Multiple regression on distance matrices: a multivariate spatial analysis tool

I explore the use of multiple regression on distance matrices (MRM), an extension of partial Mantel analysis, in spatial analysis of ecological data. MRM involves a multiple regression of a response matrix on any number of explanatory matrices, where each matrix contains distances or similarities (in terms of ecological, spatial, or other attributes) between all pair-wise combinations of n objects (sample units); tests of statistical significance are performed by permutation. The method is flexible in terms of the types of data that may be analyzed (counts, presence–absence, continuous, categorical) and the shapes of response curves. MRM offers several advantages over traditional partial Mantel analysis: (1) separating environmental distances into distinct distance matrices allows inferences to be made at the level of individual variables; (2) nonparametric or nonlinear multiple regression methods may be employed; and (3) spatial autocorrelation may be quantified and tested at different spatial scales using a series of lag matrices, each representing a geographic distance class. The MRM lag matrices model may be parameterized to yield very similar inferences regarding spatial autocorrelation as the Mantel correlogram. Unlike the correlogram, however, the lag matrices model may also include environmental distance matrices, so that spatial patterns in species abundance distances (community similarity) may be quantified while controlling for the environmental similarity between sites. Examples of spatial analyses with MRM are presented.     

Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression

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

A review of techniques for spatial modeling in geographical, conservation and landscape genetics

Most evolutionary processes occur in a spatial context and several spatial analysis techniques have been employed in an exploratory context. However, the existence of autocorrelation can also perturb significance tests when data is analyzed using standard correlation and regression techniques on modeling genetic data as a function of explanatory variables. In this case, more complex models incorporating the effects of autocorrelation must be used. Here we review those models and compared their relative performances in a simple simulation, in which spatial patterns in allele frequencies were generated by a balance between random variation within populations and spatially-structured gene flow. Notwithstanding the somewhat idiosyncratic behavior of the techniques evaluated, it is clear that spatial autocorrelation affects Type I errors and that standard linear regression does not provide minimum variance estimators. Due to its flexibility, we stress that principal coordinate of neighbor matrices (PCNM) and related eigenvector mapping techniques seem to be the best approaches to spatial regression. In general, we hope that our review of commonly used spatial regression techniques in biology and ecology may aid population geneticists towards providing better explanations for population structures dealing with more complex regression problems throughout geographic space.     

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.     

Local spatial autocorrelation in biological variables

Spatial autocorrelation (SA) methods have recently been extended to include the detection of local spatial autocorrelation at individual sampling stations. We review the formulas for these statistics and report on the results of an extensive population-genetic simulation study we have published elsewhere to test the applicability of these methods in spatially distributed biological data. We find that most biological variables exhibit global SA, and that in such cases the methods proposed for testing the significance of local SA coefficients reject the null hypothesis excessively. When global SA is absent, permutational methods for testing significance yield reliable results. Although standard errors have been published for the local SA coefficients, their employment using an asymptotically normal approach leads to unreliable results; permutational methods are preferred. In addition to significance tests of suspected non-stationary localities, we can use these methods in an exploratory manner to find and identify hotspots (places with positive local SA) and coldspots (negative local SA) in a dataset. We illustrate the application of these methods in three biological examples from plant population biology, ecology and population genetics. The examples range from the study of single variables to the joint analysis of several variables and can lead to successful demographic and evolutionary inferences about the populations studied.     

Spatial autocorrelation in biology 1. Methodology

Spatial autocorrelation analysis tests whether the observed value of a nominal, ordinal, or interval variable at one locality is independent of values of the variable at neighbouring localities. The computation of autocorrelation coefficients for nominal, ordinal, and for interval data is illustrated, together with appropriate significance tests. The method is extended to include the computation of correlograms for spatial autocorrelation. These show the autocorrelation coefficient as a function of distance between pairs of localities being considered, and summarize the patterns of geographic variation exhibited by the response surface of any given variable.
Autocorrelation analysis is applied to microgeographic variation of allozyme frequencies in the snail Helix aspersa. Differences in variational patterns in two city blocks are interpreted.
The inferences that can be drawn from correlograms are discussed and illustrated with the aid of some artificially generated patterns. Computational formulae, expected values and standard errors are furnished in two appendices.     

