Ecography 30: 609Á628, 2007 doi: 10.1111/j.2007.0906-7590.05171.x # 2007 The Authors. Journal compilation # 2007 Ecography Subject Editor: Carsten Rahbek. Accepted 3 August 2007

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

Carsten F. Dormann, Jana M. McPherson, Miguel B. Arau´jo, Roger Bivand, Janine Bolliger, Gudrun Carl, Richard G. Davies, Alexandre Hirzel, Walter Jetz, W. Daniel Kissling, Ingolf Ku¨hn, Ralf Ohlemu¨ller, Pedro R. Peres-Neto, Bjo¨rn Reineking, Boris Schro¨der, Frank M. Schurr and Robert Wilson

C. F. Dormann ([email protected]), Dept of Computational Landscape Ecology, UFZ Helmholtz Centre for Environmental Research, Permoserstr. 15, DE-04318 Leipzig, Germany. Á J. M. McPherson, Dept of Biology, Dalhousie Univ., 1355 Oxford Street Halifax NS, B3H 4J1 Canada. Á M. B. Arau´jo, Dept de Biodiversidad y Biologı´a Evolutiva, Museo Nacional de Ciencias Naturales, CSIC, C/ Gutie´rrez Abascal, 2, ES-28006 Madrid, Spain, and Centre for Macroecology, Inst. of Biology, Universitetsparken 15, DK- 2100 Copenhagen Ø, Denmark. Á R. Bivand, Economic Geography Section, Dept of Economics, Norwegian School of Economics and Business Administration, Helleveien 30, NO-5045 Bergen, Norway. Á J. Bolliger, Swiss Federal Research Inst. WSL, Zu¨rcherstrasse 111, CH-8903 Birmensdorf, Switzerland. Á G. Carl and I. Ku¨hn, Dept of Community Ecology (BZF), UFZ Helmholtz Centre for Environmental Research, Theodor-Lieser-Strasse 4, DE-06120 Halle, Germany, and Virtual Inst. Macroecology, Theodor-Lieser- Strasse 4, DE-06120 Halle, Germany. Á R. G. Davies, Biodiversity and Macroecology Group, Dept of Animal and Plant Sciences, Univ. of Sheffield, Sheffield S10 2TN, U.K. Á A. Hirzel, Ecology and Evolution Dept, Univ. de Lausanne, Biophore Building, CH- 1015 Lausanne, Switzerland. Á W. Jetz, Ecology Behavior and Evolution Section, Div. of Biological Sciences, Univ. of California, San Diego, 9500 Gilman Drive, MC 0116, La Jolla, CA 92093-0116, USA. Á W. D. Kissling, Community and Macroecology Group, Inst. of Zoology, Dept of Ecology, Johannes Gutenberg Univ. of Mainz, DE-55099 Mainz, Germany, and Virtual Inst. Macroecology, Theodor-Lieser-Strasse 4, DE-06120 Halle, Germany. Á R. Ohlemu¨ller, Dept of Biology, Univ. of York, PO Box 373, York YO10 5YW, U.K. Á P. R. Peres-Neto, Dept of Biology, Univ. of Regina, SK, S4S 0A2 Canada, present address: Dept of Biological Sciences, Univ. of Quebec at Montreal, CP 8888, Succ. Centre Ville, Montreal, QC, H3C 3P8, Canada. Á B. Reineking, Forest Ecology, ETH Zurich CHN G 75.3, Universita¨tstr. 16, CH-8092 Zu¨rich, Switzerland. Á B. Schro¨der, Inst. for Geoecology, Univ. of Potsdam, Karl- Liebknecht-Strasse 24-25, DE-14476 Potsdam, Germany. Á F. M. Schurr, Plant Ecology and Nature Conservation, Inst. of Biochemistry and Biology, Univ. of Potsdam, Maulbeerallee 2, DE-14469 Potsdam, Germany. Á R. Wilson, A´rea de Biodiversidad y Conservacio´n, Escuela Superior de Ciencias Experimentales y Tecnologı´a, Univ. Rey Juan Carlos, Tulipa´n s/n, Mo´stoles, ES-28933 Madrid, Spain.

Species distributional or trait data based on map (extent-of-occurrence) or atlas 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 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 . 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 ; (conditional and simultaneous) autoregressive models and generalised . 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

609 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 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.

Species distributional data such as species range maps errors, occasionally even inverting the slope of relation- (extent-of-occurrence), breeding bird surveys and bio- ships from non- (Ku¨hn 2007). diversity atlases are a common source for analyses of A variety of methods have consequently been devel- species-environment relationships. These, in turn, form oped to correct for the effects of spatial autocorrelation the basis for conservation and management plans for (partially reviewed in Keitt et al. 2002, Miller et al. 2007, endangered species, for calculating distributions under see below), but only a few have made it into the future climate and land-use scenarios and other forms ecological literature. The aims of this paper are to 1) of environmental risk assessment. present and explain methods that account for spatial The analysis of spatial data is complicated by a autocorrelation in analyses of spatial data; the app- phenomenon known as spatial autocorrelation. Spatial roaches considered are: autocovariate regression, spatial autocorrelation (SAC) occurs when the values of vari- eigenvector mapping (SEVM), generalised least squares ables sampled at nearby locations are not independent (GLS), conditional autoregressive models (CAR), simul- from each other (Tobler 1970). The causes of spatial taneous autoregressive models (SAR), generalised linear autocorrelation are manifold, but three factors are mixed models (GLMM) and generalised estimation particularly common (Legendre and Fortin 1989, equations (GEE); 2) describe which of these methods Legendre 1993, Legendre and Legendre 1998): 1) can be used for which error distribution, and discuss biological processes such as speciation, extinction, potential problems with implementation; 3) illustrate dispersal or species interactions are distance-related; 2) how to implement these methods using simulated data non-linear relationships between environment and spe- sets and by providing computing code (Anon. 2005). cies are modelled erroneously as linear; 3) the statistical model fails to account for an important environmental Methods for dealing with spatial determinant that in itself is spatially structured and thus causes spatial structuring in the response (Besag 1974). autocorrelation The second and third points are not always referred to as Detecting and quantifying spatial autocorrelation spatial autocorrelation, but rather spatial dependency (Legendre et al. 2002). Since they also lead to auto- Before considering the use of modelling methods that correlated residuals, these are equally problematic. A account for spatial autocorrelation, it is a sensible first fourth source of spatial autocorrelation relates to spatial step to check whether spatial autocorrelation is in fact resolution, because coarser grains lead to a spatial likely to impact the planned analyses, i.e. if model of data. In all of these cases, SAC may residuals indeed display spatial autocorrelation. Check- confound the analysis of species distribution data. ing for spatial autocorrelation (SAC) has become a Spatial autocorrelation may be seen as both an commonplace exercise in geography and ecology (Sokal opportunity and a challenge for spatial analysis. It is an and Oden 1978a, b, Fortin and Dale 2005). Established opportunity when it provides useful information for procedures include (Isaaks and Shrivastava 1989, Perry inference of process from pattern (Palma et al. 1999) et al. 2002): Moran’s I plots (also termed Moran’s I by, for example, increasing our understanding of by Legendre and Legendre 1998), Geary’s contagious biotic processes such as population growth, c and semi-variograms. In all three cases a geographic dispersal, differential mortality, social measure of similarity (Moran’s I, Geary’s c) or organization or competition dynamics (Griffith and (variogram) of data points (i and j) is plotted as a Peres-Neto 2006). In most cases, however, the presence function of the distance between them (dij). Distances of spatial autocorrelation is seen as posing a serious are usually grouped into bins. Moran’s I-based correlo- shortcoming for hypothesis testing and grams typically show a decrease from some level of SAC (Lennon 2000, Dormann 2007b), because it violates to a value of 0 (or below; in the absence the assumption of independently and identically dis- of SAC: E(I)1/(nÁ1), where n size), tributed (i.i.d.) errors of most standard statistical indicating no SAC at some distance between locations. procedures (Anselin 2002) and hence inflates type I Variograms depict the opposite, with the variance

