Spatial regression techniques for inter‐population data: studying the relationships between morphological and environmental variation

Understanding the importance of environmental dimensions behind the morphological variation among populations has long been a central goal of evolutionary biology. The main objective of this study was to review the spatial regression techniques employed to test the association between morphological and environmental variables. In addition, we show empirically how spatial regression techniques can be used to test the association of cranial form variation among worldwide human populations with a set of ecological variables, taking into account the spatial autocorrelation in data. We suggest that spatial autocorrelation must be studied to explore the spatial structure underlying morphological variation and incorporated in regression models to provide more accurate statistical estimates of the relationships between morphological and ecological variables. Finally, we discuss the statistical properties of these techniques and the underlying reasons for using the spatial approach in population studies.     

Model selection and information theory in geographical ecology

Aim   Although parameter estimates are not as affected by spatial autocorrelation as Type I errors, the change from classical null hypothesis significance testing to model selection under an information theoretic approach does not completely avoid problems caused by spatial autocorrelation. Here we briefly review the model selection approach based on the Akaike information criterion (AIC) and present a new routine for Spatial Analysis in Macroecology (SAM) software that helps establishing minimum adequate models in the presence of spatial autocorrelation.
Innovation    We illustrate how a model selection approach based on the AIC can be used in geographical data by modelling patterns of mammal species in South America represented in a grid system ( n  = 383) with 2° of resolution, as a function of five environmental explanatory variables, performing an exhaustive search of minimum adequate models considering three regression methods: non-spatial ordinary least squares (OLS), spatial eigenvector mapping and the autoregressive (lagged-response) model. The models selected by spatial methods included a smaller number of explanatory variables than the one selected by OLS, and minimum adequate models contain different explanatory variables, although model averaging revealed a similar rank of explanatory variables.
Main conclusions    We stress that the AIC is sensitive to the presence of spatial autocorrelation, generating unstable and overfitted minimum adequate models to describe macroecological data based on non-spatial OLS regression. Alternative regression techniques provided different minimum adequate models and have different uncertainty levels. Despite this, the averaged model based on Akaike weights generates consistent and robust results across different methods and may be the best approach for understanding of macroecological patterns.     

Spatial autocorrelation and the analysis of invasion processes from distribution data: a study with the crayfish <Emphasis Type="Italic">Procambarus clarkii</Emphasis>

Complex spatial dynamics are frequent in invasive species; analyzing distribution patterns can help to understand the mechanisms driving invasions. We used different spatial regression techniques to evaluate processes determining the invasion of the red swamp crayfish Procambarus clarkii. We evaluated four a priori hypotheses on processes that may determine crayfish invasion: landscape alteration, connectivity, wetland suitability for abiotic and biotic features. We assessed the distribution of P. clarkii in 119 waterbodies in a recently invaded area. We used spatially explicit statistical techniques (spatial eigenvector mapping, generalized additive models, Bayesian intrinsic conditional autoregressive models) within an information-theoretic framework to assess the support of hypotheses; we also analyzed the pattern of spatial autocorrelation of data, model residuals, and eigenvectors. We found strong agreement between the results of spatial eigenvector mapping and Bayesian autoregressive models. Procambarus clarkii was significantly associated with the largest, permanent wetlands. Additive models suggested also association with human-dominated landscapes, but tended to overfit data. The results indicate that abiotic wetlands features and landscape alteration are major drivers of the species’ distribution. Species distribution data, residuals of ordinary least squares regression, and spatial eigenvectors all showed positive and significant spatial autocorrelation at distances up to 2,500 m; this may be caused by the dispersal ability of the species. Our analyses help to understand the processes determining the invasion and to identify the areas most at risk where screening and early management efforts can be focused. The comparison of multiple spatial techniques allows a robust assessment of factors determining complex distribution patterns.     

