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.     

Comparison of the Mantel test and alternative approaches for detecting complex multivariate relationships in the spatial analysis of genetic data

The Mantel test is widely used to test the linear or monotonic independence of the elements in two distance matrices. It is one of the few appropriate tests when the hypothesis under study can only be formulated in terms of distances; this is often the case with genetic data. In particular, the Mantel test has been widely used to test for spatial relationship between genetic data and spatial layout of the sampling locations. We describe the domain of application of the Mantel test and derived forms. Formula development demonstrates that the sum-of-squares (SS) partitioned in Mantel tests and regression on distance matrices differs from the SS partitioned in linear correlation, regression and canonical analysis. Numerical simulations show that in tests of significance of the relationship between simple variables and multivariate data tables, the power of linear correlation, regression and canonical analysis is far greater than that of the Mantel test and derived forms, meaning that the former methods are much more likely than the latter to detect a relationship when one is present in the data. Examples of difference in power are given for the detection of spatial gradients. Furthermore, the Mantel test does not correctly estimate the proportion of the original data variation explained by spatial structures. The Mantel test should not be used as a general method for the investigation of linear relationships or spatial structures in univariate or multivariate data. Its use should be restricted to tests of hypotheses that can only be formulated in terms of distances.     

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.     

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.     

Euclidean nature of phylogenetic distance matrices

Phylogenies are fundamental to comparative biology as they help to identify independent events on which statistical tests rely. Two groups of phylogenetic comparative methods (PCMs) can be distinguished: those that take phylogenies into account by introducing explicit models of evolution and those that only consider phylogenies as a statistical constraint and aim at partitioning trait values into a phylogenetic component (phylogenetic inertia) and one or multiple specific components related to adaptive evolution. The way phylogenetic information is incorporated into the PCMs depends on the method used. For the first group of methods, phylogenies are converted into variance-covariance matrices of traits following a given model of evolution such as Brownian motion (BM). For the second group of methods, phylogenies are converted into distance matrices that are subsequently transformed into Euclidean distances to perform principal coordinate analyses. Here, we show that simply taking the elementwise square root of a distance matrix extracted from a phylogenetic tree ensures having a Euclidean distance matrix. This is true for any type of distances between species (patristic or nodal) and also for trees harboring multifurcating nodes. Moreover, we illustrate that this simple transformation using the square root imposes less geometric distortion than more complex transformations classically used in the literature such as the Cailliez method. Given the Euclidean nature of the elementwise square root of phylogenetic distance matrices, the positive semidefinitiveness of the phylogenetic variance-covariance matrix of a trait following a BM model, or related models of trait evolution, can be established. In that way, we build a bridge between the two groups of statistical methods widely used in comparative analysis. These results should be of great interest for ecologists and evolutionary biologists performing statistical analyses incorporating phylogenies.     

Linking variability in species composition and MODIS NDVI based on beta diversity measurements

Finding an effective method to quantify species compositional changes in time and space has been an important task for ecologists and biogeographers. Recently, exploring regional floristic patterns using data derived from satellite imagery, such as the normalized difference vegetation index (NDVI) has drawn considerable research interests among ecologists. Studies have shown that NDVI could be a fairly good surrogate for primary productivities. In this study, we used plant distribution data in the North and the South Carolina states to investigate the correlations between species composition and NDVI within defined ecoregions using Mantel test and multi-response permutation procedure (MRPP). Our analytical approach involved generating compositional dissimilarity matrices by computing pairwise beta diversities of the 145 counties in the two states for species distribution data and by computing Euclidian distances for NDVI time series data. We argue that beta diversity measurements take the pairwise dissimilarities into consideration explicitly and could provide more spatial correlation information compared with uni- or multi-dimensional regressions. Our results showed a significant positive correlation between species compositional dissimilarity matrices and NDVI distance matrices. We also found for the first time that the strength of correlation increased at a lower taxonomic rank. Same trends were discovered when incorporating variability in phenological patterns in NDVI. Our findings suggest that remotely sensed NDVI can be viable for monitoring species compositional changes at regional scales.     

Assessing phylogenetic signal with measurement error: a comparison of mantel tests, blomberg et Al.'s k, and phylogenetic distograms

In macroevolutionary studies, different approaches are commonly used to measure phylogenetic signal-the tendency of related taxa to resemble one another-including the K statistic and the Mantel test. The latter was recently criticized for lacking statistical power. Using new simulations, we show that the power of the Mantel test depends on the metrics used to define trait distances and phylogenetic distances between species. Increasing power is obtained by lowering variance and increasing negative skewness in interspecific distances, as obtained using Euclidean trait distances and the complement of Abouheif proximity as a phylogenetic distance. We show realistic situations involving "measurement error" due to intraspecific variability where the Mantel test is more powerful to detect a phylogenetic signal than a permutation test based on the K statistic. We highlight limitations of the K-statistic (univariate measure) and show that its application should take into account measurement errors using repeated measures per species to avoid estimation bias. Finally, we argue that phylogenetic distograms representing Euclidean trait distance as a function of the square root of patristic distance provide an insightful representation of the phylogenetic signal that can be used to assess both the impact of measurement error and the departure from a Brownian evolution model.     