610 between pairs of points increasing up to a certain spatial eigenvector mapping seek to capture the spatial distance, where variance levels off. Variograms are more configuration in additional covariates, which are then commonly employed in descriptive , while added into a generalised (GLM). 2) correlograms are the prevalent graphical presentation in Generalised least squares (GLS) methods fit a var- ecology (Fortin and Dale 2005). iance- matrix based on the non-independence Values of Moran’s I are assessed by a test of spatial observations. Simultaneous autoregressive (the Moran’s I standard deviate) which indicates the models (SAR) and conditional autoregressive models statistical significance of SAC in e.g. model residuals. (CAR) do the same but in different ways to GLS, and Additionally, model residuals may be plotted as a map the generalised linear mixed models (GLMM) we that more explicitly reveals particular patterns of spatial employ for non-normal data are a generalisation of autocorrelation (e.g. anisotropy or non-stationarity of GLS. 3) Generalised estimating equations (GEE) split spatial autocorrelation). For further details and for- the data into smaller clusters before also modelling the mulae see e.g. Isaaks and Shrivastava (1989) or Fortin variance-covariance relationship. For comparison, the and Dale (2005). following non-spatial models were also employed: simple GLM and trend-surface generalised additive Assumptions common to all modelling models (GAM: Hastie and Tibshirani 1990, Wood approaches considered 2006), in which geographical location was fitted using splines as a trend-surface (as a two-dimensional spline All methods assume spatial stationarity, i.e. spatial on geographical coordinates). Trend surface GAM does autocorrelation and effects of environmental correlates not address the problem of spatial autocorrelation, but to be constant across the region, and there are very few merely accounts for trends in the data across larger methods to deal with non-stationarity (Osborne et al. geographical distances (Cressie 1993). A promising tool 2007). Stationarity may or may not be a reasonable which became available only recently is the use of assumption, depending, among other things, on the to remove spatial autocorrelation (Carl and spatial extent of the study. If the main cause of spatial Ku¨hn 2007b). However, the method was published too autocorrelation is dispersal (for example in research on recently to be included here and hence awaits further animal distributions), stationarity is likely to be testing. violated, for example when moving from a floodplain We also did not include Bayesian spatial models in to the mountains, where movement may be more this review. Several recent publications have employed restricted. One method able to accommodate spatial this method and provide a good coverage of its variation in autocorrelation is geographically weighted implementation (Osborne et al. 2001, Hooten et al. regression (Fotheringham et al. 2002), a method not 2003, Thogmartin et al. 2004, Gelfand et al. 2005, considered here because of its limited use for hypothesis Ku¨hn et al. 2006, Latimer et al. 2006). The Bayesian testing (coefficient estimates depend on spatial position) approach to spatial models used in these studies is based and because it was not designed to remove spatial either on a CAR or an autologistic implementation autocorrelation (see e.g. Kupfer and Farris 2007, for a similar to the one we used as a frequentist method. The GWR correlogram). Bayesian framework allows for a more flexible incor- Another assumption is that of isotropic spatial poration of other complications (observer bias, missing autocorrelation. This that the process causing data, different error distributions) but is much more the spatial autocorrelation acts in the same way in all computer-intensive then any of the methods presented directions. Environmental factors that may cause here. anisotropic spatial autocorrelation are wind (giving a Beyond the methods mentioned above, there are wind-dispersed organism a preferential direction), water also those which correct test statistics for spatial auto- currents (e.g. carrying plankton), or directionality in correlation. These include Dutilleul’s modified t-test soil transport (carrying seeds) from mountains to plains. (Dutilleul 1993) or the CRH-correction for correla- He et al. (2003) as well as Worm et al. (2005) provide tions (Clifford et al. 1989), randomisation tests such as examples of analyses accounting for anisotropy in partial Mantel tests (Legendre and Legendre 1998), or ecological data, and several of the methods described strategies employed by Lennon (2000), Liebhold and below can be adapted for such circumstances. Gurevitch (2002) and Segurado et al. (2006) which are all useful as a robust assessment of correlation between Description of spatial statistical modelling environmental and response variables. As these methods methods do not allow a correction of the parameter estimates, however, they are not considered further in this study. The methods we describe in the following fall broadly In the following sections we present a detailed descrip- into three groups. 1) Autocovariate regression and tion of all methods employed here.

611 1. Autocovariate models 2. Spatial eigenvector mapping (SEVM)

Autocovariate models address spatial autocorrelation by Spatial eigenvector mapping is based on the idea that estimating how much the response variable at any one the spatial arrangement of data points can be translated site reflects response values at surrounding sites. This is into explanatory variables, which capture spatial effects achieved through a simple extension of generalised at different spatial resolutions. During the analysis, linear models by adding a distance-weighted function of those eigenvectors that reduce spatial autocorrelation in neighbouring response values to the model’s explana- the residuals best are chosen explicitly as spatial tory variables. This extra parameter is known as the predictors. Since each eigenvector represents a particu- autocovariate. The autocovariate is intended to capture lar spatial patterning, SAC is effectively allowed to vary spatial autocorrelation originating from endogenous in space, relaxing the assumption of both spatial processes such as conspecific attraction, limited dis- isotropy and stationarity. Plotting these eigenvectors persal, contagious population growth, and movement reveals the spatial patterning of the spatial autocorrela- of censused individuals between sites (Smith tion (see Diniz-Filho and Bini 2005, for an example). 1994, Keitt et al. 2002, Yamaguchi et al. 2003). This method could thus be very useful for data with Adding the autocovariate transforms the linear SAC stemming from larger scale observation bias. predictor of a generalised linear model from its usual The method is based on the eigenfunction decom- form, yXbo, to yXbrAo, where b is a position of spatial connectivity matrices, a relatively vector of coefficients for intercept and explana- new and still unfamiliar method for describing spatial tory variables X; and r is the coefficient of the autoco- patterns in complex data (Griffith 2000b, Borcard and variate A. Legendre 2002, Griffith and Peres-Neto 2006, Dray A at any site i may be calculated as: et al. 2006). A very similar approach, called eigenvector filtering, was presented by Diniz-Filho and Bini (2005) Ai  wijyj (the weighted sum) or based on their method to account for phylogenetic non- Xj  ki independence in biological data (Diniz-Filho et al. 1998). Eigenvectors from these matrices represent the w y ij j decompositions of Moran’s I statistic into all mutually Xj  ki Ai  (the weighted average); orthogonal maps that can be generated from a given wij connectivity matrix (Griffith and Peres-Neto 2006). Xj  ki Either binary or distance-based connectivity matrices can be decomposed, offering a great deal of flexibility where yj is the response value of y at site j among site i’s regarding topology and transformations. Given the set of ki neighbours; and wij is the weight given to site j’s influence over site i (Augustin et al. 1996, Gumpertz non-Euclidean nature of the spatial connectivity ma- et al. 1997). Usually, weight functions are related to trices (i.e. not all sampling units are connected), both geographical distance between data points (Augustin positive and negative eigenvalues are produced. The et al. 1996, Arau´jo and Williams 2000, Osborne et al. non-Euclidean part is introduced by the fact that only 2001, Brownstein et al. 2003) or environmental certain connections among sampling units, and not all, distance (Augustin et al. 1998, Ferrier et al. 2002). are considered. Eigenvectors with positive eigenvalues The weighting scheme and neighbourhood size (k) are represent positive autocorrelation, whereas eigenvectors often chosen arbitrarily, but may be optimised (by trial with negative eigenvalues represent negative autocorre- and error) to best capture spatial autocorrelation lation. For the sake of presenting a general method that (Augustin et al. 1996). Alternatively, if the cause of will work for either binary or distance matrices, we used spatial autocorrelation is known (or at least suspected), a distance-based eigenvector procedure (after Dray the choice of neighbourhood configuration may be et al. 2006) which can be summarized as follows: informed by biological parameters, such as the species’ 1) compute a pairwise Euclidean (geographic) distance dispersal capacity (Knapp et al. 2003). matrix among sampling units: D[dij]; 2) choose a Autocovariate models can be applied to binomial threshold value t and construct a connectivity matrix data (‘‘autologistic regression’’, Smith 1994, Augustin using the following rule: et al. 1996, Klute et al. 2002, Knapp et al. 2003), as well as normally and Poisson-distributed data (Luoto 0 if ij 0 if d t et al. 2001, Kaboli et al. 2006). W [wij]8 ij  < 2 Where spatial autocorrelation is thought to be [1(dij=4t) ] if dij 5t anisotropic (e.g. because seed dispersal follows prevail- : ing winds or downstream run-off), multiple autoco- where t is chosen as the maximum distance that variates can be used to capture spatial autocorrelation in maintains connections among all sampling units being different geographic directions (He et al. 2003). connected using a minimum spanning tree algorithm

