Spatially autocorrelated sampling falsely inflates measures of accuracy for presence‐only niche models

Aim Environmental niche models that utilize presence‐only data have been increasingly employed to model species distributions and test ecological and evolutionary predictions. The ideal method for evaluating the accuracy of a niche model is to train a model with one dataset and then test model predictions against an independent dataset. However, a truly independent dataset is often not available, and instead random subsets of the total data are used for ‘training’ and ‘testing’ purposes. The goal of this study was to determine how spatially autocorrelated sampling affects measures of niche model accuracy when using subsets of a larger dataset for accuracy evaluation. Location The distribution of Centaurea maculosa (spotted knapweed; Asteraceae) was modelled in six states in the western United States: California, Oregon, Washington, Idaho, Wyoming and Montana. Methods Two types of niche modelling algorithms – the genetic algorithm for rule‐set prediction (GARP) and maximum entropy modelling (as implemented with Maxent) – were used to model the potential distribution of C. maculosa across the region. The effect of spatially autocorrelated sampling was examined by applying a spatial filter to the presence‐only data (to reduce autocorrelation) and then comparing predictions made using the spatial filter with those using a random subset of the data, equal in sample size to the filtered data. Results The accuracy of predictions from both algorithms was sensitive to the spatial autocorrelation of sampling effort in the occurrence data. Spatial filtering led to lower values of the area under the receiver operating characteristic curve plot but higher similarity statistic (I) values when compared with predictions from models built with random subsets of the total data, meaning that spatial autocorrelation of sampling effort between training and test data led to inflated measures of accuracy. Main conclusions The findings indicate that care should be taken when interpreting the results from presence‐only niche models when training and test data have been randomly partitioned but occurrence data were non‐randomly sampled (in a spatially autocorrelated manner). The higher accuracies obtained without the spatial filter are a result of spatial autocorrelation of sampling effort between training and test data inflating measures of prediction accuracy. If independently surveyed data for testing predictions are unavailable, then it may be necessary to explicitly account for the spatial autocorrelation of sampling effort between randomly partitioned training and test subsets when evaluating niche model predictions.     

Testing the predictive performance of distribution models

Distribution models are used to predict the likelihood of occurrence or abundance of a species at locations where census data are not available. An integral part of modelling is the testing of model performance. We compared different schemes and measures for testing model performance using 79 species from the North American Breeding Bird Survey. The four testing schemes we compared featured increasing independence between test and training data: resubstitution, random data hold‐out and two spatially segregated data hold‐out designs. The different testing measures also addressed different levels of information content in the dependent variable: regression R² for absolute abundance, squared correlation coefficient r² for relative abundance and AUC/Somer’s D for presence/absence. We found that higher levels of independence between test and training data lead to lower assessments of prediction accuracy. Even for data collected independently, spatial autocorrelation leads to dependence between random hold‐out test data and training data, and thus to inflated measures of model performance. While there is a general awareness of the importance of autocorrelation to model building and hypothesis testing, its consequences via violation of independence between training and testing data have not been addressed systematically and comprehensively before. Furthermore, increasing information content (from correctly classifying presence/absence, to predicting relative abundance, to predicting absolute abundance) leads to decreasing predictive performance. The current tests for presence/absence distribution models are typically overly optimistic because a) the test and training data are not independent and b) the correct classification of presence/absence has a relatively low information content and thus capability to address ecological and conservation questions compared to a prediction of abundance. Meaningful evaluation of model performance requires testing on spatially independent data, if the intended application of the model is to predict into new geographic or climatic space, which arguably is the case for most applications of distribution models.     

Land-cover data improve bioclimatic models for anurans and turtles at a regional scale