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.     

Sequence ordinations: a multivariate analysis approach to analysing large sequence data sets

Ordination is a powerful method for analysing complex data setsbut has been largely ignored in sequence analysis. This papershows how to use principal coordinates analysis to find low–dimensionalrepresentations of distance matrices derived from aligned setsof sequences. The method takes a matrix of Euclidean distancesbetween all pairs of sequence and finds a coordinate space wherethe distances are exactly preserved The main problem is to finda measure of distance between aligned sequences that is Euclidean.The simplest distance function is the square root of the percentagedifference (as measured by identities) between two sequences,where one ignores any positions in the alignment where thereis a gap in any sequence. If one does not ignore positions witha gap, the distances cannot be guaranteed to be Euclidean butthe deleterious effects are trivial. Two examples of using themethod are shown. A set of 226 aligned globins were analysedand the resulting ordination very successfully represents theknown patterns of relationship between the sequences. In theother example, a set of 610 aligned 5S rRNA sequences were analysed.Sequence ordinations complement phylogenetic analyses. Theyshould not be viewed as a complete alternative.     

Streams over mountains: influence of riparian connectivity on gene flow in the Pacific jumping mouse (Zapus trinotatus)

In species affiliated with heterogeneous habitat, we expect gene flow to be restricted due to constraints placed on individual movement by habitat boundaries. This is likely to impact both individual dispersal and connectivity between populations. In this study, a GIS-based landscape genetics approach was used, in combination with fine-scale spatial autocorrelation analysis and the estimation of recent intersubpopulation migration rates, to infer patterns of dispersal and migration in the riparian-affiliated Pacific jumping mouse (Zapus trinotatus). A total of 228 individuals were sampled from nine subpopulations across a system of three rivers and genotyped at eight microsatellite loci. Significant spatial autocorrelation among individuals revealed a pattern of fine-scale spatial genetic structure indicative of limited dispersal. Geographical distances between pairwise subpopulations were defined following four criteria: (i) Euclidean distance, and three landscape-specific distances, (ii) river distance (distance travelled along the river only), (iii) overland distance (similar to Euclidean, but includes elevation), and (iv) habitat-path distance (a least-cost path distance that models movement along habitat pathways). Pairwise Mantel tests were used to test for a correlation between genetic distance and each of the geographical distances. Significant correlations were found between genetic distance and both the overland and habitat-path distances; however, the correlation with habitat-path distance was stronger. Lastly, estimates of recent migration rates revealed that migration occurs not only within drainages but also across large topographic barriers. These results suggest that patterns of dispersal and migration in Pacific jumping mice are largely determined by habitat connectivity.     

Palmar pattern frequencies as indicators of relationships among Sardinian linguistic groups of males

Palmar pattern frequencies were used to calculate distance coefficients between Sardinian linguistic groups of males for the purpose of verifying, by means of correlation matrix analyses, whether or not the dermatoglyphic traits considered lead to a reliable identification of the biological relationships on the basis of the linguistic backgrounds of these groups. With Sanghvi's as the distance measure and by using palmar pattern frequencies in the Hy area, Th/I, II, III, and IV interdigital areas and all traits together for palms combined or separated were calculated dermatoglyphic distance measures. Mantel tests of matrix correspondence showed that, by using palmar pattern frequencies in the Th/I interdigital area (palms combined), in the II, III, and IV interdigital areas, or all traits together for palms combined and separated, statistical significance between dermatoglyphic and linguistic distances can be obtained, even when the effect of geography is removed; there is no statistically significant correspondence between geographic and dermatoglyphic distance matrices, even when the effect of language is removed. The results obtained in this study by means of the Mantel test procedure demonstrate that the dermatoglyphic traits analyzed, with the exception of palmar pattern frequencies in the Hy area and in the Th/I interdigital area for plams separated when these areas are used singly, can be considered as a good set of variables to use in finding biological relationships between Sardinian linguistic groups of males examined on the basis of their linguistic backgrounds.     

Genetic affinities of Jewish populations.