EcoGenetics: An R package for the management and exploratory analysis of spatial data in landscape genetics

The integration of ecology and genetics has become established in recent decades, in hand with the development of new technologies, whose implementation is allowing an improvement of the tools used for data analysis. In a landscape genetics context, integrative management of population information from different sources can make spatial studies involving phenotypic, genotypic and environmental data simpler, more accessible and faster. Tools for exploratory analysis of autocorrelation can help to uncover the spatial genetic structure of populations and generate appropriate hypotheses in searching for possible causes and consequences of their spatial processes. This study presents EcoGenetics, an R package with tools for multisource management and exploratory analysis in landscape genetics.     

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.     

Importance of spatial autocorrelation in modeling bird distributions at a continental scale

Spatial autocorrelation in species' distributions has been recognized as inflating the probability of a type I error in hypotheses tests, causing biases in variable selection, and violating the assumption of independence of error terms in models such as correlation or regression. However, it remains unclear whether these problems occur at all spatial resolutions and extents, and under which conditions spatially explicit modeling techniques are superior. Our goal was to determine whether spatial models were superior at large extents and across many different species. In addition, we investigated the importance of purely spatial effects in distribution patterns relative to the variation that could be explained through environmental conditions. We studied distribution patterns of 108 bird species in the conterminous United States using ten years of data from the Breeding Bird Survey. We compared the performance of spatially explicit regression models with non-spatial regression models using Akaike's information criterion. In addition, we partitioned the variance in species distributions into an environmental, a pure spatial and a shared component. The spatially-explicit conditional autoregressive regression models strongly outperformed the ordinary least squares regression models. In addition, partialling out the spatial component underlying the species' distributions showed that an average of 17% of the explained variation could be attributed to purely spatial effects independent of the spatial autocorrelation induced by the underlying environmental variables. We concluded that location in the range and neighborhood play an important role in the distribution of species. Spatially explicit models are expected to yield better predictions especially for mobile species such as birds, even in coarse-grained models with a large extent.     

Spatial autocorrelation in predictors reduces the impact of positional uncertainty in occurrence data on species distribution modelling

Aim To investigate the impact of positional uncertainty in species occurrences on the predictions of seven commonly used species distribution models (SDMs), and explore its interaction with spatial autocorrelation in predictors. Methods A series of artificial datasets covering 155 scenarios including different combinations of five positional uncertainty scenarios and 31 spatial autocorrelation scenarios were simulated. The level of positional uncertainty was defined by the standard deviation of a normally distributed zero‐mean random variable. Each dataset included two environmental gradients (predictor variables) and one set of species occurrence sample points (response variable). Seven commonly used models were selected to develop SDMs: generalized linear models, generalized additive models, boosted regression trees, multivariate adaptive regression spline, random forests, genetic algorithm for rule‐set production and maximum entropy. A probabilistic approach was employed to model and simulate five levels of error in the species locations. To analyse the propagation of positional uncertainty, Monte Carlo simulation was applied to each scenario for each SDM. The models were evaluated for performance using simulated independent test data with Cohen’s Kappa and the area under the receiver operating characteristic curve. Results Positional uncertainty in species location led to a reduction in prediction accuracy for all SDMs, although the magnitude of the reduction varied between SDMs. In all cases the magnitude of this impact varied according to the degree of spatial autocorrelation in predictors and the levels of positional uncertainty. It was shown that when the range of spatial autocorrelation in the predictors was less than or equal to three times the standard deviation of the positional error, the models were less affected by error and, consequently, had smaller decreases in prediction accuracy. When the range of spatial autocorrelation in predictors was larger than three times the standard deviation of positional error, the prediction accuracy was low for all scenarios. Main conclusions The potential impact of positional uncertainty in species occurrences on the predictions of SDMs can be understood by comparing it with the spatial autocorrelation range in predictor variables.     

