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.     

Meta‐analysis of rare events under the assumption of a homogeneous treatment effect

We studied the performance of several meta‐analysis methods in rare event settings, when the treatment effect is assumed to be homogeneous and baseline prevalences are either homogeneous or heterogeneous. We conducted extensive simulations that included the three most common effect sizes with count data: the odds ratio, the relative risk, and the risk difference. We investigated several important scenarios by varying the level of rareness, the value of the trials’ arms unbalance, and the size of the treatment effect. We found that the Mantel–Haenszel method and the Binomial regression model provided the best results across all the scenarios investigated. The Peto method performed satisfactorily only when the true effect size was not too large and the degree of unbalance moderate. Inverse variance was the least reliable method. The use of a continuity correction factor slightly improved the performance of the inverse variance method but deteriorated that of the Peto and Mantel–Haenszel methods. A method based on median unbiased estimators of the probabilities provided similar results to those obtained when using the inverse variance method with a continuity correction. Therefore, when the treatment effect can be assumed to be homogeneous and for either homogeneous or heterogeneous baseline prevalences, we highly recommend using the Mantel‐Haenszel method without continuity correction (for all the effect sizes) or the Binomial regression model (for the odds ratio only) to meta‐analyze the data.     

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.     

A method to incorporate the effect of taxonomic uncertainty on multivariate analyses of ecological data

Researchers in ecology commonly use multivariate analyses (e.g. redundancy analysis, canonical correspondence analysis, Mantel correlation, multivariate analysis of variance) to interpret patterns in biological data and relate these patterns to environmental predictors. There has been, however, little recognition of the errors associated with biological data and the influence that these may have on predictions derived from ecological hypotheses. We present a permutational method that assesses the effects of taxonomic uncertainty on the multivariate analyses typically used in the analysis of ecological data. The procedure is based on iterative randomizations that randomly re‐assign non identified species in each site to any of the other species found in the remaining sites. After each re‐assignment of species identities, the multivariate method at stake is run and a parameter of interest is calculated. Consequently, one can estimate a range of plausible values for the parameter of interest under different scenarios of re‐assigned species identities. We demonstrate the use of our approach in the calculation of two parameters with an example involving tropical tree species from western Amazonia: 1) the Mantel correlation between compositional similarity and environmental distances between pairs of sites, and; 2) the variance explained by environmental predictors in redundancy analysis (RDA). We also investigated the effects of increasing taxonomic uncertainty (i.e. number of unidentified species), and the taxonomic resolution at which morphospecies are determined (genus‐resolution, family‐resolution, or fully undetermined species) on the uncertainty range of these parameters. To achieve this, we performed simulations on a tree dataset from southern Mexico by randomly selecting a portion of the species contained in the dataset and classifying them as unidentified at each level of decreasing taxonomic resolution. An analysis of covariance showed that both taxonomic uncertainty and resolution significantly influence the uncertainty range of the resulting parameters. Increasing taxonomic uncertainty expands our uncertainty of the parameters estimated both in the Mantel test and RDA. The effects of increasing taxonomic resolution, however, are not as evident. The method presented in this study improves the traditional approaches to study compositional change in ecological communities by accounting for some of the uncertainty inherent to biological data. We hope that this approach can be routinely used to estimate any parameter of interest obtained from compositional data tables when faced with taxonomic uncertainty.     

There is more than one way to skin a G matrix

Because of its importance in directing evolutionary trajectories, there has been considerable interest in comparing variation among genetic variance-covariance (G) matrices. Numerous statistical approaches have been suggested but no general analysis of the relationship among these methods has previously been published. In this study, we used data from a half-sib experiment and simulations to explore the results of applying eight tests (T method, modified Mantel test, Bartlett's test, Flury hierarchy, jackknife-manova, jackknife-eigenvalue test, random skewers, selection skewers). Whereas a randomization approach produced acceptable estimates, those from a bootstrap were typically unacceptable and we recommend randomization as the preferred method. All methods except the jackknife-eigenvalue test gave similar results although a fine-scale analysis suggested that the former group can be subdivided into two or possibly three groups, hierarchical tests, skewers and the rest (jackknife-manova, modified Mantel, T method, probably Bartlett's). An advantage of the jackknife methods is that they permit tests of association with other factors, such as in this case, temperature and sex. We recommend applying all the tests described in this article, with the exception of the T method, and provide R functions for this purpose.     

Haplotype-sharing analysis using Mantel statistics for combined genetic effects

We applied a new approach based on Mantel statistics to analyze the Genetic Analysis Workshop 14 simulated data with prior knowledge of the answers. The method was developed in order to improve the power of a haplotype sharing analysis for gene mapping in complex disease. The new statistic correlates genetic similarity and phenotypic similarity across pairs of haplotypes from case-control studies. The genetic similarity is measured as the shared length between haplotype pairs around a genetic marker. The phenotypic similarity is measured as the mean corrected cross-product based on the respective phenotypes. Cases with phenotype P1 and unrelated controls were drawn from the population of Danacaa. Power to detect main effects was compared to the X2-test for association based on 3-marker haplotypes and a global permutation test for haplotype association to test for main effects. Power to detect gene x gene interaction was compared to unconditional logistic regression. The results suggest that the Mantel statistics might be more powerful than alternative tests.     

