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.     

Correlation analysis of dissimilarity matrices

Distance-based methods have been a valuable tool for ecologists for decades. Indirectly, distance-based ordination and cluster analysis, in particular, have been widely practiced as they allow the visualization of a multivariate data set in a few dimensions. The explicitly distance-based Mantel test and multiple regression on distance matrices (MRM) add hypothesis testing to the toolbox. One concern for ecologists wishing to use these methods lies in deciding whether to combine data vectors into a compound multivariate dissimilarity to analyze them individually. For Euclidean distances on scaled data, the correlation of a pair of multivariate distance matrices can be calculated from the correlations between the two sets of individual distance matrices if one set is orthogonal, demonstrating a clear link between individual and compound distances. The choice between Mantel and MRM should be driven by ecological hypotheses rather than mathematical concerns. The relationship between individual and compound distance matrices also provides a means for calculating the maximum possible value of the Mantel statistic, which can be considerably less than 1 for a given analysis. These relationships are demonstrated with simulated data. Although these mathematical relationships are only strictly true for Euclidean distances when one set of variables is orthogonal, simulations show that they are approximately true for weakly correlated variables and Bray–Curtis dissimilarities.     

Using simulations to evaluate Mantel‐based methods for assessing landscape resistance to gene flow

Mantel‐based tests have been the primary analytical methods for understanding how landscape features influence observed spatial genetic structure. Simulation studies examining Mantel‐based approaches have highlighted major challenges associated with the use of such tests and fueled debate on when the Mantel test is appropriate for landscape genetics studies. We aim to provide some clarity in this debate using spatially explicit, individual‐based, genetic simulations to examine the effects of the following on the performance of Mantel‐based methods: (1) landscape configuration, (2) spatial genetic nonequilibrium, (3) nonlinear relationships between genetic and cost distances, and (4) correlation among cost distances derived from competing resistance models. Under most conditions, Mantel‐based methods performed poorly. Causal modeling identified the true model only 22% of the time. Using relative support and simple Mantel r values boosted performance to approximately 50%. Across all methods, performance increased when landscapes were more fragmented, spatial genetic equilibrium was reached, and the relationship between cost distance and genetic distance was linearized. Performance depended on cost distance correlations among resistance models rather than cell‐wise resistance correlations. Given these results, we suggest that the use of Mantel tests with linearized relationships is appropriate for discriminating among resistance models that have cost distance correlations <0.85 with each other for causal modeling, or <0.95 for relative support or simple Mantel r. Because most alternative parameterizations of resistance for the same landscape variable will result in highly correlated cost distances, the use of Mantel test‐based methods to fine‐tune resistance values will often not be effective.     

POOR STATISTICAL PERFORMANCE OF THE MANTEL TEST IN PHYLOGENETIC COMPARATIVE ANALYSES

The Mantel test, based on comparisons of distance matrices, is commonly employed in comparative biology, but its statistical properties in this context are unknown. Here, we evaluate the performance of the Mantel test for two applications in comparative biology: testing for phylogenetic signal, and testing for an evolutionary correlation between two characters. We find that the Mantel test has poor performance compared to alternative methods, including low power and, under some circumstances, inflated type‐I error. We identify a remedy for the inflated type‐I error of three‐way Mantel tests using phylogenetic permutations; however, this test still has considerably lower power than independent contrasts. We recommend that use of the Mantel test should be restricted to cases in which data can only be expressed as pairwise distances among taxa.     

Mantel test in population genetics

The comparison of genetic divergence or genetic distances, estimated by pairwise F_ST and related statistics, with geographical distances by Mantel test is one of the most popular approaches to evaluate spatial processes driving population structure. There have been, however, recent criticisms and discussions on the statistical performance of the Mantel test. Simultaneously, alternative frameworks for data analyses are being proposed. Here, we review the Mantel test and its variations, including Mantel correlograms and partial correlations and regressions. For illustrative purposes, we studied spatial genetic divergence among 25 populations of Dipteryx alata (“Baru”), a tree species endemic to the Cerrado, the Brazilian savannas, based on 8 microsatellite loci. We also applied alternative methods to analyze spatial patterns in this dataset, especially a multivariate generalization of Spatial Eigenfunction Analysis based on redundancy analysis. The different approaches resulted in similar estimates of the magnitude of spatial structure in the genetic data. Furthermore, the results were expected based on previous knowledge of the ecological and evolutionary processes underlying genetic variation in this species. Our review shows that a careful application and interpretation of Mantel tests, especially Mantel correlograms, can overcome some potential statistical problems and provide a simple and useful tool for multivariate analysis of spatial patterns of genetic divergence.     