612 (Legendre and Legendre 1998). Because the example application in ecology has been very limited so far data we use represent a regular grid (see below), t1 and (Jetz and Rahbek 2002, Keitt et al. 2002, Lichstein 2 thus wij is either 0 or 1Á1/4 0.9375 in our analysis. et al. 2002, Dark 2004, Tognelli and Kelt 2004). This Note that we can change 0.9375 to 1 without affecting is most likely due to the limited availability of eigenvector extraction. This would make the matrix fully appropriate software that easily facilitates the applica- compatible with a binary matrix which is the case for a tion of these kinds of models (Lichstein et al. 2002). regular grid. 3) Compute the eigenvectors of the centred With the recent development of programs that fit a similarity matrix: (IÁ11T/n)W(I 11T/n), where I is the variety of GLS (Littell et al. 1996, Pinheiro and Bates identity matrix. Due to numericalÁ precision regarding 2000, Venables and Ripley 2002) and autoregressive the eigenvector extraction of large matrices (Bai et al. models (Kaluzny et al. 1998, Bivand 2005, Rangel 1996) the method is limited to ca 7000 observations et al. 2006), however, the range of available tools for depending on platform and software (but see Griffith ecologists to analyse spatially autocorrelated normal 2000a, for solutions based on large binary connectivity data has been greatly expanded. matrices). 4) Select eigenvectors to be included as spatial predictors in a linear or generalised linear model. Here, a procedure that minimizes the amount of Generalised least squares (GLS) spatial autocorrelation in residuals was used (see Griffith As before, the underlying model is YXbo, with the and Peres-Neto 2006 and Appendix for computational error vector oN(0,a). a is called the variance- details). In this approach, eigenvectors are added to a . Instead of fitting individual values model until the spatial autocorrelation in the residuals, for the variance-covariance matrix a, a parametric measured by Moran’s I, is non-significant. Our selection is assumed. Correlation functions algorithm considered global Moran’s I (i.e. autocorrela- are isotropic, i.e. they depend only on the distance sij tion across all residuals), but could be easily amended to between locations i and j, but not on the direction. target spatial autocorrelation within certain distance Three frequently used examples of correlation functions 2 classes. The significance of Moran’s I was tested using a C(s) also used in this study are exponential (C(s)s 2 2 permutation test as implemented in Lichstein et al. exp(r/s)), Gaussian (C(s)s exp(r/s)) ) and sphe- (2002). This potentially renders the selection procedure rical (C(s)s2(12=p(r=s 1r2=s2 sin1r=s)); computationally intensive for large data sets (200 or where r is a scaling factor thatpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is estimated from the more observations), because a permutation test must be data). performed for each new eigenvector entered into the Some restrictions are placed upon the resulting model. Once the location-dependent, but data-inde- variance-covariance matrix a: a) it must be symmetric, pendent eigenvectors are selected, they are incorporated and b) it must be positive definite. This guarantees that into the ordinary regression model (i.e. linear or the matrix is invertible, which is necessary for ) as covariates. Since their the fitting process (see below). The choice of correlation relevance has been assessed during the filtering process function is commonly based on a visual investigation of model simplification is not indicated (although some the semi-variogram or correlogram of the residuals. eigenvectors will not be significant). Parameter estimation is a two-step process. First, the parameters of the correlation function (i.e. scaling factor r in the examples used here) are found by 3. Spatial models based on generalised least optimizing the so called profiled log-likelihood, which squares regression is the log-likelihood where the unknown values for b and s2 are replaced by their algebraic maximum In linear models of normally distributed data, spatial likelihood . Secondly, given the parameter- autocorrelation can be addressed by the related ap- ization of the variance-covariance matrix, the values for proaches of generalised least squares (GLS) and auto- b and s2 are found by solving a weighted ordinary least regressive models (conditional autoregressive models square problem: (CAR) and simultaneous autoregressive models (SAR)). T T T 1=2 1=2 1=2 GLS directly models the spatial covariance structure in y  Xb o the variance-covariance matrix a, using parametric X  X  X  T functions. CAR and SAR, on the other hand, model where the error term (a1=2) o is now normally the error generating process and operate with weight distributed with 0 and variance s2I. matrices that specify the strength of between neighbouring sites. Although models based on generalised least squares Autoregressive models have been known in the statistical literature for Both CAR and SAR incorporate spatial autocorrelation decades (Besag 1974, Cliff and Ord 1981), their using neighbourhood matrices which specify the

613 relationship between the response values (in the case of propagules disperse passively with river flow, leading CAR) or residuals (in the case of SAR) at each location to a directional spatial effect. The SAR lagged-response (i) and those at neighbouring locations (j) (Cressie model (SAR lag) takes the form 1993, Lichstein et al. 2002, Haining 2003). The neighbourhood relationship is formally expressed in a Y rWY Xbo nn matrix of spatial weights (W) with elements (w ) 1 ij (which is equivalent to Y(IrW) Xb(I representing a measure of the connection between 1 rW) o), where r is the autoregression parameter, locations i and j. The specification of the spatial weights W the spatial weights matrix, and b a vector represent- matrix starts by identifying the neighbourhood struc- ing the slopes associated with the predictors in the ture of each cell. Usually, a binary neighbourhood original predictor matrix X. matrix N is formed where nij 1 when observation j is a Second, spatial autocorrelation can affect both neighbour to observation i. This neighbourhood can be response and predictor variables (‘‘lagged-mixed identified by the adjacency of cells on a grid map, or by model’’, SAR mix). Ecologically, this adds a local Euclidean or great circle distance (e.g. the distance aggregation component to the spatial effect in the lag- along earth’s surface), or predefined according to a model above. In this case, another term (WXg) must specific number of neighbours (e.g. a neighbourhood also appear in the model, which describes the regression distance of 1.5 in our case includes the 8 adjacent coefficients (g) of the spatially lagged predictors (WX). neighbours). The elements of N can further be weighted The SAR lagged- takes the form to give closer neighbours higher weights and more distant neighbours lower weights. The matrix of spatial Y rWY XbWXgo weights W consists of zeros on the diagonal, and weights for the neighbouring locations (wij) in the off- Finally, the ‘‘spatial error model’’ (SAR err) assumes diagonal positions. A good introduction to the CAR that the autoregressive process occurs only in the error and SAR methodology is given by Wall (2004). term and neither in response nor in predictor variables. The model is most similar to the CAR, with no directionality in the error. In this case, the usual OLS Conditional autoregressive models (CAR) regression model (YXbo) is complemented by a The CAR model can be written as (Keitt et al. 2002): term (lWm) which represents the spatial structure (lW) in the spatially dependent error term (m). The SAR Y XbrW (Y Xb)o spatial error model thus takes the form 2 2 with oN(0, Vc). If si s for all locations i, the 2 1 Y XblWmo covariance matrix is VC s (IrW) , where W has to be symmetric. Consequently, CAR is unsuitable where l is the spatial autoregression coefficient, and the when directional processes such as stream flow effects or rest as above. SAR and CAR are related to each other, prevalent wind directions are coded as non-Euclidean but the terms rW used in both CAR and SAR are not distances, resulting in an asymmetric covariance matrix. identical. As noted above, in CAR, W must be In such situations, the closely related simultaneous symmetrical, whereas in SAR it need not be. Let rW autoregressive models (SAR) are a better option, as their of the CAR be called K and rW of the SAR be called S. W need not be symmetric (see below). For our analysis, Then any SAR is a CAR with KSSTSTS we used a row-standardised binary weights matrix for a (Haining 2003). Assuming constant variance s2, the neighbour-distance of 2 (Appendix). formal relationship between the error variance-covar- iance matrices in GLS, SAR, and CAR is as follows: 2 2 1 2 VGLS s C(s); VCAR s (IÁK) and VSAR s (IÁ Simultaneous autogressive models (SAR) S)1(IÁST)1, with K and S as defined above. Thus SAR models can take three different forms (we use the CAR and SAR models are equivalent if VCAR VSAR. notation presented in Anselin 1988), depending on The relationship between specific values in correlation where the spatial autoregressive process is believed to matrix C and weight matrix W is not straightforward, occur (see Cliff and Ord 1981, Anselin 1988, Haining however. In particular, spatial dependence parameters 2003, for details). The first SAR model assumes that the that decrease monotonically with distance do not autoregressive process occurs only in the response necessarily correspond to spatial that de- variable (‘‘lagged-response model’’), and thus includes crease monotonically with distance (Waller and Gotway a term (rW) for the spatial autocorrelation in the 2004). An extensive comparison of the impact of response variable Y, but also the standard term for the different model formulations on parameter estimation predictors and errors (Xbo) as used in an ordinary and type I error control is given by Kissling and Carl least squares (OLS) regression. Spatial autocorrelation (2007) using simulated datasets with different spatial in the response may occur, for example, where autocorrelation structures.

