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 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.     

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.     

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.     

Complementary strengths of spatially-explicit and multi-species distribution models

Species distribution models (SDMs) project the outcome of community assembly processes – dispersal, the abiotic environment and biotic interactions – onto geographic space. Recent advances in SDMs account for these processes by simultaneously modeling the species that comprise a community in a multivariate statistical framework or by incorporating residual spatial autocorrelation in SDMs. However, the effects of combining both multivariate and spatially-explicit model structures on the ecological inferences and the predictive abilities of a model are largely unknown. We used data on eastern hemlock Tsuga canadensis and five additional co-occurring overstory tree species in 35 569 forest stands across Michigan, USA to evaluate how the choice of model structure, including spatial and non-spatial forms of univariate and multivariate models, affects ecological inference about the processes that shape community composition as well as model predictive ability. Incorporating residual spatial autocorrelation via spatial random effects did not improve out-of-sample prediction for the six tree species, although in-sample model fit was higher in the spatial models. Spatial models attributed less variation in occurrence probability to environmental covariates than the non-spatial models for all six tree species, and estimated higher (more positive) residual co-occurrence values for most species pairs. The non-spatial multivariate model was better suited for evaluating habitat suitability and hypotheses about the processes that shape community composition. Environmental correlations and residual correlations among species pairs were positively related, perhaps indicating that residual correlations were due to shared responses to unmeasured environmental covariates. This work highlights the importance of choosing a non-spatial model formulation to address research questions about the species–environment relationship or residual co-occurrence patterns, and a spatial model formulation when within-sample prediction accuracy is the main goal.     

Detecting spatial regimes in ecosystems

Research on early warning indicators has generally focused on assessing temporal transitions with limited application of these methods to detecting spatial regimes. Traditional spatial boundary detection procedures that result in ecoregion maps are typically based on ecological potential (i.e. potential vegetation), and often fail to account for ongoing changes due to stressors such as land use change and climate change and their effects on plant and animal communities. We use Fisher information, an information theory‐based method, on both terrestrial and aquatic animal data (U.S. Breeding Bird Survey and marine zooplankton) to identify ecological boundaries, and compare our results to traditional early warning indicators, conventional ecoregion maps and multivariate analyses such as nMDS and cluster analysis. We successfully detected spatial regimes and transitions in both terrestrial and aquatic systems using Fisher information. Furthermore, Fisher information provided explicit spatial information about community change that is absent from other multivariate approaches. Our results suggest that defining spatial regimes based on animal communities may better reflect ecological reality than do traditional ecoregion maps, especially in our current era of rapid and unpredictable ecological change.     

Detection of the horizontal spatial structure of soil fungal communities in a natural forest

Soil microbes are considered to be a key determinant of the aboveground plant community. They are not distributed uniformly in the environment, and their activity, abundance, and ecosystem functioning could vary across localities, characterized by high β-diversity. Investigating factors that contribute to high β-diversity can help infer the possible mechanisms of microbial community assembly, and predict the scale and extent of impacts that soil microbes have on the plant community. Because soil systems consist of multiple horizons (i.e., vertical stratification) associated with different soil properties, complete understanding of high β-diversity requires consideration of both horizontal and vertical spatial structures of soil microbial communities. We studied the community composition of soil fungi from the O- and A-horizons in a Castanopsis-dominated temperate forest, and compared horizontal spatial autocorrelation in species composition between the two soil horizons (O- versus A-horizons). Pyrosequencing analysis yielded 67,129 sequencing reads, summed across all the 48 forest soil samples. Clustering analysis resulted in 597 molecular operational taxonomic units (OTUs), 68 % of which were identified as fungi, represented by four phyla. The Mantel test revealed that the O-horizon communities are spatially clustered, and the observed high β-diversity was driven not only by changes in OTUs present, but also by high turnover in identities of OTUs in soil samples. Furthermore, Mantel correlogram analysis showed that the O-horizon communities resembled each other in composition within the range of 50 m, whereas the A-horizon communities lacked such horizontal autocorrelation. These differences in the scale patchiness could arise from two processes: (1) that environmental conditions could show higher heterogeneity in finer scale at the A-horizon than at the O-horizon; and/or (2) dispersal could be more frequent at the O-horizon than the A-horizon. The present study suggests that either environmental filtering (i.e., the niche-based process) or dispersal limitation (i.e., neutral process) could characterize the observed patterns of spatial clustering in the soil fungal community.     

