling unit is globally less important than other settings such as a study's spatial scale.     

Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression

In geo-statistics, the Durbin-Watson test is frequently employed to detect the presence of residual serial correlation from least squares regression analyses. However, the Durbin-Watson statistic is only suitable for ordered time or spatial series. If the variables comprise cross-sectional data coming from spatial random sampling, the test will be ineffectual because the value of Durbin-Watson’s statistic depends on the sequence of data points. This paper develops two new statistics for testing serial correlation of residuals from least squares regression based on spatial samples. By analogy with the new form of Moran’s index, an autocorrelation coefficient is defined with a standardized residual vector and a normalized spatial weight matrix. Then by analogy with the Durbin-Watson statistic, two types of new serial correlation indices are constructed. As a case study, the two newly presented statistics are applied to a spatial sample of 29 China’s regions. These results show that the new spatial autocorrelation models can be used to test the serial correlation of residuals from regression analysis. In practice, the new statistics can make up for the deficiencies of the Durbin-Watson test.     

Genetic spatial autocorrelation can readily detect sex-biased dispersal

Sex-biased dispersal is expected to generate differences in the fine-scale genetic structure of males and females. Therefore, spatial analyses of multilocus genotypes may offer a powerful approach for detecting sex-biased dispersal in natural populations. However, the effects of sex-biased dispersal on fine-scale genetic structure have not been explored. We used simulations and multilocus spatial autocorrelation analysis to investigate how sex-biased dispersal influences fine-scale genetic structure. We evaluated three statistical tests for detecting sex-biased dispersal: bootstrap confidence intervals about autocorrelation r values and recently developed heterogeneity tests at the distance class and whole correlogram levels. Even modest sex bias in dispersal resulted in significantly different fine-scale spatial autocorrelation patterns between the sexes. This was particularly evident when dispersal was strongly restricted in the less-dispersing sex (mean distance <200 m), when differences between the sexes were readily detected over short distances. All tests had high power to detect sex-biased dispersal with large sample sizes (n ≥ 250). However, there was variation in type I error rates among the tests, for which we offer specific recommendations. We found congruence between simulation predictions and empirical data from the agile antechinus, a species that exhibits male-biased dispersal, confirming the power of individual-based genetic analysis to provide insights into asymmetries in male and female dispersal. Our key recommendations for using multilocus spatial autocorrelation analyses to test for sex-biased dispersal are: (i) maximize sample size, not locus number; (ii) concentrate sampling within the scale of positive structure; (iii) evaluate several distance class sizes; (iv) use appropriate methods when combining data from multiple populations; (v) compare the appropriate groups of individuals.     

An investigation into the influence of boundary condition specification in finite difference methods on the behaviour of passive and active neuronal models

Neuronal models provide a major aid to understanding the behaviour of individual neurons and networks of neurons. The solution of the model equations by finite difference methods is widespread because of the inherent simplicity of the technique. Error in the finite difference approach due to spatial and temporal discretisation is shown to be equivalent to a mis-specification of membrane current density. The effect of this mis-specification on the accuracy of the solution to the model equations is shown to depend on the structure of the model and its input, as well as the size of the discretisation intervals themselves. Through a theoretical analysis, illustrated by a number of examples on passive and active dendrites, this article demonstrates that the accuracy with which core current is implemented numerically at segment end-points in elementary models influences the behaviour of the numerical solution of these models, and consequently any physiological conclusions drawn from them.     

具有空间自相关残差的回归模型及其应用

提出了具有空间自相关残差的多变量回归问题,这种残差包括单点、局部和区域化三种尺度.文中提出了两种处理空间自相关的方法:把自相关加入残差协方差作参数的改进估计──相邻相关方法和把残差作区域化最优自相关估计──一般空间自相关方法.用森林对舞毒蛾危害敏感性概率的大范围估计为例,比较了传统回归方法和本文方法间的差异,交叉检验结果表明本文的方法明显优于直接回归方法,这说明加入空间自相关对空间回归问题是必要的.     