Aim We investigated whether accounting for land cover could improve bioclimatic models for eight species of anurans and three species of turtles at a regional scale. We then tested whether accounting for spatial autocorrelation could significantly improve bioclimatic models after statistically controlling for the effects of land cover. Location Nova Scotia, eastern Canada. Methods Species distribution data were taken from a recent (1999–2003) herpetofaunal atlas. Generalized linear models were used to relate the presence or absence of each species to climate and land‐cover variables at a 10‐km resolution. We then accounted for spatial autocorrelation using an autocovariate or third‐order trend surface of the geographical coordinates of each grid square. Finally, variance partitioning was used to explore the independent and joint contributions of climate, land cover and spatial autocorrelation. Results The inclusion of land cover significantly increased the explanatory power of bioclimatic models for 10 of the 11 species. Furthermore, including land cover significantly increased predictive performance for eight of the 11 species. Accounting for spatial autocorrelation improved model fit for rare species but generally did not improve prediction success. Variance partitioning demonstrated that this lack of improvement was a result of the high correlation between climate and trend‐surface variables. Main conclusions The results of this study suggest that accounting for the effects of land cover can significantly improve the explanatory and predictive power of bioclimatic models for anurans and turtles at a regional scale. We argue that the integration of climate and land‐cover data is likely to produce more accurate spatial predictions of contemporary herpetofaunal diversity. However, the use of land‐cover simulations in climate‐induced range‐shift projections introduces additional uncertainty into the predictions of bioclimatic models. Further research is therefore needed to determine whether accounting for the effects of land cover in range‐shift projections is merited.     

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.     

Effect of species rarity on the accuracy of species distribution models for reptiles and amphibians in southern California

Aim Several studies have found that more accurate predictive models of species’ occurrences can be developed for rarer species; however, one recent study found the relationship between range size and model performance to be an artefact of sample prevalence, that is, the proportion of presence versus absence observations in the data used to train the model. We examined the effect of model type, species rarity class, species’ survey frequency, detectability and manipulated sample prevalence on the accuracy of distribution models developed for 30 reptile and amphibian species. Location Coastal southern California, USA. Methods Classification trees, generalized additive models and generalized linear models were developed using species presence and absence data from 420 locations. Model performance was measured using sensitivity, specificity and the area under the curve (AUC) of the receiver‐operating characteristic (ROC) plot based on twofold cross‐validation, or on bootstrapping. Predictors included climate, terrain, soil and vegetation variables. Species were assigned to rarity classes by experts. The data were sampled to generate subsets with varying ratios of presences and absences to test for the effect of sample prevalence. Join count statistics were used to characterize spatial dependence in the prediction errors. Results Species in classes with higher rarity were more accurately predicted than common species, and this effect was independent of sample prevalence. Although positive spatial autocorrelation remained in the prediction errors, it was weaker than was observed in the species occurrence data. The differences in accuracy among model types were slight. Main conclusions Using a variety of modelling methods, more accurate species distribution models were developed for rarer than for more common species. This was presumably because it is difficult to discriminate suitable from unsuitable habitat for habitat generalists, and not as an artefact of the effect of sample prevalence on model estimation.     

Evaluating,partitioning, and mapping the spatial autocorrelation component in ecological niche modeling: a new approach based on environmentally equidistant records

Most species data display spatial autocorrelation that can affect ecological niche models (ENMs) accuracy‐statistics, affecting its ability to infer geographic distributions. Here we evaluate whether the spatial autocorrelation underlying species data affects accuracy‐statistics and map the uncertainties due to spatial autocorrelation effects on species range predictions under past and future climate models. As an example, ENMs were fitted to Qualea grandiflora (Vochysiaceae), a widely distributed plant from Brazilian Cerrado. We corrected for spatial autocorrelation in ENMs by selecting sampling sites equidistant in geographical (GEO) and environmental (ENV) spaces. Distributions were modelled using 13 ENMs evaluated by two accuracy‐statistics (TSS and AUC), which were compared with uncorrected ENMs. Null models and the similarity statistics I were used to evaluate the effects of spatial autocorrelation. Moreover, we applied a hierarchical ANOVA to partition and map the uncertainties from the time (across last glacial maximum, pre‐insustrial, and 2080 time periods) and methodological components (ENMs and autocorrelation corrections). The GEO and ENV models had the highest accuracy‐statistics values, although only the ENV model had values higher than expected by chance alone for most of the 13 ENMs. Uncertainties from time component were higher in the core region of the Brazilian Cerrado where Q. grandiflora occurs, whereas methodological components presented higher uncertainties in the extreme northern and southern regions of South America (i.e. outside of Brazilian Cerrado). Our findings show that accounting for autocorrelation in environmental space is more efficient than doing so in geographical space. Methodological uncertainties were concentrated in outside the core region of Q. grandiflora's habitat. Conversely, uncertainty due to time component in the Brazilian Cerrado reveals that ENMs were able to capture climate change effects on Q. grandiflora distributions.     