Using weighted permutation scores to detect differential gene expression with microarray data

A class of nonparametric statistical methods, including a nonparametric empirical Bayes (EB) method, the Significance Analysis of Microarrays (SAM) and the mixture model method (MMM) have been proposed to detect differential gene expression for replicated microarray experiments. They all depend on constructing a test statistic, for example, a t-statistic, and then using permutation to draw inferences. However, due to special features of microarray data, using standard permutation scores may not estimate the null distribution of the test statistic well, leading to possibly too conservative inferences. We propose a new method of constructing weighted permutation scores to overcome the problem: posterior probabilities of having no differential expression from the EB method are used as weights for genes to better estimate the null distribution of the test statistic. We also propose a weighted method to estimate the false discovery rate (FDR) using the posterior probabilities. Using simulated data and real data for time-course microarray experiments, we show the improved performance of the proposed methods when implemented in MMM, EB and SAM.     

A Test of Spatial Autocorrelation Analysis Using an Isolation-by-Distance Model

Using the isolation-by-distance model as an example, we have examined several assumptions of spatial autocorrelation analysis applied to gene frequency surfaces. Gene frequency surfaces generated by a simulation of Wright's isolation-by-distance model were shown to exhibit spatial autocorrelation, except in the panmictic case. Identical stochastic generating processes result in surfaces with characteristics that are functions of the process parameters, such as parental vagility and neighborhood size. Differences in these parameters are detectable as differences in spatial autocorrelations after only a few generations of the simulations. Separate realizations of processes with identical parameters yield similar spatial correlograms. We have examined the inferences about population structure that could have been made from these observations if they had been real, rather than simulated, populations. From such inferences, we could have drawn conclusions about the presence of selection, migration and drift in given natural systems.     

Quantitative approaches in climate change ecology

Contemporary impacts of anthropogenic climate change on ecosystems are increasingly being recognized. Documenting the extent of these impacts requires quantitative tools for analyses of ecological observations to distinguish climate impacts in noisy data and to understand interactions between climate variability and other drivers of change. To assist the development of reliable statistical approaches, we review the marine climate change literature and provide suggestions for quantitative approaches in climate change ecology. We compiled 267 peer‐reviewed articles that examined relationships between climate change and marine ecological variables. Of the articles with time series data (n = 186), 75% used statistics to test for a dependency of ecological variables on climate variables. We identified several common weaknesses in statistical approaches, including marginalizing other important non‐climate drivers of change, ignoring temporal and spatial autocorrelation, averaging across spatial patterns and not reporting key metrics. We provide a list of issues that need to be addressed to make inferences more defensible, including the consideration of (i) data limitations and the comparability of data sets; (ii) alternative mechanisms for change; (iii) appropriate response variables; (iv) a suitable model for the process under study; (v) temporal autocorrelation; (vi) spatial autocorrelation and patterns; and (vii) the reporting of rates of change. While the focus of our review was marine studies, these suggestions are equally applicable to terrestrial studies. Consideration of these suggestions will help advance global knowledge of climate impacts and understanding of the processes driving ecological change.     

Testing for marginal independence between two categorical variables with multiple responses

Questions that ask respondents to "choose all that apply" from a set of items occur frequently in surveys. Categorical variables that summarize this type of survey data are called both pick any/c variables and multiple-response categorical variables. It is often of interest to test for independence between two categorical variables. When both categorical variables can have multiple responses, traditional Pearson chi-square tests for independence should not be used because of the within-subject dependence among responses. An intuitively constructed version of the Pearson statistic is proposed to perform the test using bootstrap procedures to approximate its sampling distribution. First- and second-order adjustments to the proposed statistic are given in order to use a chi-square distribution approximation. A Bonferroni adjustment is proposed to perform the test when the joint set of responses for individual subjects is unavailable. Simulations show that the bootstrap procedures hold the correct size more consistently than the other procedures.     

The consequences of spatial structure for the design and analysis of ecological field surveys

