Multivariate analyses in microbial ecology

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1 MINIREVIEW Multivariate analyses in microbial ecology Alban Ramette Microbial habitat grou, Max Planck Institute for Marine Microbiology, Bremen, Germany OnlineOen: This article is available free online at Corresondence: Alban Ramette, Microbial habitat grou, Max Planck Institute for Marine Microbiology, Celsiusstrasse 1, Bremen, Germany. Tel.: ; fax: ; Received 17 January 27; revised 18 July 27; acceted 2 July 27. DOI:1.1111/j x Editor: Ian Head Abstract Environmental microbiology is undergoing a dramatic revolution due to the increasing accumulation of biological information and contextual environmental arameters. This will not only enable a better identification of diversity atterns, but will also shed more light on the associated environmental conditions, satial locations, and seasonal fluctuations, which could exlain such atterns. Comlex ecological questions may now be addressed using multivariate statistical analyses, which reresent a vast otential of techniques that are still underexloited. Here, well-established exloratory and hyothesis-driven aroaches are reviewed, so as to foster their addition to the microbial ecologist toolbox. Because such tools aim at reducing data set comlexity, at identifying major atterns and utative causal factors, they will certainly find many alications in microbial ecology. Keywords ordination; multivariate; modeling; statistics; gradient. Introduction Microbial ecology is undergoing a rofound change because structure function relationshis between communities and their environment are starting to be investigated at the field, regional, and even continental scales (e.g. Hughes Martiny et al., 26; Ramette & Tiedje, 27a, b). Because DNA sequences are being accumulated at an unrecedented rate due to high-throughut technologies such as yrosequencing (Edwards et al., 26a, b), single-cell genome sequencing (Zhang et al., 26), or metagenomics (Venter et al., 24; Field et al., 26; Gill et al., 26), future challenges will very likely consist of interreting the observed diversity atterns as a function of contextual environmental arameters. This would hel answer fundamental questions in microbial ecology such as whether microbial diversity resonds qualitatively and quantitatively to the same factors as macroorganism diversity (Horner-Devine et al., 24; van der Gast et al., 25; Green & Bohannan, 26; Hughes Martiny et al., 26). Most obstacles encountered by microbial ecologists when they try to summarize and further exlore large data sets concern the choice of the adequate numerical tools to further evaluate the data statistically and visually. Such tools, which have been develoed by community ecologists to work on distribution and diversity atterns of lants and animals, could be readily alied in microbial ecology. Although multivariate analyses of community diversity atterns are well described in the literature, microbial ecologists have used multivariate analyses either rarely or mostly for exloratory uroses. A brief survey of the literature confirms this trend (Table 1; Fig. 1). Table 1 indicates that bacterial studies rank third after lant and fish studies for their use of multivariate analyses. Comlex data sets are mostly exlored via rincial comonent analysis, or cluster analysis, and hyothesis-driven techniques such as redundancy analysis, canonical corresondence analysis (CCA), or Mantel tests are more rarely used (Fig. 1). Axis 1 (horizontal) clearly differentiates microscoic (bacteria, microorganisms, fungi) from macroscoic (fish, bird, lant, insect) life, and this may be related to the use of more exloratory methods (e.g. cluster analysis, PCA) in the first grou. It is imortant to state that the figures resented in Table 1 and Fig. 1 have to be taken with caution because many articles do not include a descrition of statistical aroaches in their titles or abstracts, and so Journal comilation c 27 Federation of Euroean Microbiological Societies

2 2 A. Ramette Table 1. Usage (%) of multivariate methods in different fields Exloratory analysis Hyothesis-driven analysis Keywords w Cluster PCA MDS PCoA CCA RDA MANOVA Mantel ANOSIM CVA Total number z Bacter Microb Plant Fung Fish Bird Insect A literature search was erformed with the Thomson ISI research tool with the following arameters (Doc tye, all document tyes; language, all languages; databases, SCI-EXPANDED, SSCI, A&HCI; Timesan, 19 26) on December 13, 26 in the titles and abstracts of the articles only. w Asterisks were laced at the end of each keyword to accommodate for variations. Each keyword was additionally combined with the following technical designations: cluster, cluster analysis; PCA, rincial comonent analysis; MDS, multidimensional scaling; PcoA, rincial coordinate analysis; CCA, canonical corresondence analysis; RDA, redundancy analysis; Mantel, Mantel test, or CVA, canonical variate analysis. z Total number refers to the total number of ublications identified by each keyword and all its combinations. The ordination based on corresondence analysis of the raw number is deicted in Fig MANOVA Mantel ANOSIM MDS Fish CCA BirdRDA Plant Insect PCoA PCA CVA cluster Fung Bacteria the table is certainly biased and incomlete. However, the oint of the table was both to identify some general trends in the literature and to give one examle of the usefulness of multivariate analysis to analyze a data table. Microb Fig. 1. Corresondence analysis of method usage in various scientific fields. In this symmetrical scaling of CA scores, the first two axes exlained 47.3% and 35.8% of the total inertia of Table 1, resectively. The gray areas were drawn to facilitate the interretation. Comlete row names (scientific fields; full circles) and column names (methods; white triangles) are given in Table 1. Methods (triangles) located close to each other corresond to methods often occurring together in studies. The distance between a scientific field oint and a method oint aroximates the robability of method usage in the field. 1. This review aims at resenting some common multivariate techniques in order to foster their integration into the microbial ecologist s toolbox. Indeed, it is no longer ossible to gain a full understanding of Ecology and Systematics without some knowledge of multivariate analysis. Or, contrariwise, misunderstanding of the methods can inhibit advancement of the science (James & McCulloch, 199). Such a review is ambitious because it tries to rovide a few guidelines for a very vast disciline that is still under develoment. For this reason, it cannot be exhaustive and does not retend to offer in-deth coverage of all selected toics. The review is largely insired by descritions, comments, and suggestions originating from multile, highly recommended sources (ter Braak & Prentice, 1988; James & McCulloch, 199; Legendre & Legendre, 1998; Les & Smilauer, 1999; ter Braak & Smilauer, 22; Palmer, 26), where detailed information about each technique can be obtained. In the first art, data tye and rearation are reviewed as a necessary basis for subsequent multivariate analyses. Second, common multivariate methods (i.e. cluster analysis, rincial comonent analysis, corresondence analysis, multidimensional scaling) and a few statistical methods to test for significant differences between grous or clusters are described, focusing on the methods main objectives, alications, and limitations. Beyond the mere identification of diversity atterns, microbial ecologists may wish to correlate or exlain those atterns using measured environmental arameters, and this aroach is addressed in the third art. Secial emhasis is laced on a few methods that have roven useful in ecological studies, namely redundancy analysis, CCA, linear discriminant analysis, as well as variation artitioning. The final art rovides ractical considerations to hel researchers avoid itfalls and choose the most aroriate methods. Journal comilation c 27 Federation of Euroean Microbiological Societies