Genetic relations between various Jewish (J) and non-Jewish (NJ) populations were assessed using two sets of data. The first set contained 12 pairs of matched J and NJ populations from Europe, the Middle East, and North Africa, for which 10 common polymorphic genetic systems (13 loci) were available. The second set included 22 polymorphic genetic systems (26 loci) with various numbers of populations (ranging from 21 to 51) for each system. Therefore, each system was studied separately. Nei's standard genetic distance (D) matrices obtained for these two sets of data were tested against design matrices specifying hypotheses concerning the affiliations of the tested populations. The tests against single designs were carried out by means of Mantel tests. Our results consistently show lower distances among J populations than with their NJ neighbors, most simply explained by the common origin of the former. Yet, there is evidence also of genetic similarity between J and corresponding NJ populations, suggesting reciprocal gene flow between these populations or convergent selection in a common environment. The results of our study also indicate that stochastic factors are likely to have played a role in masking the descent relationships of the J populations.     

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.     

Cultural interaction and biological distance in postclassic period Mexico

Economic, political, and cultural relationships connected virtually every population throughout Mexico during Postclassic period (AD 900–1520). Much of what is known about population interaction in prehistoric Mexico is based on archaeological or ethnohistoric data. What is unclear, especially for the Postclassic period, is how these data correlate with biological population structure. We address this by assessing biological (phenotypic) distances among 28 samples based upon a comparison of dental morphology trait frequencies, which serve as a proxy for genetic variation, from 810 individuals. These distances were compared with models representing geographic and cultural relationships among the same groups. Results of Mantel and partial Mantel matrix correlation tests show that shared migration and trade are correlated with biological distances, but geographic distance is not. Trade and political interaction are also correlated with biological distance when combined in a single matrix. These results indicate that trade and political relationships affected population structure among Postclassic Mexican populations. We suggest that trade likely played a major role in shaping patterns of interaction between populations. This study also shows that the biological distance data support the migration histories described in ethnohistoric sources. Am J Phys Anthropol 157:121–133, 2015. © 2015 Wiley Periodicals, Inc.     

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.     

Topometric analysis of diffuse astrocytomas

OBJECTIVE: To investigate differences between astrocytomas of WHO grade 2 and anaplastic astrocytomas of WHO grade 3 in terms of topometric variables characterizing individual tumor cell nuclei. STUDY DESIGN: Paraffin sections from surgical specimens from 25 astrocytomas (grade 2, n = 11; grade 3, n = 14) were analyzed by means of an image analysis system. At least 300 tumor cell nuclei were measured in the region with the highest Ki-67 proliferation index. Three different kinds of topometric variables were determined for each tumor cell nucleus: (1) several distances; (2) the variable Angle 2/1, the angle between the straight lines representing the distance to the nearest and second-nearest nucleus; and (3) the number of neighbors according to our mathematical definition. RESULTS: Most topometric variables showed distinct differences between the 2 tumor grades (multivariate analysis of variance), with 88% cases correctly reclassified by means of cross-validated discriminant analysis. The variables with the highest discriminatory power were the SD of Angle 2/1 and the ratio between the distance to the second-nearest and nearest tumor cell nucleus, with lower values for these variables in anaplastic astrocytomas. Even variables concerning neighborhood relationships showed significant differences. CONCLUSION: The results of this pilot study show that this first set of topometric variables is sufficient to detect differences between topologic characteristics of tumor cell nuclei in astrocytomas grade 2 and grade 3. Topometry seems to be an important tool for grading astrocytomas.     

Inferring landscape resistance to gene flow when genetic drift is spatially heterogeneous

In connectivity models, land cover types are assigned cost values characterizing their resistance to species movements. Landscape genetic methods infer these values from the relationship between genetic differentiation and cost distances. The spatial heterogeneity of population sizes, and consequently genetic drift, is rarely included in this inference although it influences genetic differentiation. Similarly, migration rates and population spatial distributions potentially influence this inference. Here, we assessed the reliability of cost value inference under several migration rates, population spatial patterns and degrees of population size heterogeneity. Additionally, we assessed whether considering intra-population variables, here using gravity models, improved the inference when drift is spatially heterogeneous. We simulated several gene flow intensities between populations with varying local sizes and spatial distributions. We then fit gravity models of genetic distances as a function of (i) the ‘true’ cost distances driving simulations or alternative cost distances, and (ii) intra-population variables (population sizes, patch areas). We determined the conditions making the identification of the ‘true’ costs possible and assessed the contribution of intra-population variables to this objective. Overall, the inference ranked cost scenarios reliably in terms of similarity with the ‘true’ scenario (cost distance Mantel correlations), but this ‘true’ scenario rarely provided the best model goodness of fit. Ranking inaccuracies and failures to identify the ‘true’ scenario were more pronounced when migration was very restricted (<4 dispersal events/generation), population sizes were most heterogeneous and some populations were spatially aggregated. In these situations, considering intra-population variables helps identify cost scenarios reliably, thereby improving cost value inference from genetic data.     