Residuals for proportional hazards models with interval-censored survival data

We develop diagnostic tools for use with proportional hazards models for interval-censored survival data. We propose counterparts to the Cox-Snell, Lagakos (or martingale), deviance, and Schoenfeld residuals. Many of the properties of these residuals carry over to the interval-censored case. In particular, the interval-censored versions of the Lagakos and Schoenfeld residuals may be derived as components of suitable score statistics. The Lagakos residuals may be used to check regression relationships, while the Schoenfeld residuals can help to detect nonproportional hazards in semiparametric models. The methods apply to parametric models and to the semiparametric model with discrete observation times.     

Genome-wide linkage analysis of systolic blood pressure: a comparison of two approaches to phenotype definition

Problem 1 of the Genetic Analysis Workshop 13(GAW13) contains longitudinal data of cardiovascular measurements from 330 pedigrees. The longitudinal data complicates the phenotype definition because multiple measurements are taken on each individual. To address this complication, we propose an approach that uses generalized estimating equations to obtain residuals for each time point for each person. The mean residual is then taken as the new phenotype with which to use in a variance components linkage analysis. We compare our phenotype definition approach to an approach that first reduces the multiple measurements to a single measurement and then models these summary statistics as regression terms in a variance components analysis. For each approach, multipoint linkage analysis was performed using the residuals and the SOLAR computer program. Our results show little difference between the methods based on the LOD scores.     

Red herrings revisited: spatial autocorrelation and parameter estimation in geographical ecology

There have been numerous claims in the ecological literature that spatial autocorrelation in the residuals of ordinary least squares (OLS) regression models results in shifts in the partial coefficients, which bias the interpretation of factors influencing geographical patterns. We evaluate the validity of these claims using gridded species richness data for the birds of North America, South America, Europe, Africa, the ex‐USSR, and Australia. We used richness in 110×110 km cells and environmental predictor variables to generate OLS and simultaneous autoregressive (SAR) multiple regression models for each region. Spatial correlograms of the residuals from each OLS model were then used to identify the minimum distance between cells necessary to avoid short‐distance residual spatial autocorrelation in each data set. This distance was used to subsample cells to generate spatially independent data. The partial OLS coefficients estimated with the full dataset were then compared to the distributions of coefficients created with the subsamples. We found that OLS coefficients generated from data containing residual spatial autocorrelation were statistically indistinguishable from coefficients generated from the same data sets in which short‐distance spatial autocorrelation was not present in all 22 coefficients tested. Consistent with the statistical literature on this subject, we conclude that coefficients estimated from OLS regression are not seriously affected by the presence of spatial autocorrelation in gridded geographical data. Further, shifts in coefficients that occurred when using SAR tended to be correlated with levels of uncertainty in the OLS coefficients. Thus, shifts in the relative importance of the predictors between OLS and SAR models are expected when small‐scale patterns for these predictors create weaker and more unstable broad‐scale coefficients. Our results indicate both that OLS regression is unbiased and that differences between spatial and nonspatial regression models should be interpreted with an explicit awareness of spatial scale.     

Detection of microspheres in vivo using multispectral optoacoustic tomography

We introduce a new approach to detect individual microparticles that contain NIR fluorescent dye by multispectral optoacoustic tomography in the context of the hemoglobin-rich environment within murine liver. We encapsulated a near infrared (NIR) fluorescent dye within polystyrene microspheres, then injected them into the ileocolic vein, which drains to the liver. NIR absorption was determined using multispectral optoacoustic tomography. To quantitate the minimum diameter of microspheres, we used both colorimetric and spatial information to segment the regions in which the microspheres appear. Regional diameter was estimated by doubling the maximum regional distance. We found that the minimum microsphere size threshold for detection by multispectral optoacoustic tomography images is 78.9 µm.     