Multi-scale variation in spatial heterogeneity for microbial community structure in an eastern Virginia agricultural field

To better understand the distribution of soil microbial communities at multiple spatial scales, a survey was conducted to examine the spatial organization of community structure in a wheat field in eastern Virginia (USA). Nearly 200 soil samples were collected at a variety of separation distances ranging from 2.5 cm to 11 m. Whole-community DNA was extracted from each sample, and community structure was compared using amplified fragment length polymorphism (AFLP) DNA fingerprinting. Relative similarity was calculated between each pair of samples and compared using geostatistical variogram analysis to study autocorrelation as a function of separation distance. Spatial autocorrelation was found at scales ranging from 30 cm to more than 6 m, depending on the sampling extent considered. In some locations, up to four different correlation length scales were detected. The presence of nested scales of variability suggests that the environmental factors regulating the development of the communities in this soil may operate at different scales. Kriging was used to generate maps of the spatial organization of communities across the plot, and the results demonstrated that bacterial distributions can be highly structured, even within a habitat that appears relatively homogeneous at the plot and field scale. Different subsets of the microbial community were distributed differently across the plot, and this is thought to be due to the variable response of individual populations to spatial heterogeneity associated with soil properties.     

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.     

Genetic structure of the indigenous populations of Siberia

This study explores the genetic structure of Siberian indigenous populations on the basis of standard blood group and protein markers and DNA variable number of tandem repeats (VNTR) variation. Four analytical methods were utilized in this study: Harpending and Jenkin's R-matrix; Harpending and Ward's method of correlating genetic heterozygosity (H) to the distance from the centroid of the gene frequency array (r_ii); spatial autocorrelation, and Mantel tests. Because of the underlying assumptions of the various methods, the numbers of populations used in the analyses varied from 15 to 62. Since spatial autocorrelation is based upon separate correlations between alleles, a larger number of standard blood markers and populations were used. Fewest Siberian populations have been sampled for VNTRs, thus, only a limited comparison was possible. The four analytical procedures employed in this study yielded complementary results suggestive of the effects of unique historical events, evolutionary forces, and geography on the distribution of alleles in Siberian indigenous populations. The principal components analysis of the R-matrix demonstrated the presence of populational clusters that reflect their phylogenetic relationship. Mantel comparisons of matrices indicate that an intimate relationship exists between geography, languages, and genetics of Siberian populations. Spatial autocorrelation patterns reflect the isolation-by-distance model of Malecot and the possible effects of long-distance migration. Am J Phys Anthropol 104:177–192, 1997. © 1997 Wiley-Liss, Inc.     

准噶尔盆地荒漠景观不同地貌类型鼠、蚤多样性及分布特征

为揭示鼠、蚤空间分布特征与变化规律,本研究以准噶尔盆地鼠疫自然疫源地为靶区,基于鼠类和蚤类的样点采集数据,计算不同地貌的鼠、蚤生态学指标并分析其相关性。基于不同行政区生态学指标计算结果,借助Moran′s I指数、重心模型、标准差椭圆等分析方法探究不同行政区鼠、蚤生态学指标的聚类特征,开展鼠、蚤的空间分布特征及变化规律的相关研究。结果表明：(1)通过对不同地貌鼠、蚤生态指标的研究,可得出鼠、蚤的物种多样性和生态优势度呈负相关,表明在物种多样性较高的群落中,鼠、蚤生态优势度表现不明显。鼠类物种多样性较高的地貌类型与蚤类物种多样性呈正相关,证实鼠类(宿主)物种数量增加,蚤类(寄生)物种的数量也在增加。低海拔地区鼠、蚤群落的相似性总体上大于中海拔地区群落相似性,且相似性系数q值与Cody指数呈相反变化趋势;(2)不同鼠、蚤指标单变量Moran′s I指数表明,鼠类数量、子午沙鼠数量、蚤类均匀度的全局Moran′s I指数大于0,且P值小于0.05,表现出空间集聚现象。单变量局部空间自相关分析结果表明,部分鼠、蚤指标存在多种聚类模式,其中最为典型的聚类模式是高—高聚类模式。不同鼠、蚤指标双变量...     