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.     

Spectral Analysis of Binary Time Series: Square Waves vs. Sinusoidal Functions

The analysis of signals consisting of discrete and irregular data causes methodological problems for the Fourier spectral Analysis: Since it is based on sinusoidal functions, rectangular signals with unequal periodicities cannot easily be replicated. The Walsh spectral Analysis is based on the so called "Walsh functions", a complete set of orthonormal, rectangular waves and thus seems to be the method of choice for analysing signals consisting of binary or ordinal data. The paper compares the Walsh spectral analysis and the Fourier spectral analysis on the basis of simulated and real binary data sets of various length. Simulated data were derived from signals with defined cyclic patterns that were noised by randomly generated signals of the same length. The Walsh and Fourier spectra of each set were determined and up to 25% of the periodogram coefficients were utilized as input for an inverse transform. Mean square approximation error (MSE) was calculated for each of the series in order to compare the goodness of fit between the original and the reconstructed signal. The same procedure was performed with real data derived from a behavioral observation in pigs. The comparison of the two methods revealed that, in the analysis of discrete and binary time series, Walsh spectral analysis is the more appropriate method, if the time series is rather short. If the length of the signal increases, the difference between the two methods is less substantial.     

Spectral Analysis of Binary Time Series: Square Waves vs. Sinusoidal Functions

The analysis of signals consisting of discrete and irregular data causes methodological problems for the Fourier spectral Analysis: Since it is based on sinusoidal functions, rectangular signals with unequal periodicities cannot easily be replicated. The Walsh spectral Analysis is based on the so called "Walsh functions", a complete set of orthonormal, rectangular waves and thus seems to be the method of choice for analysing signals consisting of binary or ordinal data. The paper compares the Walsh spectral analysis and the Fourier spectral analysis on the basis of simulated and real binary data sets of various length. Simulated data were derived from signals with defined cyclic patterns that were noised by randomly generated signals of the same length. The Walsh and Fourier spectra of each set were determined and up to 25% of the periodogram coefficients were utilized as input for an inverse transform. Mean square approximation error (MSE) was calculated for each of the series in order to compare the goodness of fit between the original and the reconstructed signal. The same procedure was performed with real data derived from a behavioral observation in pigs. The comparison of the two methods revealed that, in the analysis of discrete and binary time series, Walsh spectral analysis is the more appropriate method, if the time series is rather short. If the length of the signal increases, the difference between the two methods is less substantial.     

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.     

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.     

Validation of criteria for the selection of AFLP markers to assess the genetic variation of a breeders’ collection of evergreen azaleas

Fluorescent AFLP and automated data analysis were employed to assess the genetic conformity within a breeders’ collection of evergreen azaleas. The study included 75 genotypes of Belgian pot azaleas (Rhododendron simsii Planch. hybrids), Kurume and Hirado azaleas and wild ancestor species from the Tsutsusi subgenus. Fluorescent detection and addition of an internal size standard to each lane enabled the automated scoring of each fragment arising from a single AFLP primer combination (PC). The use of three PCs generated an initial data set with a total of 648 fragments ranging from 70 bp to 450 bp. Different marker selection thresholds for average fluorescent signal intensity and marker frequency were used to create eight extra restricted data subsets. Pairwise plant genetic similarity was calculated for the nine data sets using the Simple Matching coefficient (symmetrical, including double-zeros) and Jaccard coefficient (asymmetrical, excluding double zeros). The averages, the ranges and the correlation to one other (Mantel analysis) were compared for the obtained similarity matrices. This revealed the sensitivity of ordinations obtained by both similarity coefficients for the presence of weak or intensive markers or for the degree of polymorphism of the markers. For 34 cultivars, pedigree information (at maximum to the fifth ancestor generation) was available. Genetic similarity by descent (kinship coefficient) was turned into a genetic distance and correlated to the genetic conformity, as revealed by the different selections of AFLP markers (Mantel analysis). Use of a Simple Matching coefficient with no or moderate selection to signal intensity and excluding rare and abundant markers gave the best correlation with pedigree. Finally, the ordination of the studied genotypes by means of dendrograms and principal co-ordinate analysis was confronted with known or accepted relationships based on geographical origin, parentage and morphological characters. Genotypes could be assigned to three distinct groups: pot azaleas, Kurume azaleas and Hirado azaleas. Wild ancestor species appeared to be more related to the Japanese azaleas. Intermediate cultivars could be typified as crossings with Kurume or Hirado azaleas or with wild species. Received: 3 September 1998 / Accepted: 25 March 1999     

Is a multivariate consensus representation of genetic relationships among populations always meaningful?