Towards an integrated computational tool for spatial analysis in macroecology and biogeography

Because most macroecological and biodiversity data are spatially autocorrelated, special tools for describing spatial structures and dealing with hypothesis testing are usually required. Unfortunately, most of these methods have not been available in a single statistical package. Consequently, using these tools is still a challenge for most ecologists and biogeographers. In this paper, we present sam (Spatial Analysis in Macroecology), a new, easy-to-use, freeware package for spatial analysis in macroecology and biogeography. Through an intuitive, fully graphical interface, this package allows the user to describe spatial patterns in variables and provides an explicit spatial framework for standard techniques of regression and correlation. Moran's I autocorrelation coefficient can be calculated based on a range of matrices describing spatial relationships, for original variables as well as for residuals of regression models, which can also include filtering components (obtained by standard trend surface analysis or by principal coordinates of neighbour matrices). sam also offers tools for correcting the number of degrees of freedom when calculating the significance of correlation coefficients. Explicit spatial modelling using several forms of autoregression and generalized least-squares models are also available. We believe this new tool will provide researchers with the basic statistical tools to resolve autocorrelation problems and, simultaneously, to explore spatial components in macroecological and biogeographical data. Although the program was designed primarily for the applications in macroecology and biogeography, most of sam 's statistical tools will be useful for all kinds of surface pattern spatial analysis. The program is freely available at http://www.ecoevol.ufg.br/sam (permanent URL at http://purl.oclc.org/sam/ ).     

Pattern‐oriented modelling of population genetic structure

Although several statistical approaches can be used to describe patterns of genetic variation and infer stochastic differentiation, selective responses, or interruptions of gene flow due to physical or environmental barriers, it is worthwhile to note that similar processes, controlled by several parameters in theoretical models, frequently give rise to similar patterns. Here, we develop a Pattern‐Oriented Modelling (POM) approach that allows us to determine how a complex set of parameters potentially driving empirical genetic differentiation among populations generate alternative scenarios that can be fitted to observed data. We generated 10 000 random combinations of parameters related to population size, gene flow and response to gradients (both driven by dispersal and selection) in a spatially explicit model, and analysed simulated patterns with F_ST statistics and mean correlograms using Moran's I spatial autocorrelation coefficients. These statistics were compared with observed patterns for a tree species endemic to the Brazilian Cerrado. For a best match with observed F_ST (equal to 0.182), the important parameters driving simulated scenario are mainly related to population structure, including low population size with closed populations (low N_m), strong distance decay of gene flow, in addition to a strong effect of the initial variance of allele frequencies. These scenarios present a low autocorrelation of allele frequencies. Best matching of correlograms, on the other hand, appears in simulations with a large population size, high N_m and low population differentiation and F_ST (as well as more gene flow). Thus, targeting the two statistics (correlograms and F_ST) shows that best matches with empirical data with two distinct sets of parameters in the simulations, because observed patterns involve both a relatively high F_ST and significant autocorrelation. This conflict can be resolved by assuming that initial variance in allele frequencies can be interpreted as reflecting deep‐time historical variation and evolutionary dynamics of allele frequencies, creating a relatively high level of population differentiation, whereas current patterns in gene flow creates spatial autocorrelation. This make sense in terms of the previous knowledge on population differentiation in D. alata, especially if patterns are explained by a combination of isolation‐by‐distance and allelic surfing due to range expansion after the last glacial maximum. This reveals the potential for more complex applications of POM in population genetics. © 2014 The Linnean Society of London, Biological Journal of the Linnean Society, 2014, 113 , 1152–1161.     

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.     

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.     

Conditionally autoregressive models improve occupancy analyses of autocorrelated data: An example with environmental DNA