Permutation tests for the correlation among genetic distances and measures of heterosis

It is often found that heterosis tends to increase with genetic distance of the parents, though the correlation is not usually very close. It is therefore important to test the null hypothesis that the correlation is zero. The present work shows that standard procedures tend to yield too liberal tests, owing to the lack of independence among genetic distances and among heterosis estimates. A valid alternative is to use a permutation test, which was first suggested by Mantel [(1967) Cancer Res 27: 209–220). This test is well-known among plant breeders and geneticists, who often use it to test the correlation among two distance matrices. Its use is not restricted to the comparison of distance matrices. This is demonstrated in the present work, using two published datasets on marker-based genetic distances of maize inbreds or populations and heterosis of their crosses. It is shown that the test is also applicable in the presence of missing data.     

Geographic variation of flower traits in Narcissus papyraceus (Amaryllidaceae): do pollinators matter?

Aim The geographic clinal variation of traits in organisms can indicate the possible causes of phenotypic evolution. We studied the correlates of flower trait variation in populations of a style‐dimorphic plant, Narcissus papyraceus Ker‐Gawl., within a region of high biogeographical significance, the Strait of Gibraltar. This species shows a geographic gradient in the style‐morph ratio, suggested to be driven by pollinator shifts. We tested whether parallel geographic variation of perianth traits also exists, concomitant with vegetative trait variation or genetic similarity of plant populations. Location The Strait of Gibraltar region (SG hereafter, including both south‐western Iberian Peninsula and north‐western Morocco). Methods We used univariate and multivariate analyses of flower and vegetative traits in 23 populations. We applied Mantel tests and partial Mantel correlations on vegetative and flower traits and geographic locations of populations to test for spatial effects. We used Moran’s autocorrelation analyses to explore the spatial structure within the range, and performed the analyses with and without the Moroccan samples to test for the effects of the SG on spatial patterns. Amplified fragment length polymorphism data were used to estimate the genetic distance between populations and to ascertain its relationship with morphometric distance. Results There was high variation between and within populations in both flower and vegetative traits. Mantel correlations between geographic and morphometric distances were not significant, but the exclusion of Moroccan populations revealed some distance effect. Partial Mantel correlation did not detect a significant correlation between flower and vegetative morphometric distances after controlling for geographic distance. There were opposite trends in spatial autocorrelograms of flower and vegetative traits. The genetic distance between pairs of populations was directly correlated with geographic distance; however, flower morphometric and genetic distances were not significantly correlated. Main conclusions The SG had some influence on phenotypes, although the causes remain to be determined. The opposite trend of variation in flower and vegetative traits, and the lack of correlation between genetic distance and dissimilarity of flower phenotypes favour the hypothesis of pollinator‐mediated selection on flower morphology, although this may affect only particular traits and populations rather than overall phenotypes. Although stochastic population processes may have a small effect, other factors may account for the high flower variation within and between populations.     

Genome-wide association to fine-scale ecological heterogeneity within a continuous population of Biscutella laevigata (Brassicaceae)

Gene flow, drift and selection can be detected through different signatures across the genome and the landscape. Genetic discontinuities along with their correlation to environmental features can be used to tease out isolation-by-distance and isolation-by-time from processes related to selection. Using spatial statistics (spatial autocorrelation methods, canonical correspondence analysis and partial Mantel tests) dealing with genome-wide amplified fragment length polymorphism (AFLP) under unlikely Hardy-Weinberg assumptions, this study investigates 124 individuals within a continuous population of the autopolyploid Biscutella laevigata (Brassicaceae). Fine-scale spatial genetic structure was strong and the mosaic-like distribution of AFLP genotypes was consistently associated with habitat factors, even when controlled for geographical distances. The use of multivariate analyses enabled separation of the factors responsible for the repartition of the genetic variance and revealed a composite effect of isolation by distance, phenological divergence and local adaptation to habitats characterised by different solar radiation regimes. These results suggest that the immigrant inviability barrier facilitated the maintenance of adapted subpopulations to distinct environmental conditions at the local scale.     