614 Spatial generalised linear mixed models (GLMM) Firstly, consider the generalised linear model E(y) Spatial generalised linear mixed models are generalised m, mg1 (Xb) where y is a vector of response linear models (GLMs) in which the linear predictor variables, m the expected value, g1 the inverse of the may contain random effects and within-group errors link function, X the matrix of predictors, and b the may be spatially autocorrelated (Breslow and Clayton vector of regression parameters. Minimization of a 1993, Venables and Ripley 2002). Formally, if Yij is quadratic form leads to the GLM equation the j-th observation of the response variable in group i, (Diggle et al. 1995, Dobson 2002, Myers et al. 2002)

T 1 1 D V (ym)0; E[Yij zi]g (hij) and hij xijbzijzi; j where DT is the transposed matrix of D of partial where g is the link function, h is the linear predictor, b derivatives D1m/1b. Secondly, note that the variance and z are coefficients for fixed and random effects, of the response can be replaced by a variance-covariance respectively, and x and z are the explanatory variables matrix V which takes into account that observations associated with these effects. Conditional on the are not independent. In GEEs, the sample is split up random effects z, the standard GLM applies and the into m clusters and the complete dataset is ordered in a within-group distribution of Y can be described using way that in all clusters data are arranged in the same 1 the same error distributions as in GLM. sequence: E(yj)mj ; mj g (Xj b): Then the var- Since the GLMM is often implemented based on iance-covariance matrix has block diagonal form, since so-called penalized quasi-likelihood (PQL) methods responses of different clusters are assumed to be (Breslow and Clayton 1993, Venables and Ripley uncorrelated. One can consequently transform the score 2002) around the GLS-algorithm (McCullough and equation into the following form Nelder 1989), we can use it in a similar way, i.e. fitting m the structure of the variance-covariance-matrix to the DTV1 (y m )0; data (see GLS above), albeit with a different error j j j j Xj1 distribution. In cases where spatial data are available from several disjunct regions, GLMMs can thus be used which sums over all clusters j. This equation is called to fit overall fixed effects while spatial correlation the generalised estimating equation or the quasi-score structures are nested within regions, allowing the equation. accommodation of regional differences in e.g. auto- For spatial dependence the following correlation correlation distances, and assuming autocorrelation structures for V are important: 1) Fixed. The correla- only between observations within the same region tion structure is completely specified by the user and (Orme et al. 2005, Davies et al. 2006, Stephenson will not change during an iterative procedure. Referred et al. 2006). to here as GEE. 2) User defined. Correlation para- meters are to be estimated, but one can specify that certain parameters must be equal, e.g. that the strength 4. Spatial generalised estimating equations (GEE) of correlation is always the same at a certain distance. Referred to here as geese. Liang and Zeger (1986) developed the generalised First, we consider the GEE model with fixed estimating equation (GEE) approach which is an correlation structure. In order to predetermine the extension of generalised linear models (GLMs). When correlation structure we have good reasons to assume responses are measured repeatedly through time or that the correlation decreases exponentially with in- space, the GEE method takes correlations within creasing spatial distance in ecological applications. clusters of sampling units into account by means of a Therefore, we use the function parameterised correlation matrix, while correlations d  ij between clusters are assumed to be zero. In a spatial a a1 context such clusters can be interpreted as geographical for computation of correlation parameters a. Here dij is regions, if distances between different regions are large the distance between centre points of grid cells i and j enough (Albert and McShane 1995). We modified the and a1 is the correlation parameter for nearest approach of Liang and Zeger to use these GEE models neighbours. The parameter a1 is estimated by Moran’s for spatial, two-dimensional datasets sampled in rec- I of GLM residuals. In this way we obtain a full nn tangular grids (see Carl and Ku¨hn 2007a, for more correlation matrix with known parameters. Thus details). Fortunately, estimates of regression parameters clustering is not necessary. are fairly robust against misspecification of the correla- In the user defined case we build a specific variance- tion matrix (Dobson 2002). The GEE approach is covariance matrix in block diagonal form with 5 especially suited for parameter estimation rather than unknown correlation parameters (corresponding to prediction (Augustin et al. 2005). the five different distance classes in a 33 grid) which

615 have to be calculated iteratively. The dispersion para- cells i and j (dij 0 if ij). On our lattice, the distance meter as a correction of can be calculated between the mid-points of neighbouring cells is dij 1. as well. Then, V(vij) is a matrix defined as vij exp(r dij); r (r]0) is a parameter that determines the decline of inter-cell correlation in errors with inter- Example analysis using simulated data cell distance. The strength of spatial autocorrelation increases with increasing values of r (there is no spatial To illustrate and compare the various approaches that autocorrelation if r0). Here, we used a value of r are available to incorporate SAC into the analysis of 0.3, which resulted in strongly correlated errors in species distribution data, we constructed artificial neighbouring cells (vij 0.74, if dij 1), but a steep datasets with known properties. The datasets represent decline of autocorrelation with increasing distance. A virtual species distribution data (for example species weights matrix W was calculated (by Choleski decom- T atlases) and environmental (such as climatic) covariates, position) using VW W. Finally, the spatially corre- T available on a lattice of 1108 square cells imposed on lated errors are given by oW j, with j drawn from the surface of a virtual island (Fig. 3). the standard .