To determine the relationships among closely related populations or species, two methods are commonly used in the literature: phylogenetic reconstruction or multivariate analysis. The aim of this article is to assess the reliability of multivariate analysis. We describe a method that is based on principal component analysis and Mantel correlations, using a two-step process: The first step consists of a single-marker analysis and the second step tests if each marker reveals the same typology concerning population differentiation. We conclude that if single markers are not congruent, the compromise structure is not meaningful. Our model is not based on any particular mutation process and it can be applied to most of the commonly used genetic markers. This method is also useful to determine the contribution of each marker to the typology of populations. We test whether our method is efficient with two real data sets based on microsatellite markers. Our analysis suggests that for closely related populations, it is not always possible to accept the hypothesis that an increase in the number of markers will increase the reliability of the typology analysis.     

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.     

Binomial transformation applied to presence-absence community data

Community data is often transformed or standardized to meet the requirements and assumptions of multivariate analysis. While these methods are usually appropriate for abundance data, they are seldom applied to presence-absence data. Here, a method of transforming a binary matrix using the binomial probability is described. Number of trials (n), number of successes (x) and probability of success (p) are necessary to compute the binomial probability. Successes were defined as the number of sites where the species occurrence can be considered; trials were equal and greater than the number of successes. The actual occurrence of each species along the gradient was considered the probability of success. The Mantel statistic associated with the binomially transformed distance matrix and the distance matrix based on binary data were used to choose an appropriate binomial transformation. The chosen binomial transformation gave greater value to species indicating habitat typologies. Binomially transformed data rendered results closer to expectations.     

Stability across environments of the coffee variety near infrared spectral signature

Previous study on food plants has shown that near infrared (NIR) spectral methods seem effective for authenticating coffee varieties. We confirm that result, but also show that inter-variety differences are not stable from one harvest to the next. We put forward the hypothesis that the spectral signature is affected by environmental factors. The purpose of this study was to find a way of reducing this environmental variance to increase the method's reliability and to enable practical application in breeding. Spectral collections were obtained from ground green coffee samples from multilocation trials. Two harvests of bean samples from 11 homozygous introgressed lines, and the cv 'Caturra' as the control, supplied from three different sites, were compared. For each site, squared Euclidean distances among the 12 varieties were estimated from the NIR spectra. Matrix correlation coefficients were assessed by the Mantel test. We obtained very good stability (high correlations) for inter-variety differences across the sites when using the two harvests data. If only the most heritable zones of the spectrum were used, there was a marked improvement in the efficiency of the method. This improvement was achieved by treating the spectrum as succession of phenotypic variables, each resulting from an environmental and genetic effect. Heritabilities were calculated with confidence intervals. A near infrared spectroscopy signature, acquired over a set of harvests, can therefore effectively characterize a coffee variety. We indicated how this typical signature can be used in breeding to assist in selection.     

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.     

Does Ordinal Cover Estimation Offer Reliable Quality Data Structures in Vegetation Ecological Studies?

Using visual estimation of species cover in ordinal interval classes may reduce costs in vegetation studies. In phytosociology, species cover within plots is usually estimated according to the well-known Braun-Blanquet scale and ordinal data from this scale are usually treated using common exploratory analysis tools that are adequate for ratio-scale variables only. This paper addresses whether the visual estimation of ordinal cover data and the treatment of these data with multivariate procedures tailored for ratio-scale data would lead to a significant loss of information with respect to the use of more accurate methods of data collection and analysis. To answer these questions we used three data sets sampled by different authors in different sites of Tuscany (central Italy) in which the species cover is measured with the point quadrat method. For each data set we used a Mantel test to compare the dissimilarity matrices obtained from the original point-quadrat cover data with those obtained from the corresponding ordinal interval classes. The results suggest that the ordinal data are suitable to represent the plot-to-plot dissimilarity structure of all data sets in a reasonable way and that in using such data there is no need to apply dissimilarity coefficients specifically tailored for ordinal scales.     

New targeted approaches for the quantification of data‐independent acquisition mass spectrometry

The use of data‐independent acquisition (DIA) approaches for the reproducible and precise quantification of complex protein samples has increased in the last years. The protein information arising from DIA analysis is stored in digital protein maps (DIA maps) that can be interrogated in a targeted way by using ad hoc or publically available peptide spectral libraries generated on the same sample species as for the generation of the DIA maps. The restricted availability of certain difficult‐to‐obtain human tissues (i.e., brain) together with the caveats of using spectral libraries generated under variable experimental conditions limits the potential of DIA. Therefore, DIA workflows would benefit from high‐quality and extended spectral libraries that could be generated without the need of using valuable samples for library production. We describe here two new targeted approaches, using either classical data‐dependent acquisition repositories (not specifically built for DIA) or ad hoc mouse spectral libraries, which enable the profiling of human brain DIA data set. The comparison of our results to both the most extended publically available human spectral library and to a state‐of‐the‐art untargeted method supports the use of these new strategies to improve future DIA profiling efforts.