Geographical patterns in prediction errors of species distribution models

Aim To describe and explain geographical patterns of false absence and false presence prediction errors that occur when describing current plant species ranges with species distribution models. Location Europe. Methods We calibrated species distribution models (generalized linear models) using a set of climatic variables and gridded distribution data for 1065 vascular plant species from the Atlas Florae Europaeae. We used randomly selected subsets for each species with a constant prevalence of 0.5, modelled the distribution 1000 times, calculated weighted averages of the model parameters and used these to predict the current distribution in Europe. Using a threshold of 0.5, we derived presence/absence maps. Comparing observed and modelled species distribution, we calculated the false absence rates, i.e. species wrongly modelled as absent, and the false presence rates, i.e. species wrongly modelled as present, on a 50 × 50 km grid. Subsequently, we related both error rates to species range properties, land use and topographic variability within grid cells by means of simultaneous autoregressive models to correct for spatial autocorrelation. Results Grid‐cell‐specific error rates were not evenly distributed across Europe. The mean false absence rate was 0.16 ± 0.12 (standard deviation) and the mean false presence rate was 0.22 ± 0.13. False absence rates were highest in central Spain, the Alps and parts of south‐eastern Europe, while false presence rates were highest in northern Spain, France, Italy and south‐eastern Europe. False absence rates were high when range edges of species accumulated within a grid cell and when the intensity of human land use was high. False presence rates were positively associated with relative occurrence area and accumulation of range edges. Main conclusions Predictions for various species are not only accompanied by species‐specific but also by grid‐cell‐specific errors. The latter are associated with characteristics of the grid cells but also with range characteristics of occurring species. Uncertainties of predictive species distribution models are not equally distributed in space, and we would recommend accompanying maps of predicted distributions with a graphical representation of predictive performance.     

The effects of scale and sample size on the accuracy of spatial predictions of tiger beetle (Cicindelidae) species richness

We apply geostatistical modeling techniques to investigate spatial patterns of species richness. Unlike most other statistical modeling techniques that are valid only when observations are independent, geostatistical methods are designed for applications involving spatially dependent observations. When spatial dependencies, which are sometimes called autocorrelations, exist, geostatistical techniques can be applied to produce optimal predictions in areas (typically proximate to observed data) where no observed data exist. Using tiger beetle species (Cicindelidae) data collected in western North America, we investigate the characteristics of spatial relationships in species numbers data, First, we compare the accuracy of spatial predictions of species richness when data from grid squares of two different sizes (scales) are used to form the predictions. Next we examine how prediction accuracy varies as a function of areal extent of the region under investigation. Then we explore the relationship between the number of observations used to build spatial prediction models and prediction accuracy. Our results indicate that, within the taxon of tiger beetles and for the two scales we investigate, the accuracy of spatial predictions is unrelated to scale and that prediction accuracy is not obviously related lo the areal extent of the region under investigation. We also provide information about the relationship between sample size and prediction accuracy, and, finally, we show that prediction accuracy may be substantially diminished if spatial correlations in the data are ignored.     

Using Historical Atlas Data to Develop High-Resolution Distribution Models of Freshwater Fishes

Understanding the spatial pattern of species distributions is fundamental in biogeography, and conservation and resource management applications. Most species distribution models (SDMs) require or prefer species presence and absence data for adequate estimation of model parameters. However, observations with unreliable or unreported species absences dominate and limit the implementation of SDMs. Presence-only models generally yield less accurate predictions of species distribution, and make it difficult to incorporate spatial autocorrelation. The availability of large amounts of historical presence records for freshwater fishes of the United States provides an opportunity for deriving reliable absences from data reported as presence-only, when sampling was predominantly community-based. In this study, we used boosted regression trees (BRT), logistic regression, and MaxEnt models to assess the performance of a historical metacommunity database with inferred absences, for modeling fish distributions, investigating the effect of model choice and data properties thereby. With models of the distribution of 76 native, non-game fish species of varied traits and rarity attributes in four river basins across the United States, we show that model accuracy depends on data quality (e.g., sample size, location precision), species’ rarity, statistical modeling technique, and consideration of spatial autocorrelation. The cross-validation area under the receiver-operating-characteristic curve (AUC) tended to be high in the spatial presence-absence models at the highest level of resolution for species with large geographic ranges and small local populations. Prevalence affected training but not validation AUC. The key habitat predictors identified and the fish-habitat relationships evaluated through partial dependence plots corroborated most previous studies. The community-based SDM framework broadens our capability to model species distributions by innovatively removing the constraint of lack of species absence data, thus providing a robust prediction of distribution for stream fishes in other regions where historical data exist, and for other taxa (e.g., benthic macroinvertebrates, birds) usually observed by community-based sampling designs.     