Site occupancy‐detection models (SODMs) are statistical models widely used for biodiversity surveys where imperfect detection of species occurs. For instance, SODMs are increasingly used to analyse environmental DNA (eDNA) data, taking into account the occurrence of both false‐positive and false‐negative errors. However, species occurrence data are often characterized by spatial and temporal autocorrelation, which might challenge the use of standard SODMs. Here we reviewed the literature of eDNA biodiversity surveys and found that most of studies do not take into account spatial or temporal autocorrelation. We then demonstrated how the analysis of data with spatial or temporal autocorrelation can be improved by using a conditionally autoregressive SODM, and show its application to environmental DNA data. We tested the autoregressive model on both simulated and real data sets, including chronosequences with different degrees of autocorrelation, and a spatial data set on a virtual landscape. Analyses of simulated data showed that autoregressive SODMs perform better than traditional SODMs in the estimation of key parameters such as true‐/false‐positive rates and show a better discrimination capacity (e.g., higher true skill statistics). The usefulness of autoregressive SODMs was particularly high in data sets with strong autocorrelation. When applied to real eDNA data sets (eDNA from lake sediment cores and freshwater), autoregressive SODM provided more precise estimation of true‐/false‐positive rates, resulting in more reasonable inference of occupancy states. Our results suggest that analyses of occurrence data, such as many applications of eDNA, can be largely improved by applying conditionally autoregressive specifications to SODMs.     

Can niche‐based distribution models outperform spatial interpolation?

Aim Distribution modelling relates sparse data on species occurrence or abundance to environmental information to predict the population of a species at any point in space. Recently, the importance of spatial autocorrelation in distributions has been recognized. Spatial autocorrelation can be categorized as exogenous (stemming from autocorrelation in the underlying variables) or endogenous (stemming from activities of the organism itself, such as dispersal). Typically, one asks whether spatial models explain additional variability (endogenous) in comparison to a fully specified habitat model. We turned this question around and asked: can habitat models explain additional variation when spatial structure is accounted for in a fully specified spatially explicit model? The aim was to find out to what degree habitat models may be inadvertently capturing spatial structure rather than true explanatory mechanisms. Location We used data from 190 species of the North American Breeding Bird Survey covering the conterminous United States and southern Canada. Methods We built 13 different models on 190 bird species using regression trees. Our habitat‐based models used climate and landcover variables as independent variables. We also used random variables and simulated ranges to validate our results. The two spatially explicit models included only geographical coordinates or a contagion term as independent variables. As another angle on the question of mechanism vs. spatial structure we pitted a model using related bird species as predictors against a model using randomly selected bird species. Results The spatially explicit models outperformed the traditional habitat models and the random predictor species outperformed the related predictor species. In addition, environmental variables produced a substantial R² in predicting artificial ranges. Main conclusions We conclude that many explanatory variables with suitable spatial structure can work well in species distribution models. The predictive power of environmental variables is not necessarily mechanistic, and spatial interpolation can outperform environmental explanatory variables.     

Challenges in analysis and interpretation of microsatellite data for population genetic studies

Advancing technologies have facilitated the ever‐widening application of genetic markers such as microsatellites into new systems and research questions in biology. In light of the data and experience accumulated from several years of using microsatellites, we present here a literature review that synthesizes the limitations of microsatellites in population genetic studies. With a focus on population structure, we review the widely used fixation (F_ST) statistics and Bayesian clustering algorithms and find that the former can be confusing and problematic for microsatellites and that the latter may be confounded by complex population models and lack power in certain cases. Clustering, multivariate analyses, and diversity‐based statistics are increasingly being applied to infer population structure, but in some instances these methods lack formalization with microsatellites. Migration‐specific methods perform well only under narrow constraints. We also examine the use of microsatellites for inferring effective population size, changes in population size, and deeper demographic history, and find that these methods are untested and/or highly context‐dependent. Overall, each method possesses important weaknesses for use with microsatellites, and there are significant constraints on inferences commonly made using microsatellite markers in the areas of population structure, admixture, and effective population size. To ameliorate and better understand these constraints, researchers are encouraged to analyze simulated datasets both prior to and following data collection and analysis, the latter of which is formalized within the approximate Bayesian computation framework. We also examine trends in the literature and show that microsatellites continue to be widely used, especially in non‐human subject areas. This review assists with study design and molecular marker selection, facilitates sound interpretation of microsatellite data while fostering respect for their practical limitations, and identifies lessons that could be applied toward emerging markers and high‐throughput technologies in population genetics.     

