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.     

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.     

Influence of edaphic factors on the spatial structure of inland halophytic communities: a case study in China

Abstract. In order to understand the influence of edaphic factors on the spatial structure of inland halophytic plant communities, a 2.6 km² study site, located on the lower fringe of the alluvial fan of the Hutubi River, in an arid region of China, was sampled and mapped. 105 patches were found to be homogeneous in species composition. Plant species and their coverage were recorded in each patch. 45 patches were randomly selected for the measurement of edaphic variables. A map with quadrat locations and boundaries of patches was digitized into a GIS and related to the vegetation and edaphic data matrices. CCA was used to evaluate the relative importance of edaphic factors in explaining the variation of the species assemblages and to identify the ecological preferences of species. The spatial structure of the communities and the main edaphic factors were analyzed using correlograms, Mantel correlograms and clustering under constraint of spatial contiguity. Gradient analysis showed that there are two distinct vegetation gradients in the study area, one of which is determined mainly by soil moisture (determined by depth to the water table), and the other by soil salinity (determined by electrical conductivity and hydrolytic alkalinity of the first soil layer). However, spatial analyses showed that at the sampling scale the halophytic communities in the study area are structured along one main spatial gradient determined by the water table level. Similar spatial autocorrelation structures between the factors related to the first soil layer and the communities, given our sampling scale, could not be detected. Our results suggest that the relative importance of the effects of different edaphic factors on the spatial structure of halophytic communities is scale-dependent. The partitioning of species variation indicates that in addition to edaphic factors, other factors, such as biotic interactions, may play an important role in structuring these communities.     

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.     

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.     

Spatial analysis of two-species interactions

Summary In this paper, I present and discuss some methods for the analysis of univariate and bivariate spatial point pattern data. Examples of such data in ecology include x-y coordinates of organisms in mapped field plots. I illustrate the methods with analyses of data from mapped field plots on Mount St. Helens, Washington state, USA. The statistical methods I emphasize are graphical methods that rely on analysis of distances between organisms. Hypothesis testing for methods like these is easily done using Monte Carlo methods, which I also discuss. For both univariate and bivariate analyses, I find that second-order methods such as K-function plots are often preferable to first-order methods (i.e., QQ-plots). However, for multivariate analyses, these second-order methods are more sensitive to small sample sizes than first-order analyses.     

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.     

Spatial ordination of vegetation data using a generalization of Wartenberg's multivariate spatial correlation

Question: Are there spatial structures in the composition of plant communities? Methods: Identification and measurement of spatial structures is a topic of great interest in plant ecology. Univariate measurements of spatial autocorrelation such as Moran's I and Geary's c are widely used, but extensions to the multivariate case (i.e. multi‐species) are rare. Here, we propose a multivariate spatial analysis based on Moran's I (MULTISPATI) by introducing a row‐sum standardized spatial weight matrix in the statistical triplet notation. This analysis, which is a generalization of Wartenberg's approach to multivariate spatial correlation, would imply a compromise between the relations among many variables (multivariate analysis) and their spatial structure (autocorrelation). MULTISPATI approach is very flexible and can handle various kinds of data (quantitative and/or qualitative data, contingency tables). A study is presented to illustrate the method using a spatial version of Correspondence Analysis. Location: Territoire d'Etude et d'Expérimentation de Trois‐Fontaines (eastern France). Results: Ordination of vegetation plots by this spatial analysis is quite robust with reference to rare species and highlights spatial patterns related to soil properties.     

Comparing patterns of geographic variation in cave crickets by combining geostatistic methods and Mantel tests

Geographic variation patterns of biological characters and environmental variables are compared by using a procedure employing multivariate analyses, production of contour maps by the kriging method with enclosed validation of estimates, and Mantel tests to assess the significance of comparisons. As biological material we chose a sample of Dolichopoda cave crickets populations from Central-Southern Italy. The kriging technique provides estimates of the interpolation error for each true and estimated point. This profitable feature offers the opportunity to use, with ascertained levels of confidence, the estimated z -scores for further analysis and to compare data collected within the same area, but not exactly coincident in location or number. In such a way, we were able to use for subsequent comparisons by means of Mantel tests the maximum number of data points for all data sets, which originally differed in sampling sites. The interpretation of the contour maps and their statistical comparison suggested that allozymes and epiphallus shape data sets follow the phylogenetic pathways within the Dolichopoda populations, whereas variation in leg elongation is almost entirely under the control of an environmental gradient, synthetically described by the cave temperature.