Patterns of phenotypic covariation and correlation in modern humans as viewed from morphological integration

Proportionality of phenotypic and genetic distance is of crucial importance to adequately focus on population history and structure, and it depends on the proportionality of genetic and phenotypic covariance. Constancy of phenotypic covariances is unlikely without constancy of genetic covariation if the latter is a substantial component of the former. If phenotypic patterns are found to be relatively stable, the most probable explanation is that genetic covariance matrices are also stable. Factors like morphological integration account for such stability. Morphological integration can be studied by analyzing the relationships among morphological traits. We present here a comparison of phenotypic correlation and covariance structure among worldwide human populations. Correlation and covariance matrices between 47 cranial traits were obtained for 28 populations, and compared with design matrices representing functional and developmental constraints. Among-population differences in patterns of correlation and covariation were tested for association with matrices of genetic distances (obtained after an examination of 10 Alu-insertions) and with Mahalanobis distances (computed after craniometrical traits). All matrix correlations were estimated by means of Mantel tests. Results indicate that correlation and covariance structure in our species is stable, and that among-group correlation/covariance similarity is not related to genetic or phenotypic distance. Conversely, genetic and morphological distance matrices were highly correlated. Correlation and covariation patterns were largely associated with functional and developmental factors, which probably account for the stability of covariance patterns.     

Phylogenetic Inference with Weighted Codon Evolutionary Distances

We develop a new approach to estimate a matrix of pairwise evolutionary distances from a codon-based alignment based on a codon evolutionary model. The method first computes a standard distance matrix for each of the three codon positions. Then these three distance matrices are weighted according to an estimate of the global evolutionary rate of each codon position and averaged into a unique distance matrix. Using a large set of both real and simulated codon-based alignments of nucleotide sequences, we show that this approach leads to distance matrices that have a significantly better treelikeness compared to those obtained by standard nucleotide evolutionary distances. We also propose an alternative weighting to eliminate the part of the noise often associated with some codon positions, particularly the third position, which is known to induce a fast evolutionary rate. Simulation results show that fast distance-based tree reconstruction algorithms on distance matrices based on this codon position weighting can lead to phylogenetic trees that are at least as accurate as, if not better, than those inferred by maximum likelihood. Finally, a well-known multigene dataset composed of eight yeast species and 106 codon-based alignments is reanalyzed and shows that our codon evolutionary distances allow building a phylogenetic tree which is similar to those obtained by non-distance-based methods (e.g., maximum parsimony and maximum likelihood) and also significantly improved compared to standard nucleotide evolutionary distance estimates.     

Genetic structure of La Cabrera, Spain, from surnames and migration matrices

The genetic structure of La Cabrera (province of Léon, Spain), a highly isolated and inbred population (alpha3 = 0.00482), is analyzed by applying multivariate methods (nonmetric multidimensional scaling, Mantel test, Monmonier's algorithm) to different biodemographic data sets. Isonymy, parent-offspring migration (total, males, females), and marital migration matrices were obtained from 5,714 marriages recorded in 37 parishes (clustered in 4 municipalities) between 1880 and 1989. The aim of the study is to investigate the relationships between the genetic and geographic structures of the area. Endemicity values (diagonal of parent-offspring migration matrices), calculated for both sexes at two hierarchical levels (parishes and municipalities), show that female mobility follows the virilocal migration model at the higher (municipalities) level and the uxorilocal model at the parish level. Analysis of isonymy and parent-offspring migration matrices shows high correspondence between the genetic structure and geographic location of the parishes. In fact, the main reproductive barriers, constructed using Monmonier's algorithm, generally coincide with geographic barriers, highlighting increasing isolation patterns from northwest to southeast. Moreover, the analysis of isonymous relationships, which are influenced by earlier population movements, identifies three parishes whose outlier positions are explained by historical-cultural or geographic reasons. The positive and highly significant values (0.32 < or = r < or = 0.51;p < or = 0.001) given by the Mantel tests underline the dependence of the genetic structure on geographic distance. In confirmation of the endemicity results, the lowest correlation value (r = 0.32) is given by the female migration matrix. When the outlier parishes are omitted from the analysis, the correlation between isonymy and geographic distance increases from 0.35 to 0.46, and the values from the other migration matrices remain unchanged. In conclusion, the combination of different data sets and methods allows a wider and more coherent reconstruction of the historical changes in the genetic structure of the La Cabrera population.