Incorporating spatial autocorrelation into species distribution models alters forecasts of climate‐mediated range shifts

Species distribution models (SDMs) are widely used to forecast changes in the spatial distributions of species and communities in response to climate change. However, spatial autocorrelation (SA) is rarely accounted for in these models, despite its ubiquity in broad‐scale ecological data. While spatial autocorrelation in model residuals is known to result in biased parameter estimates and the inflation of type I errors, the influence of unmodeled SA on species' range forecasts is poorly understood. Here we quantify how accounting for SA in SDMs influences the magnitude of range shift forecasts produced by SDMs for multiple climate change scenarios. SDMs were fitted to simulated data with a known autocorrelation structure, and to field observations of three mangrove communities from northern Australia displaying strong spatial autocorrelation. Three modeling approaches were implemented: environment‐only models (most frequently applied in species' range forecasts), and two approaches that incorporate SA; autologistic models and residuals autocovariate (RAC) models. Differences in forecasts among modeling approaches and climate scenarios were quantified. While all model predictions at the current time closely matched that of the actual current distribution of the mangrove communities, under the climate change scenarios environment‐only models forecast substantially greater range shifts than models incorporating SA. Furthermore, the magnitude of these differences intensified with increasing increments of climate change across the scenarios. When models do not account for SA, forecasts of species' range shifts indicate more extreme impacts of climate change, compared to models that explicitly account for SA. Therefore, where biological or population processes induce substantial autocorrelation in the distribution of organisms, and this is not modeled, model predictions will be inaccurate. These results have global importance for conservation efforts as inaccurate forecasts lead to ineffective prioritization of conservation activities and potentially to avoidable species extinctions.     

Spatial autocorrelation analysis allows disentangling the balance between neutral and niche processes in metacommunities

One of the most popular approaches for investigating the roles of niche and neutral processes driving metacommunity patterns consists of partitioning variation in species data into environmental and spatial components. The logic is that the distance decay of similarity in communities is expected under neutral models. However, because environmental variation is often spatially structured, the decay could also be attributed to environmental factors that are missing from the analysis. Here, we use a spatial autocorrelation analysis protocol, previously developed to detect isolation‐by‐distance in allele frequencies, to evaluate patterns of species abundances under neutral dynamics. We show that this protocol can be linked with variation partitioning analyses. Moreover, in an attempt to test the neutral model, we derive three predictions to be applied both to original species abundances and to abundances predicted by a pure spatial model species abundances will be uncorrelated; Moran's I correlograms will reveal similar short‐distance autocorrelation patterns; an increasing degree of non‐neutrality will tend to generate patterns of correlation among abundances within groups of species with similar correlograms (i.e. within species with neutral and non‐neutral dynamics). We illustrate our protocol by analyzing spatial patterns in abundance of 28 terrestrially breeding anuran species from Central Amazonia. We recommend that researchers should investigate spatial autocorrelation patterns of abundances predicted by pure spatial models to identify similar patterns of spatial autocorrelation at short distances and lack of correlation between species abundances. Therefore, the hypothesis that spatial patterns in abundances are primarily due to pure neutral dynamics (rather than to missing spatiallystructured environmental factors) can be confirmed after taking environmental variables into account.     

Universal scaling of species‐abundance distributions across multiple scales

The scale‐dependent species abundance distribution (SAD) is fundamental in ecology, but few spatially explicit models of this pattern have thus far been studied. Here we show spatially explicit neutral model predictions for SADs over a wide range of spatial scales, which appear to match empirical patterns qualitatively. We find that the assumption of a log‐series SAD in the metacommunity made by spatially implicit neutral models can be justified with a spatially explicit model in the large area limit. Furthermore, our model predicts that SADs on multiple scales are characterized by a single, compound parameter that represents the ratio of the survey area to the species’ average biogeographic range (which is in turn set by the speciation rate and the dispersal distance). This intriguing prediction is in line with recent empirical evidence for a universal scaling of the species‐area curve. Hence we hypothesize that empirical SAD patterns will show a similar universal scaling for many different taxa and across multiple spatial scales.     