Generation of artificial distribution of simulated data

The basis for the virtual island is a subset of the volcano For each error distribution, ten data sets were created, data set in R, which consists of a digital elevation model each using a random realisation of the spatially for Auckland’s Maunga Whau Volcano in New Zealand autocorrelated errors, using random draws of ji. These (Anon. 2005). Two uncorrelated (Pearson’s r0.013, data sets were then submitted to statistical analyses in p0.668) environmental variables were created based which the response variables were modelled using a on the altitude-component of this data set: ‘‘rain’’ and number of different linear models for the normally ‘‘djungle’’. These data are available as electronic distributed data, and generalized linear models with the appendix and are depicted in Fig. 3. While ‘‘rain’’ is and -link for the , a rather deterministic function of altitude (including a and and log-link for the count 1 rain-shadow in the east), ‘‘djungle’’ is dominated by a data: E(yi)g (abrainigdjunglei), where g high noise component. Data are given in the Appendix. are the corresponding link functions (identity for the On this lattice the species distribution data, yi (with i normal distribution). The variable ‘‘djungle’’ was an indicator for cell (i1, 2..., 1108)), were simulated entered into all of the statistical models as an additional as a function of one of the two artificial environmental predictor of the response. This was done to be able to predictors, raini. Onto this functional relationship, we assess the models’ ability to distinguish random noise added a spatially correlated noise component we refer to from meaningful variables. as error oi. The covariate raini can for example be Simulations and analyses were primarily carried out thought of as estimates of the total annual amount of (see Appendix for implementation details and R-code) rainfall in cell i. We simulated the three most using the statistical programming software R (Anon. commonly available types of species distribution data; 2005), with packages gee (Carey 2002), geepack (Yan continuous, binary and , using the normal 2002, 2004), spdep (Bivand 2005), ncf (Bjørnstad and distribution and approximations of the Poisson and Falck 2000) and MASS (Venables and Ripley 2002). binomial distributions respectively. The following Calculations for the spatial eigenvector mapping were models were used to simulate the artificial data: 1) originally performed in Matlab using routines later normally distributed data: yi 800.015raini ported to R (spdep) by Roger Bivand and Pedro Peres- 10oi. 2) Binary data: yi 0 if pi B0.5, and yi 1 Neto. Additional functions (Appendix) to work gen- eralised estimating equations on a 2-D lattice were if, pi ]0.5, where pi qi  qi(1qi)oi; and  written by Gudrun Carl (Carl and Ku¨hn 2007a). See e3 0:003 raini pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /q  : /y round(k  also Table 1 for alternative software. i 30:003 rain 3) Poisson data: i i 1  e i As most of the statistical methods tested allow for 30:001 raini kioi); where /ki e ; and round is an operator some flexibility in the precise structure of their spatial pusedffiffiffiffi to round values to the nearest integer. This led to component, several models per method were calculated simulated data with no over- or underdispersion. for each simulated dataset. This allowed us to identify A weight matrix W was used to simulate the spatially the model configuration that most successfully ac- correlated errors oi using weights according to the counted for spatial autocorrelation in the data at distance between data points. Let D(dij) be the hand, by, for example, varying the distance over which (Euclidean) distance matrix for the distances between spatial autocorrelation was assumed to occur, or its

616 Table 1. Methods correcting for spatial autocorrelation and their software implementations. This list is not exhaustive but represents the major software developments in use.

Method R-package1 Computational intensity2 Other3

Autocovariate regression spdep low Autoregressive models4 (CAR, SAR) spdep medium GeoDa, Matlab*, SAM, SpaceStat, S-plus$ Bayesian analysis very high WinBUGS/GeoBUGS Generalised linear mixed model MASS very high SAS (glimmix) Generalised estimating equations gee, geepack low SAS Generalised least squares4 MASS, nlme high SAS, SAM Spatial eigenvector mapping spdep very high Matlab, SAM

1 for most R-packages (Bhttp://www.r-project.org) an equivalent for S-plus is available. 2 low, medium, high and very high refer roughly to a few seconds, several minutes, a few hours and several hours of CPU-time per model (1108 data points on a Pentium 4 dual core, 3.8GHz, 2GB RAM). 3 GeoDa: freeware: Bhttp://www.geoda.uiuc.edu. 4 for normally distributed error only. Matlab: Bhttp://www.mathworks.com, with EigMapSel Á a matlab compiled software to perform the eigenvector selection procedure for generalised linear models (normal, logistic and poisson) Á available in ESA’s Electronic Data Archive (Griffith and Peres-Neto 2006). SAM: spatial analysis for macroecology; freeware under: Bhttp://www.ecoevol.ufg.br/sam/. SAS: statistical analysis system; commercial software: Bhttp://www.sas.com. SpaceStat: commercial software: Bhttp://www.terraseer.com/products/spacestat.html. S-plus: commercial software: Bhttp://www.insightful.com. WinBUGS/GeoBUGS: freeware: Bhttp://www.mrc-bsu.cam.ac.uk/bugs/winbugs/contents.shtml. *requires the free ‘‘Spatial Econometric toolbox’’: Bhttp://www.spatial-econometrics.com. $requires additional module ‘‘spatial’’. functional form. Inferior models were discarded, so that to remove all spatial autocorrelation from the data. In the results section below reports only on the best our case we know that this is due neither to the configuration for each approach. We used residuals omission of an important variable nor an incorrect based on fitted values and which were as such calculated functional relationship, but a simulated aggregation from both the spatial and the non-spatial model mechanism in the errors. In comparison, all spatial components. For each, we report the following details: models managed to decrease spatial autocorrelation in 1) model coefficients (and their standard errors); since the residuals (Fig. 2), although not all were able to the true parameters are known, we can directly judge completely eliminate it. Geese performed worst in this the quality of coefficient estimation; 2) removal of SAC regard. Our simulations are not comprehensive enough, (global Moran’s I, i.e. Moran’s I computed across however, to allow us to deduce what the influence of neighbourhood up to a distance of 20, and correlo- this incomplete removal of SAC might be on parameter grams, which plot Moran’s I for different distance estimation or hypothesis testing. classes); 3) spatial distribution of residuals (map). Another Á though inconsistent Á difference between the spatial and non-spatial models especially with binary data was that standard errors of the coefficient Results of simulations estimates for ‘‘rain’’ and ‘‘djungle’’ were often larger for the spatial models (Fig. 1). For normal and Poisson It is worth pointing out that the main aim of this study data, differences in coefficient estimates between spatial is to illustrate the different methods by applying them and non-spatial models were relatively small, and to the same data sets. The ten realisations of one type of was not affected. Only autocovariate spatial autocorrelation do not allow us to provide a model and SAR lag provided consistently incorrect comprehensive evaluation of the relative merits of each estimates of the spatially autocorrelated parameter of the methods considered. Such evaluation is beyond ‘‘rain’’. the scope of this review paper, and will depend on the Most model approaches performed well with respect data set and question under study. Nonetheless, some to type I and II error rates for the normal and Poisson interesting results emerged from our simulations. data, correctly identifying ‘‘rain’’ as a significant effect Spatial and non-spatial models differed conside- (Table 2). An exception was autocovariate regression, rably in terms of the spatial signature in their residuals which severely and consistently underestimated the (Table 2; Fig. 2 and 3). Residual maps for OLS/GLM effects of rain (Table 2, Fig. 1). Model performance and GAM exhibit clusters of large residuals of the same was worse for data with a binomial error structure than sign (Fig. 3), indicating that these models were not able for models with normal or Poisson error structure.

617 Table 2. Model quality: spatial autocorrelation in the model residuals (given as global Moran’s I) and mean estimates for the coefficients ‘‘rain’’ and ‘‘djungle’’ (91 SE across the 10 simulations). True coefficient values are given in the first row for each distribution in italics. ***, ** and ns refer to significance levels of pB0.001, B0.01 and 0.1, respectively, across the 10 realisations. See Fig. 1 for abbreviations.

Moran’s I Coefficients

‘‘rain’’ ‘‘djungle’’

Normal 0.015 0.0 GLM 0.01690.026 0.014390.0010*** 0.022090.0508ns GAM 0.00290.002 0.012590.0028*** 0.013090.0370ns autocov 0.00190.000 0.0004 90.0007ns 0.014190.0309ns GLS exp 0.00190.000 0.014090.0033*** 0.016290.0248ns CAR 0.00090.000 0.014590.0022*** 0.015690.0324ns SAR err 0.00190.000 0.014490.0028*** 0.015690.0253ns GEE 0.00190.001 0.014190.0031*** 0.016290.0255ns geese 0.00290.005 0.014190.0017*** 0.025690.0276ns SEVM 0.00190.001 0.013290.0009*** 0.016390.0257ns Binomial 0.003 0.0 GLM 0.00690.011 0.002290.0003*** 0.005290.0130ns GAM 0.00290.001 0.000690.0013** 0.001690.0162ns autocov 0.00190.001 0.000690.0004ns 0.002990.0167ns GLMM 0.00190.001 0.004290.0006*** 0.002590.0096ns GEE 0.00190.001 0.002190.0007*** 0.002490.0093ns geese 0.00090.003 0.002190.0004*** 0.004890.0101ns SEVM 0.00190.000 0.002890.0006*** 0.008490.0190ns Poisson 0.001 0.0 GLM 0.01890.024 0.001090.0000*** 0.000690.0018ns GAM 0.00290.002 0.000590.0001*** 0.000590.0018ns autocov 0.01090.010 0.000190.0001* 0.001090.0018ns GLMM 0.00190.001 0.001090.0001*** 0.000690.0009ns GEE 0.00190.001 0.001090.0001*** 0.000690.0009ns geese 0.00590.005 0.001090.0001*** 0.000890.0010ns SEVM 0.00190.001 0.001090.0001*** 0.000890.0019ns