Quantitative matrix comparisons in ecological and evolutionary investigations

The pair-wise statistical comparison of data matrices is a methodological problem which must be dealt with in a variety of disciplines. In this report we demonstrate an innovative approach using the Mantel test (a nonparametric, multivariate evaluation of test matrices) to quantitatively contrast observed color polymorphisms in male Poecilia reticulata collected from 41 samples sites in Trinidad against four evolutionary models: (1) response to an environmental gradient, (2) localized environmental patches, (3) isolation by distance, and (4) historical factors. To represent these models we derived pair-wise distances between study sites for the following data: (1) altitude, (2) density of predators, and (3) kilometric distances. To represent model four above, we generated an imposed asymptotic distance matrix for geographically contiguous sites, and a Gabriel connectivity matrix for stream-connected sites (e.g. those within the same watershed). We found that differences in color polymorphisms covary significantly with differences in predator densities and in altitudes, suggesting that male color polymorphisms track clinally distributed communities of visually hunting predators. These data substantiate previously published results from field and laboratory experiments. The utility of the Mantel procedure is that it permits a quantitative evaluation of ecological and evolutionary problems which have previously been difficult to approach statistically.     

Species delimitation and geography

Despite the importance of the geographical arrangement of populations for the inference of species boundaries, only a few approaches that integrate spatial information into species delimitation have thus far been developed. Persistent differentiation of sympatric groups of individuals is the best criterion for species status. Species delimitation becomes more prone to error if allopatric metapopulations are considered because it is often difficult to assess whether observed differences between allopatric metapopulations would be sufficient to prevent the fusion of these metapopulations upon contact. We propose a novel approach for testing the hypothesis that the multilocus genetic distances between individuals or populations belonging to two different candidate species are not larger than expected based on their geographical distances and the relationship of genetic and geographical distances within the candidate species. A rejection of this null hypothesis is an argument for classifying the two studied candidate species as distinct species. Case studies show that the proposed tests are suitable to distinguish between intra‐ and interspecific differentiation. The regression approach proposed here is more appropriate for testing species hypotheses with regard to isolation by distance than (partial) Mantel tests. Our tests assume a linear relationship between genetic and (transformed) geographical distances. This assumption can be compromised by a high genetic variability within populations as found in a case study with microsatellite markers.     

Genetic relationships among some new cat populations sampled in Europe: A spatial autocorrelation analysis

The allelic frequencies of nine Mendelizing genetic characteristics that control coat colour, tabby and length and some skeletal abnormalities have been studied in four feral domestic cat populations, two in the north of Catalonia (Girona and Roses & L’Estartit, northeastern Spain) and two Adriatic Italian populations (Rimini and Venice). Using different genetic and multivariate analyses (Nei’s and Cavalli-Sforza and Edwards’s genetic distances, phenograms and cladograms using different algorithms, strict consensus trees, canonical population, principal coordinates and nonmetric multidimensional scaling analyses), I show the genetic relationships between these populations and other Western European cat populations previously studied. In the Western European area comprising Catalonia, Italy, France and Great Britain, I found significant spatial structure for thet ^b, l andW alleles and for the average correlogram for the seven alleles studied as a whole using a spatial autocorrelation analysis. The genetic distance matrices between these European cat populations also showed a significant correlation with the geographical distance between these populations using Mantel’s test. These analyses showed that in each of these countries, local cat populations have characteristic genetic profiles which were different to neighbouring populations in nearby countries. At least in this area of Western Europe, the geographical distances between cat populations (although the gene flow can be relatively high) is an important factor which can explain differences in allele frequencies between these populations.     

ARE PARTIAL MANTEL TESTS ADEQUATE?

Partial Mantel tests were designed to test for correlation among three matrices of pairwise distances. We show through an example that these tests may be inadequate, because the associated P-value is not indicative of the type I error.     

Hidden patterns of phylogenetic non‐stationarity overwhelm comparative analyses of niche conservatism and divergence