Non-stationarity and local approaches to modelling the distributions of wildlife

Despite a growing interest in species distribution modelling, relatively little attention has been paid to spatial autocorrelation and non-stationarity. Both spatial autocorrelation (the tendency for adjacent locations to be more similar than distant ones) and non-stationarity (the variation in modelled relationships over space) are likely to be common properties of ecological systems. This paper focuses on non-stationarity and uses two local techniques, geographically weighted regression (GWR) and varying coefficient modelling (VCM), to assess its impact on model predictions. We extend two published studies, one on the presence–absence of calandra larks in Spain and the other on bird species richness in Britain, to compare GWR and VCM with the more usual global generalized linear modelling (GLM) and generalized additive modelling (GAM). For the calandra lark data, GWR and VCM produced better-fitting models than GLM or GAM. VCM in particular gave significantly reduced spatial autocorrelation in the model residuals. GWR showed that individual predictors became stationary at different spatial scales, indicating that distributions are influenced by ecological processes operating over multiple scales. VCM was able to predict occurrence accurately on independent data from the same geographical area as the training data but not beyond, whereas the GAM produced good results on all areas. Individual predictions from the local methods often differed substantially from the global models. For the species richness data, VCM and GWR produced far better predictions than ordinary regression. Our analyses suggest that modellers interpolating data to produce maps for practical actions (e.g. conservation) should consider local methods, whereas they should not be used for extrapolation to new areas. We argue that local methods are complementary to global methods, revealing details of habitat associations and data properties which global methods average out and miss.     

Methods to account for spatial autocorrelation in the analysis of species distributional data: a review

Species distributional or trait data based on range map (extent‐of‐occurrence) or atlas survey data often display spatial autocorrelation, i.e. locations close to each other exhibit more similar values than those further apart. If this pattern remains present in the residuals of a statistical model based on such data, one of the key assumptions of standard statistical analyses, that residuals are independent and identically distributed (i.i.d), is violated. The violation of the assumption of i.i.d. residuals may bias parameter estimates and can increase type I error rates (falsely rejecting the null hypothesis of no effect). While this is increasingly recognised by researchers analysing species distribution data, there is, to our knowledge, no comprehensive overview of the many available spatial statistical methods to take spatial autocorrelation into account in tests of statistical significance. Here, we describe six different statistical approaches to infer correlates of species’ distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations. A comprehensive comparison of the relative merits of these methods is beyond the scope of this paper. To demonstrate each method's implementation, however, we undertook preliminary tests based on simulated data. These preliminary tests verified that most of the spatial modeling techniques we examined showed good type I error control and precise parameter estimates, at least when confronted with simplistic simulated data containing spatial autocorrelation in the errors. However, we found that for presence/absence data the results and conclusions were very variable between the different methods. This is likely due to the low information content of binary maps. Also, in contrast with previous studies, we found that autocovariate methods consistently underestimated the effects of environmental controls of species distributions. Given their widespread use, in particular for the modelling of species presence/absence data (e.g. climate envelope models), we argue that this warrants further study and caution in their use. To aid other ecologists in making use of the methods described, code to implement them in freely available software is provided in an electronic appendix.     

Validation of species–climate impact models under climate change

Increasing concern over the implications of climate change for biodiversity has led to the use of species–climate envelope models to project species extinction risk under climate‐change scenarios. However, recent studies have demonstrated significant variability in model predictions and there remains a pressing need to validate models and to reduce uncertainties. Model validation is problematic as predictions are made for events that have not yet occurred. Resubstituition and data partitioning of present‐day data sets are, therefore, commonly used to test the predictive performance of models. However, these approaches suffer from the problems of spatial and temporal autocorrelation in the calibration and validation sets. Using observed distribution shifts among 116 British breeding‐bird species over the past ～20 years, we are able to provide a first independent validation of four envelope modelling techniques under climate change. Results showed good to fair predictive performance on independent validation, although rules used to assess model performance are difficult to interpret in a decision‐planning context. We also showed that measures of performance on nonindependent data provided optimistic estimates of models' predictive ability on independent data. Artificial neural networks and generalized additive models provided generally more accurate predictions of species range shifts than generalized linear models or classification tree analysis. Data for independent model validation and replication of this study are rare and we argue that perfect validation may not in fact be conceptually possible. We also note that usefulness of models is contingent on both the questions being asked and the techniques used. Implementations of species–climate envelope models for testing hypotheses and predicting future events may prove wrong, while being potentially useful if put into appropriate context.     