A spatially explicit method for evaluating accuracy of species distribution models

Aim Models predicting the spatial distribution of animals are increasingly used in wildlife management and conservation planning. There is growing recognition that common methods of evaluating species distribution model (SDM) accuracy, as a global overall value of predictive ability, could be enhanced by spatially evaluating the model thereby identifying local areas of relative predictive strength and weakness. Current methods of spatial SDM model assessment focus on applying local measures of spatial autocorrelation to SDM residuals, which require quantitative model outputs. However, SDM outputs are often probabilistic (relative probability of species occurrence) or categorical (species present or absent). The goal of this paper was to develop a new method, using a conditional randomization technique, which can be applied to directly spatially evaluate probabilistic and categorical SDMs. Location Eastern slopes, Rocky Mountains, Alberta, Canada. Methods We used predictions from seasonal grizzly bear (Ursus arctos) resource selection functions (RSF) models to demonstrate our spatial evaluation technique. Local test statistics computed from bear telemetry locations were used to identify areas where bears were located more frequently than predicted. We evaluated the spatial pattern of model inaccuracies using a measure of spatial autocorrelation, local Moran’s I. Results We found the model to have non‐stationary patterns in accuracy, with clusters of inaccuracies located in central habitat areas. Model inaccuracies varied seasonally, with the summer model performing the best and the least error in areas with high RSF values. The landscape characteristics associated with model inaccuracies were examined, and possible factors contributing to RSF error were identified. Main conclusions The presented method complements existing spatial approaches to model error assessment as it can be used with probabilistic and categorical model output, which is typical for SDMs. We recommend that SDM accuracy assessments be done spatially and resulting accuracy maps included in model metadata.     

Analysis of determinants of mammalian species richness in South America using spatial autoregressive models

Classically, hypotheses concerning the distribution of species have been explored by evaluating the relationship between species richness and environmental variables using ordinary least squares (OLS) regression. However, environmental and ecological data generally show spatial autocorrelation, thus violating the assumption of independently distributed errors. When spatial autocorrelation exists, an alternative is to use autoregressive models that assume spatially autocorrelated errors. We examined the relationship between mammalian species richness in South America and environmental variables, thereby evaluating the relative importance of four competing hypotheses to explain mammalian species richness. Additionally, we compared the results of ordinary least squares (OLS) regression and spatial autoregressive models using Conditional and Simultaneous Autoregressive (CAR and SAR, respectively) models. Variables associated with productivity were the most important at determining mammalian species richness at the scale analyzed. Whereas OLS residuals between species richness and environmental variables were strongly autocorrelated, those from autoregressive models showed less spatial autocorrelation, particularly the SAR model, indicating its suitability for these data. Autoregressive models also fit the data better than the OLS model (increasing R² by 5–14%), and the relative importance of the explanatory variables shifted under CAR and SAR models. These analyses underscore the importance of controlling for spatial autocorrelation in biogeographical studies.     

Limited transferability of stream‐fish distribution models among river catchments: reasons and implications

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Land use improves spatial predictions of mountain plant abundance but not presence‐absence

Question: Does a land‐use variable improve spatial predictions of plant species presence‐absence and abundance models at the regional scale in a mountain landscape? Location: Western Swiss Alps. Methods: Presence‐absence generalized linear models (GLM) and abundance ordinal logistic regression models (LRM) were fitted to data on 78 mountain plant species, with topo‐climatic and/or land‐use variables available at a 25‐m resolution. The additional contribution of land use when added to topo‐climatic models was evaluated by: (1) assessing the changes in model fit and (2) predictive power, (3) partitioning the deviance respectively explained by the topo‐climatic variables and the land‐use variable through variation partitioning, and (5) comparing spatial projections. Results: Land use significantly improved the fit of presence‐absence models but not their predictive power. In contrast, land use significantly improved both the fit and predictive power of abundance models. Variation partitioning also showed that the individual contribution of land use to the deviance explained by presence‐absence models was, on average, weak for both GLM and LRM (3.7% and 4.5%, respectively), but changes in spatial projections could nevertheless be important for some species. Conclusions: In this mountain area and at our regional scale, land use is important for predicting abundance, but not presence‐absence. The importance of adding land‐use information depends on the species considered. Even without a marked effect on model fit and predictive performance, adding land use can affect spatial projections of both presence‐absence and abundance models.     