Aim Despite the importance of the niche concept in ecological and evolutionary theory, there are still many discussions about its definition and operational evaluation, especially when dealing with niche divergence and conservatism in an explicit phylogenetic context. Here we evaluate patterns of niche evolution in 67 New World Carnivora species, measured using Hellinger distances based on MAXENT models of species distribution. We show how inferences on niche conservatism or divergence depend on the way phylogenetic patterns are analysed using matrix comparison techniques. Innovation Initially we used the simplest approach of Mantel tests to compare Hellinger distances ( N ) derived from MAXENT and phylogenetic distances ( P ) among species. Then we extended the Mantel test to generate a multivariate correlogram, in which phylogenetic patterns are analysed at multiple levels in the phylogeny and can reveal nonlinearity in the relationship between divergence and time. Finally, we proposed a new approach to generate ‘local’ (or ‘specific’) leverages of components for Mantel correlation, evaluating the non‐stationarity in the relationship between N and P for each species. This new approach was used to show if some lineages are more prone to niche shift or conservatism than others. Main conclusions Standard Mantel tests indicated a poor correspondence between N and P matrices, discarding the idea of niche conservatism for Carnivora, but the correlogram supports that closely related species tend to be more similar than expected by chance. Moreover, the variance among Hellinger distances between pairs of closely phylogenetically related species is much larger than for the entire clade. Phylogenetic non‐stationarity analysis shows that in some Carnivora families the niche tends to divergence (Mustelidae and Canidae), whereas in others it tends to conservatism (Procyonidae and Mustelidae) at short phylogenetic distances. Our analyses clearly show that misleading results may appear if niche divergence is analysed only by simple matrix correlations not taking into account complex patterns of phylogenetic nonlinearity and non‐stationarity.     

Identifying the environmental factors that determine the genetic structure of populations

The study of population genetic structure is a fundamental problem in population biology because it helps us obtain a deeper understanding of the evolutionary process. One of the issues most assiduously studied in this context is the assessment of the relative importance of environmental factors (geographic distance, language, temperature, altitude, etc.) on the genetic structure of populations. The most widely used method to address this question is the multivariate Mantel test, a nonparametric method that calculates a correlation coefficient between a dependent matrix of pairwise population genetic distances and one or more independent matrices of environmental differences. Here we present a hierarchical Bayesian method that estimates F(ST) values for each local population and relates them to environmental factors using a generalized linear model. The method is demonstrated by applying it to two data sets, a data set for a population of the argan tree and a human data set comprising 51 populations distributed worldwide. We also carry out a simulation study to investigate the performance of the method and find that it can correctly identify the factors that play a role in the structuring of genetic diversity under a wide range of scenarios.     

Resistance is futile: effects of landscape features on gene flow of the northern bobwhite

Habitat for the northern bobwhite (Colinus virginianus) has declined and changed drastically in spatial structure throughout the last century. Undoubtedly such changes have impacted bobwhite and may have altered their ability to access available habitat. We investigated whether landscape resistance, geographic distance, or interstate highway barriers were related to dispersal and gene flow of bobwhite in central and southern Illinois. Landscape resistance was determined from two empirically informed models depicting habitat suitability for bobwhite. During 2007–2008, hunters submitted bobwhite tissue samples from which we amplified 11 microsatellites. The relationship between individual genetic distances and spatial variables was analyzed with Mantel tests and causal modeling was used to verify the spatial variables influencing gene flow. Genetic distance was correlated with geographic distance but showed no relationship with interstate highway barriers. Habitat suitability did not enhance gene flow, and was inversely related in some partial Mantel tests. We suggest that bobwhite dispersal from suitable habitat patches may be less frequent than from suboptimal habitats. Bobwhite may be able to access suitable habitat across gaps of unsuitable habitat but distance limits their dispersal. Because available habitat for bobwhites may have a greater likelihood of being colonized when closer to occupied habitat, we suggest that lands closer to occupied habitat should be targeted for conservation or habitat improvement efforts.     

Optimizing the trade‐off between spatial and genetic sampling efforts in patchy populations: towards a better assessment of functional connectivity using an individual‐based sampling scheme

Genetic data are increasingly used in landscape ecology for the indirect assessment of functional connectivity, that is, the permeability of landscape to movements of organisms. Among available tools, matrix correlation analyses (e.g. Mantel tests or mixed models) are commonly used to test for the relationship between pairwise genetic distances and movement costs incurred by dispersing individuals. When organisms are spatially clustered, a population‐based sampling scheme (PSS) is usually performed, so that a large number of genotypes can be used to compute pairwise genetic distances on the basis of allelic frequencies. Because of financial constraints, this kind of sampling scheme implies a drastic reduction in the number of sampled aggregates, thereby reducing sampling coverage at the landscape level. We used matrix correlation analyses on simulated and empirical genetic data sets to investigate the efficiency of an individual‐based sampling scheme (ISS) in detecting isolation‐by‐distance and isolation‐by‐barrier patterns. Provided that pseudo‐replication issues are taken into account (e.g. through restricted permutations in Mantel tests), we showed that the use of interindividual measures of genotypic dissimilarity may efficiently replace interpopulation measures of genetic differentiation: the sampling of only three or four individuals per aggregate may be sufficient to efficiently detect specific genetic patterns in most situations. The ISS proved to be a promising methodological alternative to the more conventional PSS, offering much flexibility in the spatial design of sampling schemes and ensuring an optimal representativeness of landscape heterogeneity in data, with few aggregates left unsampled. Each strategy offering specific advantages, a combined use of both sampling schemes is discussed.     