When applied to such data, autocovariate regression (9 than the spherical model, the Gaussian model and the false negatives) and GAM (3 false negatives) were rather GEE using a different exponential function were prone to type II errors (results not shown). Moreover, equivalent, as were methods that did not specify the the spurious effect of djungle would have been retained correlation structure in such a way (e.g. SAR, Fig. 2). in the model in several cases (based on a signi- However, parameterisation for GEE resulted from the ficance level of a0.05: 6 normal, 2 binomial and 1 Moran’s I correlogram, mimicking the distance decay Poisson model of those presented in Table 2), resulting function, though not using the original correlation in type I errors (rejecting a null hypothesis although it function. was true). The ability of simultaneous autoregressive models (SAR) to correctly estimate parameters depended Limitations of our simulations heavily on SAR model structure. For instance, using a lagged response model in our artificial dataset yielded Our example analysis above was meant to illustrate the much poorer coefficient estimates for ‘‘rain’’ than using application of the presented methods to species dis- an error model (Fig. 1). This was to be expected, since tribution data. As such, it remained a cartoon of the our artificial distribution data was created such that its complexity and difficulties posed by real data. Among spatial structure most closely resembled that of the SAR the potential factors that may influence the analysis error model. of species distribution data with respect to spatial We used an exponential distance decay function to autocorrelation, we like to particularly mention the generate the spatial error (see above). Hence, we would following. also expect those methods to perform best in which a Missing environmental variables. As mentioned in correlation function can be defined accordingly the introduction, SAC can be caused by omitting an (i.e. GLS, GLMM and GEE). While indeed the important variable from the model or misspecifying its exponential GLS yielded better coefficient estimates functional relationship with the response (Legendre

618 normal GLM GAM autocov GLS exp GLS sph GLS gau CAR SAR err SAR lag SAR mix GEE geese SEVM -0.030 -0.025 -0.020 -0.015 -0.010 -0.005 0.000 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 binomial GLM GAM autocov GLMM exp GLMM sph GLMM gau GEE geese SEVM -0.012 -0.010 -0.008 -0.006 -0.004 -0.002 0.000 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 Poisson GLM GAM autocov GLMM exp GLMM sph GLMM gau GEE geese SEVM -1.4e-3 -1.2e-3 -1.0e-3 -8.0e-4 -6.0e-4 -4.0e-4 -2.0e-4 0.0 -0.002 -0.001 0.000 0.001 0.002 0.003 estimates for 'rain' estimates for 'djungle'

Fig. 1. Comparison of non-spatial (GLM) and spatial modelling approaches for data with normally, binomially and Poisson- distributed errors. Box, whiskers and dots refer to 25/75%, 10/90% and outliers of estimates across 10 realisations of the same parameter set. Vertical lines indicate true values of parameter. GEE and geese refer to generalised estimating equations with fixed and user-defined correlation structures, respectively. GAM represents a trend-surface regression. GLS and GLMM refer to generalised least squares-based models with exponential, spherical and Gaussian correlation structure, respectively. SAR (simultaneous autoregressive model) was analysed as error, lag and mixed models. SEVM stands for the spatial eigenvector mapping approach.

1993). This is certainly often a problem in real data, found (Lennon 2000, but see Hawkins et al. 2007 for where the ecological determinants of a species’ niche are an opposite view). not necessarily known and good spatial coverage may Mapping bias or mapping heterogeneity can cause not be available for all the important factors. Also, spatial autocorrelation in real data. If real data resulted moderate collinearity among environmental variables from several different regional mapping schemes with may lead models to exclude one or more variables different protocols or from different people performing which would be important in explaining the species’ the mapping, data can differ systematically across a grid spatial patterning. with being more similar within a region and more Biased spatial error. The autocorrelated error we different across. added in our simulated data had no bias in geographical Spatial autocorrelation at different spatial scales. (stationarity of the error) or parameter space. Hence our Several of the methods presented build a ‘‘correction non-spatial models performed similar to the spatial ones structure’’ across all spatial scales (i.e. the variance- with regards to parameter estimation, as opposed to covariance matrices in GLS-based models as well as removal of SAC. This may or may not be very different the spatial eigenvectors), but others do not (the in real data, where both non-stationarity (Ver Hoef autocovariate and the cluster in geese have one specific et al. 1993, Brunsdon et al. 1996, Foody 2004, spatial scale). Even the former may be dominated by Osborne et al. 2007) and bias in parameter space (e.g. patterns at one spatial scale, underestimating effects of less complete data coverage in warmer regions) can be another.

619 normal Discussion 0.8 The analysis of species distribution data has reached 0.6 high statistical sophistication in recent years (Elith et al. 2006). However, even the most advanced and compu- ter-intensive statistical procedures are no guarantee for 0.4 improving our understanding of the determinants of species distributions, nor of our ability to predict 0.2 species distributions under altered environmental con- ditions (Arau´jo and Rahbek 2006, Dormann 2007c). 0.0 One critical step in statistical modelling is the identi- fication of the correct model structure. As pointed out -0.2 for experimental ecology in 1984 by Hurlbert, designs analysed without consideration of the nested nature of binomial subsampling are fundamentally flawed. Spatial auto- 0.6 correlation is a subtle, less obvious form of subsampling (Fortin and Dale 2005): samples from within the range of spatial autocorrelation around a data point will add 0.4 little independent information (depending on the strength of autocorrelation), but unduly inflate sample 0.2 size, and thus degrees of freedom of model residuals, thereby influencing statistical inference. We have presented an overview of different model- Moran's I 0.0 ling approaches for the analysis of species distribution data in which environmental correlates of the distribu- -0.2 tion are inferred. All these methods can be implemented in freely available software packages (Table 1). In GLM choosing between the methods, the type of error -0.4 GAM Poisson distribution in the response variable will be an autocov important criterion. For normal data, GLS-based GLS/GLMM exp methods (GLS, SAR, CAR) can be used efficiently. 0.6 CAR SAR err The most flexible methods, addressing SAC for GEE different error distributions, are spatial GLMMs, 0.4 geese GEEs and SEVM. The autocovariate method, too, is SEVM flexible, but performed very poorly with regards to coefficient estimation in our analyses. We encourage 0.2 users to try a number of methods, since there is often not enough mechanistic information to choose one specific method a priori. One can use AIC or alike to 0.0 compare models (Link and Barker 2006). Note that a ‘‘proper’’ (perfectly correctly specified) model would not require the kind of correction the above methods -0.2 0 5 10 15 20 undertake (Ripley in comments to Besag 1974). In the distance [no. of cells] absence of a perfect model, however, doing something is better than doing nothing (Keitt et al. 2002). Fig. 2. Correlograms of one realisation for each of the three With the exception of autocovariate regression, different distributions (normal, binomial, Poisson) and the differences in parameter estimates and inference be- methods compared. See Fig. 1 for abbreviations. tween spatial and non-spatial models were small for our simulated data. This was possibly a result of the type of spatial autocorrelation in, and the simplistic nature of, Finally, small sample sizes make the estimation of these data (see section ‘‘Limitations of our simula- model parameters unstable. Adding the additional tions’’). However, spatial autocorrelation can also parameters for spatial models will further destabilise reflect failure to include an important environmental model parameterisation. Also, patterns of SAC in small driver in the analysis or inadequate capture of its non- data sets will hinge on very few data points which may linear effect, so that its spatial autocorrelation cannot be distort the spatial correction. accounted for by non-spatial models (Besag et al. 1991,

620 Fig. 3. The two environmental covariates, raw distributional data and residual maps for the different methods for one realisation of the data with normally distributed errors. Blue indicates positive, red negative values. Quadrat size is proportional to the value of the residual (or raw data value), but scaling is different for every plot (since a comparison of quadrat sizes would simply be a comparison of model fit). See Fig. 1 for abbreviations.