A likelihood-based approach to estimating and testing for isolation by distance

Simple regression of genetic similarities between pairs of populations on their corresponding geographic distances is frequently used to detect the presence of isolation by distance (IBD). However, these pairwise values are obviously not independent and there is no parametric procedure for estimating and testing for the IBD intercepts and slopes based on standard regression theory. Nonparametric tests, such as the Mantel test, and resampling techniques, such as bootstrapping, have been exploited with limited success. Here, I describe a likelihood-based analysis to allow for simultaneously detecting patterns of correlated residuals and estimating and testing for the presence of IBD. It is shown, through the analysis of two molecular datasets in pine species, that different covariance structures of the residuals exist. More over, the likelihood ratio tests under these covariance structures are less sensitive to the presence of IBD than the Mantel test and the simple regression analysis but more sensitive than the bootstrap and jackknife samples over independent populations or population pairs. Because the likelihood analysis directly models and accounts for nonindependence of residuals, it should legitimately detect the presence of IBD, thereby allowing for accurate inferences about evolutionary and demographic processes influencing the extent and patterns of IBD.     

High-resolution spatial normalization for microarrays containing embedded technical replicates

MOTIVATION: Microarray data are susceptible to a wide-range of artifacts, many of which occur on physical scales comparable to the spatial dimensions of the array. These artifacts introduce biases that are spatially correlated. The ability of current methodologies to detect and correct such biases is limited. RESULTS: We introduce a new approach for analyzing spatial artifacts, termed 'conditional residual analysis for microarrays' (CRAM). CRAM requires a microarray design that contains technical replicates of representative features and a limited number of negative controls, but is free of the assumptions that constrain existing analytical procedures. The key idea is to extract residuals from sets of matched replicates to generate residual images. The residual images reveal spatial artifacts with single-feature resolution. Surprisingly, spatial artifacts were found to coexist independently as additive and multiplicative errors. Efficient procedures for bias estimation were devised to correct the spatial artifacts on both intensity scales. In a survey of 484 published single-channel datasets, variance fell 4- to 12-fold in 5% of the datasets after bias correction. Thus, inclusion of technical replicates in a microarray design affords benefits far beyond what one might expect with a conventional 'n = 5' averaging, and should be considered when designing any microarray for which randomization is feasible. AVAILABILITY: CRAM is implemented as version 2 of the hoptag software package for R, which is included in the Supplementary information.     

Region-Based Association Analysis of Human Quantitative Traits in Related Individuals

Regional-based association analysis instead of individual testing of each SNP was introduced in genome-wide association studies to increase the power of gene mapping, especially for rare genetic variants. For regional association tests, the kernel machine-based regression approach was recently proposed as a more powerful alternative to collapsing-based methods. However, the vast majority of existing algorithms and software for the kernel machine-based regression are applicable only to unrelated samples. In this paper, we present a new method for the kernel machine-based regression association analysis of quantitative traits in samples of related individuals. The method is based on the GRAMMAR+ transformation of phenotypes of related individuals, followed by use of existing kernel machine-based regression software for unrelated samples. We compared the performance of kernel-based association analysis on the material of the Genetic Analysis Workshop 17 family sample and real human data by using our transformation, the original untransformed trait, and environmental residuals. We demonstrated that only the GRAMMAR+ transformation produced type I errors close to the nominal value and that this method had the highest empirical power. The new method can be applied to analysis of related samples by using existing software for kernel-based association analysis developed for unrelated samples.     

Geographic profiling as a novel spatial tool for targeting the control of invasive species

Geographic profiling (GP) was originally developed as a statistical tool in criminology, where it uses the spatial locations of linked crimes (for example murder, rape or arson) to identify areas that are most likely to include the offender's residence. The technique has been successful in this field, and is now widely used by police forces and investigative agencies around the world. Here, we show that this novel technique can also be used to identify source populations of invasive species, using their current locations as input, as a prelude to targeted control measures. Our study has two main parts. In the first, we use computer simulations to compare GP to other simple measures of spatial central tendency (centre of minimum distance, spatial mean, spatial median), as well as to a more sophisticated single parameter kernel density model. GP performs significantly better than any of these other approaches. In the second part of the study, we analyse historical data from the Biological Records Centre (BRC) for 53 invasive species in Great Britain, ranging from marine invertebrates to woody trees, and from a wide variety of habitats (including littoral habitats, woodland and man‐made habitats). For 52 of these 53 data sets, GP outperforms spatial mean, spatial median and centre of minimum distance as a search strategy, particularly as the number of sources (or potential sources) increases. We analyse one of these data sets, for Heracleum mantegazzianum, in more detail, and show that GP also outperforms the kernel density model. Finally, we compare fitted parameter values between different species, groups and habitat types, with a view to identifying general values that might be used for novel invasions where data are lacking. We suggest that geographic profiling could potentially form a useful component of integrated control strategies relating to a wide variety of invasive species.     