The generalized linear model for spatial data: assessing the effects of environmental covariates on population density in the field

With interest in spatial ecology growing, correlational field studies are likely to become increasingly important. Unfortunately, ecological field data often do not follow the assumptions of classical statistics, so techniques like the popular and powerful multiple linear regression and its variants are often unreliable, and results can be misleading. The generalized linear model (GLM) is a flexible extension of linear regression that has proved especially useful for discrete data. In this paper, the technique is adapted to accommodate spatially correlated, discrete data. Specifically, to demonstrate the approach, Japanese beetle grub [Popillia japonica Newman (Coleoptera, Scarabaeidae)] population density in the field is modeled as a function of soil organic matter content. The response variable (grub counts in small soil samples) was a spatially autocorrelated, discrete random variable. Three classes of GLMs of the association between soil organic matter content and grub density were compared: (i) regression (assuming normally distributed response variable), (ii) GLM assuming negative binomial counts, and (iii) GLM based on the assumption that the counts conformed to Taylor's power law (TPL). Because the grubs were distributed in patches rather than at random, models that explicitly accounted for the spatial autocorrelation of grub counts were constructed, and compared with models that assumed independent observations. The fitted values for the discrete GLMs [viz., (ii) and (iii)] differed noticeably from the fitted values from multiple regression; but fitted values among the negative binomial and TPL GLMs were virtually identical, regardless of whether the spatial covariance was incorporated into a model, whether a spherical or exponential variogram model was used, or whether variance function parameters were estimated over a large or small scale. However, P‐values for the overall significance of the models depended heavily on whether the GLM assumed a discrete or continuous response variable, and whether or not spatial autocorrelation in the response variable was accounted for. On average, P‐values were 45‐fold higher in the spatial GLMs than in the non‐spatial and 23‐fold higher in the discrete GLMs than in the continuous.     

Spatial autocorrelation and red herrings in geographical ecology

Aim Spatial autocorrelation in ecological data can inflate Type I errors in statistical analyses. There has also been a recent claim that spatial autocorrelation generates ‘red herrings’, such that virtually all past analyses are flawed. We consider the origins of this phenomenon, the implications of spatial autocorrelation for macro‐scale patterns of species diversity and set out a clarification of the statistical problems generated by its presence. Location To illustrate the issues involved, we analyse the species richness of the birds of western/central Europe, north Africa and the Middle East. Methods Spatial correlograms for richness and five environmental variables were generated using Moran's I coefficients. Multiple regression, using both ordinary least‐squares (OLS) and generalized least squares (GLS) assuming a spatial structure in the residuals, were used to identify the strongest predictors of richness. Autocorrelation analyses of the residuals obtained after stepwise OLS regression were undertaken, and the ranks of variables in the full OLS and GLS models were compared. Results Bird richness is characterized by a quadratic north–south gradient. Spatial correlograms usually had positive autocorrelation up to c. 1600 km. Including the environmental variables successively in the OLS model reduced spatial autocorrelation in the residuals to non‐detectable levels, indicating that the variables explained all spatial structure in the data. In principle, if residuals are not autocorrelated then OLS is a special case of GLS. However, our comparison between OLS and GLS models including all environmental variables revealed that GLS de‐emphasized predictors with strong autocorrelation and long‐distance clinal structures, giving more importance to variables acting at smaller geographical scales. Conclusion Although spatial autocorrelation should always be investigated, it does not necessarily generate bias. Rather, it can be a useful tool to investigate mechanisms operating on richness at different spatial scales. Claims that analyses that do not take into account spatial autocorrelation are flawed are without foundation.     

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.