Legendre et al. 2002). In either case, spatial autocorre- distribution models may well bias coefficient estima- lation can make a large difference for statistical infe- tion, Diniz-Filho et al. (2003) and Hawkins et al. rence based on spatial data (for review see Dormann, (2007) found non-spatial model to be robust and 2007a, b, c; for drastic cases of this effect see Tognelli unbiased for several data sets. So far, no extensive and Kelt 2004 and Ku¨hn 2007). How to interpret simulation study has been carried out to investigate how these differences, especially the shifts in parameter spatial versus non-spatial methods perform under estimates between spatial and non-spatial models different forms and causes of SAC. Implementing a commonly observed in real data, remains controversial. lagged autocorrelation structure to simplistic data did While Lennon (2000) and others (Tognelli and not reveal a bias in parameter estimation in OLS Kelt 2004, Jetz et al. 2005, Dormann 2007b, Ku¨hn (Kissling and Carl 2007), consistent with the results of 2007) argue that spatial autocorrelation in species Hawkins et al. (2007).

621 One of the two most striking findings of our Tricks and tips analyses is the high error rate of the autocovariate method. Most methods for normally distributed data Each of the above methods has its quirks and some yielded coefficient estimates for ‘‘rain’’ that were require fine-tuning by the analyst. Without attempting acceptable, including the non-spatial ordinary least to cover these comprehensively, we here hint at some square regression (Fig. 1). However, two models areas for each method type which require attention. performed poorly: both the autocovariate regression In autocovariate regression, neighbourhood size and and the lag version of the simultaneous autoregressive type of weighting function are potentially sensitive model showed a very consistent and strong bias, leading parameters, which can be optimised through trial and to severe underestimation (in absolute terms) of model error. It seems, however, that small neighbourhood sizes (such as the next one to two cells) often turn out best, coefficients. A similar pattern was also found for the and that the type of weighting function has relatively non-normally distributed errors, identifying autocovari- little effect. This was the case in our analysis as well as in ate regression as a consistently worse performer than the published studies investigating different neighbourhood other approaches. The poor performance of the auto- sizes (for review see Dormann 2007b). Another covariate regression approach in our study with regards important aspect of autocovariate models is the to parameter estimation contrasts with earlier evalua- approach chosen to dealing with , which tions of this method (Augustin et al. 1996, Huffer and may lead to cells without neighbours (‘‘islands’’). Since Wu 1998, Hoeting et al. 2000, He et al. 2003), but is the issue arises for all modelling methods, we shall in line with more recent ones (Dormann 2007a, Carl briefly discuss it here. Missing data can be overcome by and Ku¨hn 2007a). These earlier studies used more a) omission (Klute et al. 2002, Moore and Swihart sophisticated parameter estimation techniques, suggest- 2005); b) strategic choice of neighbourhood structure ing that the inferiority of autocovariate models in our (Smith 1994); c) estimating missing response values by simulation may partly result from our simplistic (but initially ignoring spatial autocorrelation and regressing not unusual) implementation of the method. Moreover, known response values against explanatory variables two of the earlier studies were undertaken in the context other than the autocovariate (Augustin et al. 1996, of many missing values: Augustin et al. (1996) used Teterukovskiy and Edenius 2003, Segurado and Arau´jo only 20% of sites in their study area for model training; 2004); and d) as in c), but then refining it through Hoeting et al. (2000) used between 3.8 and 5.8%. This an iterative procedure known as the Gibbs sampler may have diminished the influence of any autocovariate (Casella and George 1992). This procedure is compu- and perhaps explains why in these studies the auto- tationally intensive, but has been found to yield the best covariate did not overwhelm other model coefficients results (Augustin et al. 1996, Wu and Huffer 1997, (as it did in ours). A final reason for the discrepancy in Osborne et al. 2001, Teterukovskiy and Edenius 2003, findings may be that our artificial data simulated spatial Brownstein et al. 2003, He et al. 2003). Simulation autocorrelation in the error structure, whereas other studies further suggest that a) parameter estimation is simulations created spatial structure directly in the poor when the autocovariate effect is strong relative to response values, which more closely reflects the assump- the effect of other explanatory variables (Wu and tions underlying autocovariate models. Huffer 1997, Huffer and Wu 1998); b) the precision The second interesting finding is the overall higher of parameter estimates varies with species prevalence, variability of results for binary data. While for normal- i.e. the number of presence records relative to the total sample size (Hoeting et al. 2000); and c) autocovariate and Poisson-distributed residuals all model approaches models adequately distinguish between meaningful (apart from autocovariate regression) yielded similar explanatory variables and random covariates (Hoeting results and little variance across the ten realisations et al. 2000) (but not in our study). Both simulation and (Fig. 1), a different pattern emerged for binary empirical studies also indicate that autocovariate models (binomial) data. We attribute this to the relatively low achieve better fit than equivalent models lacking the information content of binary data (Breslow and autocovariate term (Augustin et al. 1996, Hoeting et al. Clayton 1993, Venables and Ripley 2002), making 2000, Osborne et al. 2001, He et al. 2003, McPherson parameterisation of the model very dependent on those and Jetz 2007). data points that determine the point of inflexion of the For spatial eigenvector mapping, computational logistic curve (McCullough and Nelder 1989). This speed becomes an issue for large datasets. Although phenomenon has been noted before (McCullough and the calculation of eigenvectors itself is rapid, optimising Nelder 1989), and remains relevant for species dis- the model by permutation-based testing combinations tribution models, where the majority of studies are of spatial eigenvectors is computer-intensive. Diniz- based on the analysis of presence-absence data (Guisan Filho and Bini (2005) argue that the identity of the and Zimmermann 2000, Guisan and Thuiller 2005). selected eigenvectors is indicative of the spatial scales at