How well do multivariate data sets match? The advantages of a Procrustean superimposition approach over the Mantel test

The Mantel test provides a means to test the association between distance matrices and has been widely used in ecological and evolutionary studies. Recently, another permutation test based on a Procrustes statistic (PROTEST) was developed to compare multivariate data sets. Our study contrasts the effectiveness, in terms of power and type I error rates, of the Mantel test and PROTEST. We illustrate the application of Procrustes superimposition to visually examine the concordance of observations for each dimension separately and how to conduct hypothesis testing in which the association between two data sets is tested while controlling for the variation related to other sources of data. Our simulation results show that PROTEST is as powerful or more powerful than the Mantel test for detecting matrix association under a variety of possible scenarios. As a result of the increased power of PROTEST and the ability to assess the match for individual observations (not available with the Mantel test), biologists now have an additional and powerful analytical tool to study ecological and evolutionary relationships.     

Effects of landscape and history on diversification of a montane, stream-breeding amphibian

Aim The aim of this study was to understand the roles of landscape features in shaping patterns of contemporary and historical genetic diversification among populations of the Andean tree frog (Hypsiboas andinus) across spatial scales. Location Andes mountains, north‐western Argentina, South America. Methods Mitochondrial DNA control region sequences were utilized to assess genetic differentiation among populations and calculate population pair‐wise genetic distances. Three models of movement, namely traditional straight‐line distance and two effective distances based on habitat classification, were examined to determine which of these explained the most variation in pair‐wise population genetic differentiation. The two habitat classifications were based on digital vegetation and hydrology layers that were generated from a 90‐m resolution digital elevation model (DEM) and known relationships between elevation and habitat. Mantel tests were conducted to test for correlations between geographic and genetic distance matrices and to estimate the percentage variation explained by each type of geographic distance. To investigate the location of possible barriers to gene flow, we used Monmonier’s maximum difference algorithm as implemented in barrier 2.2. Results At both geographic scales, effective distances explained more variation in genetic differentiation than did straight‐line distance. The least‐cost distances based on the simple classification performed better than the more detailed habitat classification. We controlled for the effects of historical range fragmentation determined from previous nested clade analyses, and therefore evaluated the effect of different distances on the genetic variation attributable to more recent factors. Effective distances identified populations that were highly divergent as a result of isolation in unsuitable habitats. The proposed locations of barriers to gene flow identified using Monmonier’s maximum difference algorithm corresponded well with earlier analyses and supported findings from our partial Mantel tests. Main conclusions Our results indicate that landscape features have been important in both historical and contemporary genetic structuring of populations of H. andinus at both large and small spatial scales. A landscape genetic perspective offers novel insights not provided by traditional phylogeographic studies: (1) effective distances can better explain patterns of differentiation in populations, especially in heterogeneous landscapes where barriers to dispersal may be common; and (2) least‐cost path analysis can help to identify corridors of movement between populations that are biologically more realistic.     

Dissimilarity of stream insect assemblages: effects of multiple scales and spatial distances

In addition to the effects of environmental conditions, biotic assemblages may exhibit spatial structure depending on the scale of study. We tested whether the dissimilarity of stream insect assemblages is related to two types of spatial distances (stream corridor and overland distance), and evaluated the relative importance of diversity components at multiple spatial scales. Field data included assemblages of Ephemeroptera, Plecoptera, and Trichoptera found in 16 streams in four microbasins. We evaluated the relationship of the dissimilarity of assemblages with the distance types, using Mantel tests. In addition, we evaluated the relationships among the diversity components at multiple spatial scales, using additive partitioning analysis. The biological dissimilarities were correlated only with the geographical distances. Additive partitioning showed that the values of richness observed in the β1 (among Surber), β2 (among riffles), β3 (among streams), and β4 (among microbasins) were higher than those expected. The highest variation of the richness was found in β3 (30.6%). We conclude that stream faunas are distributed in patches all over the studied spatial extent, causing a weak relationship of biological dissimilarity with distance but important beta components when compared to a completely homogenous distribution of the fauna.