Modelling the spatial distribution of White Stork Ciconia ciconia breeding populations in Southeast Europe

Capsule Spatial environmental modelling well predicted nesting distribution of the White stork in Southeast Europe and can be used in conservation planning with respect to climate change.

Aims To create spatial models for predicting White Stork presence and densities in the Southeast Europe to identify areas of suitable habitat for White Storks.

Methods We quantified the habitat used by nesting White storks in Southeast Europe. Using spatial modelling, we defined a set of free and available online environmental variables that predict the breeding localities of the species. We employed pseudo-absences and the kriging of the residuals in order to create predictive models of nest presence and density.

Results The presence–absence model was found to be precise in predicting the presence of nests. Both density and presence of breeding pairs were best explained negatively by elevation, slope, minimum temperature during May, and distance to the nearest human settlement and positively by topographic wetness index, total area of human settlement and spring precipitation.

Conclusion Our robust and easily repeatable models offer a conservation tool to reveal suitable but unoccupied localities for breeding White Storks pairs which may inform our understanding of how climate change might affect the species' distribution in the future. For example, protecting White Storks on the Dalmatian coast may become even more significant in the future, because the Dalmatian coast is predicted as the only suitable breeding area in Croatia later this century.     

A framework for testing assumptions about foraging scales,body mass,and niche separation using telemetry data

Ecological theory predicts that if animals with very similar dietary requirements inhabit the same landscape, then they should avoid niche overlap by either exploiting food resources at different times or foraging at different spatial scales. Similarly, it is often assumed that animals that fall in different body mass modes and share the same body plan will use landscapes at different spatial scales. We developed a new methodological framework for understanding the scaling of foraging (i.e. the range and distribution of scales at which animals use their landscapes) by applying a combination of three well‐established methods to satellite telemetry data to quantify foraging patch size distributions: (1) first‐passage time analysis; (2) a movement‐based kernel density estimator; and (3) statistical comparison of resulting histograms and tests for multimodality. We demonstrate our approach using two sympatric, ecologically similar species of African ducks with quite different body masses: Egyptian Geese (actually a shelduck), and Red‐billed Teal. Contrary to theoretical predictions, the two species, which are sympatric throughout the year, foraged at almost identical spatial scales. Our results show how ecologists can use GPS tracking data to explicitly quantify and compare the scales of foraging by different organisms within an animal community. Our analysis demonstrates both a novel approach to foraging data analysis and the need for caution when making assumptions about the relationships among niche separation, diet, and foraging scale.     

Investigating the endemic transmission of the hepatitis C virus

The hepatitis C virus (HCV) infects at least 3% of people worldwide and is a leading global cause of liver disease. Although HCV spread epidemically during the 20th century, particularly by blood transfusion, it has clearly been present in human populations for several centuries. Here we attempt to redress the paucity of investigation into how long-term endemic transmission of HCV has been maintained. Such transmission not only represents the 'natural' route of infection but also contributes to new infections today. As a first step, we investigate the hypothesis that HCV can be mechanically transmitted by biting arthropods. Firstly, we use a combined bioinformatic and geographic approach to build a spatial database of endemic HCV infection and demonstrate that this can be used to geographically compare endemic HCV with the range distributions of potential vector species. Second, we use models from mathematical epidemiology to investigate if the parameters that describe the biting behaviour of vectors are consistent with a proposed basic reproduction number (R0) for HCV, and with the sustained transmission of the virus by mechanical transmission. Our analyses indicate that the mechanical transmission of HCV is plausible and that much further research into endemic HCV is needed.