In ecological field surveys, observations are gathered at different spatial locations. The purpose may be to relate biological response variables (e.g., species abundances) to explanatory environmental variables (e.g., soil characteristics). In the absence of prior knowledge, ecologists have been taught to rely on systematic or random sampling designs. If there is prior knowledge about the spatial patterning of the explanatory variables, obtained from either previous surveys or a pilot study, can we use this information to optimize the sampling design in order to maximize our ability to detect the relationships between the response and explanatory variables?
The specific questions addressed in this paper are: a) What is the effect (type I error) of spatial autocorrelation on the statistical tests commonly used by ecologists to analyse field survey data? b) Can we eliminate, or at least minimize, the effect of spatial autocorrelation by the design of the survey? Are there designs that provide greater power for surveys, at least under certain circumstances? c) Can we eliminate or control for the effect of spatial autocorrelation during the analysis? To answer the last question, we compared regular regression analysis to a modified t‐test developed by Dutilleul for correlation coefficients in the presence of spatial autocorrelation.
Replicated surfaces (typically, 1000 of them) were simulated using different spatial parameters, and these surfaces were subjected to different sampling designs and methods of statistical analysis. The simulated surfaces may represent, for example, vegetation response to underlying environmental variation. This allowed us 1) to measure the frequency of type I error (the failure to reject the null hypothesis when in fact there is no effect of the environment on the response variable) and 2) to estimate the power of the different combinations of sampling designs and methods of statistical analysis (power is measured by the rate of rejection of the null hypothesis when an effect of the environment on the response variable has been created).
Our results indicate that: 1) Spatial autocorrelation in both the response and environmental variables affects the classical tests of significance of correlation or regression coefficients. Spatial autocorrelation in only one of the two variables does not affect the test of significance. 2) A broad‐scale spatial structure present in data has the same effect on the tests as spatial autocorrelation. When such a structure is present in one of the variables and autocorrelation is found in the other, or in both, the tests of significance have inflated rates of type I error. 3) Dutilleul's modified t‐test for the correlation coefficient, corrected for spatial autocorrelation, effectively corrects for spatial autocorrelation in the data. It also effectively corrects for the presence of deterministic structures, with or without spatial autocorrelation.
The presence of a broad‐scale deterministic structure may, in some cases, reduce the power of the modified t‐test.     

Multiple species effects and spatial autocorrelation in detecting species associations

Abstract. The traditional approach to the analysis of species association within a community, based upon co-occurrence in sampling units such as quadrats, has been to test all pairs of species, using a 2 × 2 contingency table for each pair. It has long been recognised that all these tests are not independent of each other, but there is an additional problem in that the association between any particular pair may depend on the combination of the other species that are present or on the environmental factors that determine that combination. We use a 2^k contingency table to examine this problem and find that pairwise associations are not independent of the other species. The second problem that we consider is the effect of spatial autocorrelation in the data which makes the statistical tests too liberal. In the absence of a derived solution for a deflation factor to correct the test statistic calculated from a 2^k table, we describe a Monte Carlo approach that provides an approximate solution to this problem. In our data the amount of deflation that is necessary for a 2^k table is small compared to the amount required for the 2 × 2 tables used to test pairwise association.     

Disentangling drivers of spatial autocorrelation in species distribution models

Species distribution models (SDMs) are frequently used to understand the influence of site properties on species occurrence. For robust model inference, SDMs need to account for the spatial autocorrelation of virtually all species occurrence data. Current methods do not routinely distinguish between extrinsic and intrinsic drivers of spatial autocorrelation, although these may have different implications for conservation. Here, we present and test a method that disentangles extrinsic and intrinsic drivers of spatial autocorrelation using repeated observations of a species. We focus on unknown habitat characteristics and conspecific interactions as extrinsic and intrinsic drivers, respectively. We model the former with spatially correlated random effects and the latter with an autocovariate, such that the spatially correlated random effects are constant across the repeated observations whereas the autocovariate may change. We tested the performance of our model on virtual species data and applied it to observations of the corncrake Crex crex in the Netherlands. Applying our model to virtual species data revealed that it was well able to distinguish between the two different drivers of spatial autocorrelation, outperforming models with no or a single component for spatial autocorrelation. This finding was independent of the direction of the conspecific interactions (i.e. conspecific attraction versus competitive exclusion). The simulations confirmed that the ability of our model to disentangle both drivers of autocorrelation depends on repeated observations. In the case study, we discovered that the corncrake has a stronger response to habitat characteristics compared to a model that did not include spatially correlated random effects, whereas conspecific interactions appeared to be less important. This implies that future conservation efforts should primarily focus on maximizing habitat availability. Our study shows how to systematically disentangle extrinsic and intrinsic drivers of spatial autocorrelation. The method we propose can help to correctly identify the main drivers of species distributions.     

Spind: a package for computing spatially corrected accuracy measures

Using an appropriate accuracy measure is essential for assessing prediction accuracy in species distribution modelling. Therefore, model evaluation as an analytical uncertainty is a challenging problem. Although a variety of accuracy measures for the assessment of prediction errors in presence/absence models is available, there is a lack of spatial accuracy measures, i.e. measures that are sensitive to the spatial arrangement of the predictions. We present ‘spind’, a new software package (based on the R software program) that provides spatial performance measures for grid‐based models. These accuracy measures are generalized, spatially corrected versions of the classical ones, thus enabling comparisons between them. Our method for evaluation consists of the following steps: 1) incorporate additional autocorrelation until spatial autocorrelation in predictions and actuals is balanced, 2) cross‐classify predictions and adjusted actuals in a 4 × 4 contingency table, 3) use a refined weighting pattern for errors, and 4) calculate weighted Kappa, sensitivity, specificity and subsequently ROC, AUC, TSS to get spatially corrected indices. To illustrate the impact of our spatial method we present an example of simulated data as well as an example of presence/absence data of the plant species Dianthus carthusianorum across Germany. Our analysis includes a statistic for the comparison of spatial and classical (non‐spatial) indices. We find that our spatial indices tend to result in higher values than classical ones. These differences are statistically significant at medium and high autocorrelation levels. We conclude that these spatial accuracy measures may contribute to evaluate prediction errors in presence/absence models, especially in case of medium or high degree of similarity of adjacent data, i.e. aggregated (clumped) or continuous species distributions.     