中国板栗居群间等位酶基因频率的空间分布

中国板栗21个自然群间等位酶遗传变异的空间自相关分析及F－统计分析结果表明：其多数等位基因频率在居群间呈随机分布模式,缺乏一定的空间结构;而部分等位基因表现为渐变或双向渐变的非随机分布模式,又具有特定空间结构。中国板栗遗传变异空间结构模式的形成可能是长距离基因流、自然气候、人类活动、地理距离隔离等诸因素综合作用的结果。文中还就居群等位基因分布格局的成因进行了讨论：在第四纪冰川后,中国板栗以长江流域中下游的孑遗中心为起点,等位基因分别沿着向北和向南的不同方向迁移形成现在的居群结构;季风气候和人类活动干扰是削弱居群分化的主要因素,而基于环境梯度的选择,是形成由北向南渐变分布的原因。     

Disentangling the Spatio-Environmental Drivers of Human Settlement: An Eigenvector Based Variation Decomposition

The relative importance of deterministic and stochastic processes driving patterns of human settlement remains controversial. A main reason for this is that disentangling the drivers of distributions and geographic clustering at different spatial scales is not straightforward and powerful analytical toolboxes able to deal with this type of data are largely deficient. Here we use a multivariate statistical framework originally developed in community ecology, to infer the relative importance of spatial and environmental drivers of human settlement. Using Moran’s eigenvector maps and a dataset of spatial variation in a set of relevant environmental variables we applied a variation partitioning procedure based on redundancy analysis models to assess the relative importance of spatial and environmental processes explaining settlement patterns. We applied this method on an archaeological dataset covering a 15 km² area in SW Turkey spanning a time period of 8000 years from the Late Neolithic/Early Chalcolithic up to the Byzantine period. Variation partitioning revealed both significant unique and commonly explained effects of environmental and spatial variables. Land cover and water availability were the dominant environmental determinants of human settlement throughout the study period, supporting the theory of the presence of farming communities. Spatial clustering was mainly restricted to small spatial scales. Significant spatial clustering independent of environmental gradients was also detected which can be indicative of expansion into unsuitable areas or an unexpected absence in suitable areas which could be caused by dispersal limitation. Integrating historic settlement patterns as additional predictor variables resulted in more explained variation reflecting temporal autocorrelation in settlement locations.     

微生物生态研究中BIOLOG方法数据分析及R语言实现

微生物生态研究中,对微生物群落结构、群落特征以及其与环境因素的关系的揭示,一直受到广泛关注;适当的数据分析方法有助于更清晰地刻画微生物群落结构特征,明确其与环境因素的关系。结合实例,对微生物生态研究中基于BIOLOG微平板技术的数据分析方法进行梳理,分别介绍数据读取整理、特征指数计算、非限制性排序、限制性排序、聚类分析、环境向量拟合、蒙特尔检验等常用数据操作及生态分析方法;针对不同方法结论,结合研究目标和生态理论给出具有统计学意义的解释,并评价不同方法特点及适用场景;分析过程以R语言实现,并提供全部代码。结果表明,BIOLOG方法产生数据能从多个角度表征微生物群落功能特征,并结合环境指标梯度进行分析;但BIOLOG数据可能不满足正态性分布,在基于正态分布的分析前应提前进行检验;排序分析时应慎用主成分分析,可优先采用其他基于距离矩阵的排序方法;R语言能够简化BIOLOG数据读取及操作,易于完成各类统计分析。本研究能够对微生物生态研究者科学选择应用统计分析方法、提高数据处理效率提供直接参考。