622 which spatial autocorrelation takes effect, making this perform better than SAR lag or even SAR mix models method potentially very interesting for ecologists. The when tackling simulated data containing autocorrela- implementation used in our analysis requires little tion in lagged predictors (or response and predictors), as arbitration and hence should be explored more widely. recently demonstrated in a more comprehensive assess- Note that SEVM, in the way that was applied here, is ment of SAR models using different spatially auto- based on a different modelling philosophy. Its declared correlated datasets (Kissling and Carl 2007). aim is to remove residual spatial autocorrelation, unlike Generalised estimating equations require high sto- all other methods described above, which simply rage capacity for solving the GEE score equation provide a mathematical way to incorporate SAC into without clustering as we used it in our fixed model. the analysis. Application of the fixed model will therefore be limited For the GLS-based methods (GLS and the spatial for models on data with larger sample size, but the GLMM), estimation of the correlation structure func- method is very suitable for missing data and non-lattice tions (i.e. the parameter r) can be rather unstable. As a data. The need in storage capacity is considerably consequence some models yield r0 (i.e. no spatial reduced by cluster models, such as our user-defined autocorrelation incorporated) or r: , with the GLS model. But clustering requires attention to three steps model returning what is in fact a non-spatial GLM or in the analysis: cluster size, within-cluster correlation nonsensical results, respectively. This problem can be structure and allocation of cells to clusters. To find the overcome by inclusion of a ‘‘nugget’’ term that reduces best cluster size for the analysis, we recommend the correlation at infinitesimally small distances to a investigating clusters of 22, 33 and 44. In real value below 1, or, even better, a specification of r based data, these cluster sizes have been sufficient to remove on a semi-variogram of the residuals (Littell et al. 1996, spatial autocorrelation (Carl and Ku¨hn 2007a). Several Kaluzny et al. 1998). The common justification for a different correlation structures should be computed nugget term are measurement errors (on top of the initially, e.g. to allow for anisotropy. Finally, allocation spatially correlated error); including a nugget effect can of cells to clusters can start in different places. stabilize the estimation of the correlation function Depending on the starting point (e.g. top right or (Venables and Ripley 2002). north west), cells will be placed in different clusters. Autoregressive models (SAR and CAR) require a Choosing different starting points will give the analyst decision on the weighting scheme for the weights an idea of the (in our experience limited) importance of matrix, for which there is not always an a priori reason. this issue. Computing time is short. The main options are row standardised coding (sums over all rows add up to N), globally standardised coding (sums over all links add up to N), dividing globally standardised neighbours by their number (sums over all Autocorrelation in a predictive setting links add up to unity), or the variance-stabilising coding scheme proposed by Tiefelsdorf et al. (1999, pp. 167Á Spatial autocorrelation may arise for a number of 168), i.e. sums over all links to N. In our analysis, the ecological reasons, including external environmental row standardised coding was most often the superior and historical factors limiting the mobility of organ- choice, which is in line with other studies (Kissling and isms, intrinsic organism-specific dispersal mechanisms Carl 2007), but the binary and the variance-stabilising and other behavioural factors causing the spatial coding scheme also resulted in good models. SAR and aggregation of populations and species in the land- CAR models did not differ much in our analysis. scapes. In addition to these factors, spatial autocorrela- According to Cressie (1993), CAR models should be tion can also be caused by observer bias and differences preferred in terms of estimation and interpretation, in sampling schemes and sampling effort. Overall, although SAR models are preferred in the econo- spatial autocorrelation occurs at all spatial scales from metric context (Anselin 1988). Either approach the micrometre to hundreds of kilometres (Dormann can be relatively slow for large data sets (sample 2007b), possibly for a whole suite of reasons. Since size10 000) due to the estimation of the determinant these reasons are mostly unknown, one cannot readily of (IÁrW) for each step of the iteration. Note that derive a spatial correlation structure for an entirely new, Bayesian CAR models do not require the computation unobserved region. Augustin et al. (1996) and others of such a determinant and can therefore be particularly (Hoeting et al. 2000, Teterukovskiy and Edenius 2003, suitable for data on large lattices (Gelfand and Reich et al. 2004) have, however, successfully used the Vounatsou 2003). For SAR models, identification of Gibbs sampler (Casella and George 1992) to derive the correct model structure is recommended and model for unobserved areas within the study region selection procedures can help to reduce bias (Kissling (interpolation), and He et al. (2003) extrapolated and Carl 2007). The Lagrange-test (Appendix) can autologistic predictions through time to examine also help here. However, SAR error models generally possible effects of climate change.

623 Interpolation, i.e. the prediction of values within the However, Bayesian methods for the analyses of species parameter and spatial range, can be achieved by several distribution data are more flexible; they can be more of the presented methods. An advantage of GLS is that easily extended to include more complex structures the spatially correlated error can be predicted for sites (Latimer et al. 2006). Models can for example be where no observations are available, based on the values extended to a multivariate setting when several (corre- of observed sites (e.g. ). The same holds true for lated) counts of different species in each grid cell are to the spatial GLMM. For autocovariate regression and be modelled, or when both count and normally spatial eigenvector mapping, in contrast, interpolation distributed data are to be modelled within the same is more complicated, requiring use of the aforemen- framework (Thogmartin et al. 2004, Ku¨hn et al. 2006). tioned Gibbs-sampler. Bayesian methods are also a generally more suitable tool When models are projected into new geographic for inference in data sets with many missing values, or areas or time periods the handling of spatial auto- when accounting for detection (Gelfand correlation becomes more problematic (if not impos- et al. 2005, Ku¨hn et al. 2006). sible). Extrapolation in time, for example, is necessarily uncertain, particularly if biotic interactions Á and with them spatial autocorrelation patterns Á could change as Wishlist each species responds differentially to climate change. However; most of the statistical methods used for In this study, we introduced a wide range of statistical prediction in time neglect important processes such as tools to deal with spatial autocorrelation in species migration, dispersal, competition, predation (Pearson distribution data. Unfortunately, none of these tools and Dawson 2003, Dormann 2007c), or at least assume directly represents dynamic aspects of ecological reality many of them to remain constant. One might therefore (e.g. dispersal, species interaction): all the methods argue that, while taking the autocorrelation structure as examined remain phenomenological rather than me- constant adds one more assumption, the use of spatial chanistic. Therefore they are unable to disentangle parameters at least helps to derive better models. stochastic and process-introduced spatial autocorrela- Extrapolation in space, in contrast, is not recom- tion. Disentangling these sources of spatial autocorrela- mended: the variance-covariance matrix parameterised tion in the data would be particularly important for the in GLS approaches, for example, may look very analysis of species that are not at equilibrium with their different in other regions, even for the same organism. environmental drivers (e.g. newly introduced species Hence, extrapolation can only be based on the expanding in range or species that have undergone coefficient estimates, not on the spatial component of population declines due to overexploitation). Moreover, the model. Extrapolation is further complicated by it would be desirable to extend the statistical approaches model complexity. The use of non-linear predictors and used here to model multivariate response variables, such interactions between environmental variables will in- as species composition (see Ku¨hn et al. 2006, for an crease model fit, but compromises transferability of example). Similarly, presence-only data, as commonly models in time and space (Beerling et al. 1995, Sykes found for museum specimens, cannot be analysed with 2001, Gavin and Hu 2006). Our study therefore did the above methods, nor are we aware of any method not compare methods’ abilities to either make predic- suitable for such data. While in principle it is possible tions to new geographic areas or extrapolate beyond the to incorporate temporal and/or phylogenetic compo- range of environmental parameters. nents into species distribution models (e.g. into GEEs, GLMMs and Bayes), this has not yet been attempted. It also would be desirable to have methods available that allow for the strengths of spatial autocorrelation to vary Bayesian approaches in space (non-stationarity), since stationarity is a basic and strong assumption of all the methods used here Our review focused on frequentist methods. Bayesian (except perhaps SEVM). Finally, the issue of variable methods, which allow prior beliefs about data to be selection under spatial autocorrelation has received incorporated in the calculation of expected values, offer virtually no coverage in the statistical literature, and an alternative. Experience and a good understanding of hence the effect of spatial autocorrelation on the the influence of prior distributions and convergence identification of the best-fitting model, or candidate assessment of Markov chains are crucial in Bayesian set of most likely models, still remains unclear. analyses. Thus, if therefore the question of interest can be addressed using more robust, less computationally intensive methods, there is no real need to apply Authors’ contributions and acknowledgements Á Data were the ‘‘Bayesian machinery’’ (Brooks 2003). The spatial created by CFD, GC and FS. Analyses and manuscript analyses as presented in this paper can be sections describing each method were carried out as follows: done straightforwardly using non-Bayesian methods. autocovariate regression: JMM; SEVM: PRPN and RB;

624 GAM: JB, RO and CFD; GLS: BR and WJ; CAR: BS; SAR: Besag, J. et al. 1991. Bayesian image restoration with two WDK; GLMM: FMS and RGD; GEE: GC and IK. Further applications in spatial statistics (with discussion). Á Ann. analyses, figure and table preparation and initial drafting were Inst. Stat. Math. 43: 1Á59. carried out by CFD. All authors contributed to writing the Bivand, R. 2005. spdep: spatial dependence: weighting final manuscript. schemes, statistics and models. Á R package version 0.3Á We also would like to thank Pierre Legendre, Carsten 17. Rahbek, Alexandre Diniz-Filho, Jack Lennon and Thiago Bjørnstad, O. N. and Falck, W. 2000. Nonparametric spatial Rangel for comments on an earlier version. This contribution covariance functions: estimation and testing. Á Environ. is based on the international workshop ‘‘Analysing Spatial Ecol. Stat. 8: 53Á70. Distribution Data: Principles, Applications and Software’’ Borcard, D. and Legendre, P. 2002. 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