On the use of the variogram in checking for independence in spatial data

The variogram is a standard tool in the analysis of spatial data, and its shape provides useful information on the form of spatial correlation that may be present. However, it is also useful to be able to assess the evidence for the presence of any spatial correlation. A method of doing this, based on an assessment of whether the true function underlying the variogram is constant, is proposed. Nonparametric smoothing of the squared differences of the observed variables, on a suitably transformed scale, is used to estimate variogram shape. A statistic based on a ratio of quadratic forms is proposed and the test is constructed by investigating the distributional properties of this statistic under the assumption of an independent Gaussian process. The power of the test is investigated. Reference bands are proposed as a graphical follow-up. An example is discussed.     

Microhabitat analyses support relationships between niche breadth and range size when spatial autocorrelation is strong

Multiple evidence of positive relationships between nice breadth and range size (NB–RS) suggested that this can be a general ecological pattern. However, correlations between niche breadth and range size can emerge as a by-product of strong spatial structure of environmental variables. This can be problematic because niche breadth is often assessed using broad-scale macroclimatic variables, which suffer heavy spatial autocorrelation. Microhabitat measurements provide accurate information on species tolerance, and show limited autocorrelation. The aim of this study was to combine macroclimate and microhabitat data to assess NB–RS relationships in European plethodontid salamanders (Hydromantes), and to test whether microhabitat variables with weak autocorrelation can provide less biased NB–RS estimates across species. To measure macroclimatic niche, we gathered comprehensive information on the distribution of all Hydromantes species, and combined them with broad-scale climatic layers. To measure microhabitat, we recorded salamander occurrence across > 350 caves and measured microhabitat features influencing their distribution: humidity, temperature and light. We assessed NB–RS relationships through phylogenetic regression; spatial null-models were used to test whether the observed relationships are a by-product of autocorrelation. We observed positive relationships between niche breadth and range size at both the macro- and microhabitat scale. At the macroclimatic scale, strong autocorrelation heavily inflated the possibility to observe positive NB–RS. Spatial autocorrelation was weaker for microhabitat variables. At the microhabitat level, the observed NB–RS was not a by-product of spatial structure of variables. Our study shows that heavy autocorrelation of variables artificially increases the possibility to detect positive relationships between bioclimatic niche and range size, while fine-scale data of microhabitat provide more direct measure of conditions selected by ectotherms, and enable less biased measures of niche breadth. Combining analyses performed at multiple scales and datasets with different spatial structure provides more complete niche information and effectively tests the generality of niche breadth–range size relationships.     

Plant dispersal in the sub-Antarctic inferred from anisotropic genetic structure

Climatic conditions and landscape features often strongly affect species' local distribution patterns, dispersal, reproduction and survival and may therefore have considerable impacts on species' fine-scale spatial genetic structure (SGS). In this study, we demonstrate the efficacy of combining fine-scale SGS analyses with isotropic and anisotropic spatial autocorrelation techniques to infer the impact of wind patterns on plant dispersal processes. We genotyped 1304 Azorella selago (Apiaceae) specimens, a wind-pollinated and wind-dispersed plant, from four populations distributed across sub-Antarctic Marion Island. SGS was variable with Sp values ranging from 0.001 to 0.014, suggesting notable variability in dispersal distance and wind velocities between sites. Nonetheless, the data supported previous hypotheses of a strong NW-SE gradient in wind strength across the island. Anisotropic autocorrelation analyses further suggested that dispersal is strongly directional, but varying between sites depending on the local prevailing winds. Despite the high frequency of gale-force winds on Marion Island, gene dispersal distance estimates (σ) were surprisingly low (<10 m), most probably because of a low pollen dispersal efficiency. An SGS approach in association with isotropic and anisotropic analyses provides a powerful means to assess the relative influence of abiotic factors on dispersal and allow inferences that would not be possible without this combined approach.     

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

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

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.