Predicting the Geographic Distribution of a Species from Presence‐Only Data Subject to Detection Errors

Summary Several models have been developed to predict the geographic distribution of a species by combining measurements of covariates of occurrence at locations where the species is known to be present with measurements of the same covariates at other locations where species occurrence status (presence or absence) is unknown. In the absence of species detection errors, spatial point‐process models and binary‐regression models for case‐augmented surveys provide consistent estimators of a species’ geographic distribution without prior knowledge of species prevalence. In addition, these regression models can be modified to produce estimators of species abundance that are asymptotically equivalent to those of the spatial point‐process models. However, if species presence locations are subject to detection errors, neither class of models provides a consistent estimator of covariate effects unless the covariates of species abundance are distinct and independently distributed from the covariates of species detection probability. These analytical results are illustrated using simulation studies of data sets that contain a wide range of presence‐only sample sizes. Analyses of presence‐only data of three avian species observed in a survey of landbirds in western Montana and northern Idaho are compared with site‐occupancy analyses of detections and nondetections of these species.     

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.     

Assessing the accuracy of species distribution models to predict amphibian species richness patterns

1. Evaluating the distribution of species richness where biodiversity is high but has been insufficiently sampled is not an easy task. Species distribution modelling has become a useful approach for predicting their ranges, based on the relationships between species records and environmental variables. Overlapping predictions of individual distributions could be a useful strategy for obtaining estimates of species richness and composition in a region, but these estimates should be evaluated using a proper validation process, which compares the predicted richness values and composition with accurate data from independent sources. 2. In this study, we propose a simple approach to estimate model performance for several distributional predictions generated simultaneously. This approach is particularly suitable when species distribution modelling techniques that require only presence data are used. 3. The individual distributions for the 370 known amphibian species of Mexico were predicted using maxent to model data on their known presence (66,113 presence-only records). Distributions were subsequently overlapped to obtain a prediction of species richness. Accuracy was assessed by comparing the overall species richness values predicted for the region with observed and predicted values from 118 well-surveyed sites, each with an area of c. 100 km(2), which were identified using species accumulation curves and nonparametric estimators. 4. The derived models revealed a remarkable heterogeneity of species richness across the country, provided information about species composition per site and allowed us to obtain a measure of the spatial distribution of prediction errors. Examining the magnitude and location of model inaccuracies, as well as separately assessing errors of both commission and omission, highlights the inaccuracy of the predictions of species distribution models and the need to provide measures of uncertainty along with the model results. 5. The combination of a species distribution modelling method like maxent and species richness estimators offers a useful tool for identifying when the overall pattern provided by all model predictions might be representing the geographical patterns of species richness and composition, regardless of the particular quality or accuracy of the predictions for each individual species.     

Accounting for spatial autocorrelation in null models of tree species association

A commonly used null model for species association among forest trees is a well‐mixed community (WMC). A WMC represents a non‐spatial, or spatially implicit, model, in which species form nearest‐neighbor pairs at a rate equal to the product of their community proportions. WMC models assume that the outcome of random dispersal and demographic processes is complete spatial randomness (CSR) in the species’ spatial distributions. Yet, stochastic dispersal processes often lead to spatial autocorrelation (SAC) in tree species densities, giving rise to clustering, segregation, and other nonrandom patterns. Although methods exist to account for SAC in spatially‐explicit models, its impact on non‐spatial models often remains unaccounted for. To investigate the potential for SAC to bias tests based upon non‐spatial models, we developed a spatially‐heterogeneous (SH) modelling approach that incorporates measured levels of SAC. Using the mapped locations of individuals in a tropical tree community, we tested the hypothesis that the identity of nearest‐neighbors represents a random draw from neighborhood species pools. Correlograms of Moran's I confirmed that, for 50 of 51 dominant species, stem density was significantly autocorrelated over distances ranging from 50 to 200 m. The observed patterns of SAC were consistent with dispersal limitation, with most species occurring in distinct patches. For nearly all of the 106 species in the community, the frequency of pairwise association was statistically indistinguishable from that projected by the null models. However, model comparisons revealed that non‐spatial models more strongly underestimated observed species‐pair frequencies, particularly for conspecific pairs. Overall, the CSR models projected more significant facilitative interactions than did SH models, yielding a more liberal test of niche differences. Our results underscore the importance of accounting for stochastic spatial processes in tests of association, regardless of whether spatial or non‐spatial models are employed.