3 Multivariate analyses in microbial ecology 3 Data tyes and data rearation Data sets The initial multivariate data set may consist of a table of objects (e.g. samles, sites, time eriods) in rows and measured variables for those objects in columns. This table structure is the standard used in the resent review. When the latter variables are biological taxa, the columns will simly be designated as secies thereafter. It is critical to clearly identify what corresonds to objects and variables in the data set. Indeed, objects in one study may be secies or oerational taxonomic units (OTU) for which catabolic rofiles, gene resence or olymorhism, etc. are measured. In another study where samles from different sites are comared based on, for instance, community fingerrinting techniques, objects can now be samles and secies variables. This distinction is imortant because rocedures that analyze relationshis among objects or among variables are different. Objects are defined a riori by the samling strategy before making observations and variable measurements. Besides, most multivariate analyses assume indeendence between objects (or samles), i.e. observations made on an object are not a riori deendent on those made on another object. Variables, however, can be found to be intercorrelated to various degrees, but this is not necessarily known in advance. Initial data sets can also consist of distance matrices where airwise dissimilarities between objects are calculated. The original table of raw data is not always available, e.g. for DNA DNA hybridization values, hylogenetic distances, and thus secific multivariate techniques have to be considered to deal with data matrices. Data transformations In multivariate data tables, measured variables can be binary, quantitative, qualitative, rank-ordered, classes, frequencies, or even a mixture of those tyes. If variables do not have a uniform scale (e.g. environmental arameters measured in different units or scales) or an adequate format, variables have to be transformed before erforming further analyses. Each qualitative variable has to be recoded as a set of numerical variables that relace it in the numerical calculations. One way to do so is to create a series of dummy variables that corresond to all the states of the qualitative variable. For instance, if the variable season has to be recoded, four associated variables will be constructed, and for each object the value 1 will be given to the corresonding season when it occurs, and for the three other seasons when it is absent. Many statistical ackages automatically erform this recoding. Standardization rovides dimensionless variables and removes the undue influence of magnitude differences between scales or units. A common rocedure is to aly the z-score transformation to the values of each variable. For each variable, it consists of (1) comuting the difference between the original value and the mean of the variable (i.e. centering) and of (2) dividing this difference by the SD of the variable. Normalizing transformations aim at correcting the distribution shaes of certain variables, which deart from normality. One thus tries to obtain a homogeneous variances for variables, conditions under which multivariate rocedures often erform better. Different mathematical transformations can be used to normalize the x values of a variable: for instance, the arcsin ( x) transformation can be alied to ercentages or roortions, log(x1c) to variables dearting strongly from a normal distribution, and (x1c) to less roblematic cases, with c being a constant that is added to avoid mathematically undefined comutations. The c constant is generally chosen so that the smallest nonzero value is obtained by comuting x1c in the former functions. The constant should also be of the same order of magnitude as the variable (Legendre & Legendre, 1998). To make community comosition (either resence absence or abundance) data containing many zeros suitable for analysis by linear methods such as rincial comonent analysis (PCA) or canonical redundancy analysis (RDA), the Hellinger transformation [Eq. (1)] is one of five transformations that give good results (Legendre & Gallagher, 21). The chord transformation is a useful transformation that also gives less weight to rare secies in the secies table [Eq. (2)]. The transformations are given by rffiffiffiffiffiffi yij ¼ y ij ð1þ y iþ y ij yij ¼ sffiffiffiffiffiffiffiffiffiffi P yij 2 j¼1 ð2þ where y ij is the original secies value for site i and secies j, y i1 reresents the sum of all secies values for site i (i.e. sum er row), is the number of secies in the table (number of columns), and y ijreresents the resulting, transformed secies value (Legendre & Gallagher, 21). These transformations are articularly recommended when rare secies are not truly rare, i.e. when they mostly occur because the samling was erformed blindly, as generally done in soil or marine microbial ecology. Further data transformations can be found in Sokal & Rohlf (1995) and Legendre & Legendre (1998). The way to deal with missing data is a disciline on its own (Legendre & Legendre, 1998). Briefly, one can either delete rows or columns containing the missing value(s), or try to relace the missing values by mathematical estimates inferred from values obtained from other objects in the data Journal comilation c 27 Federation of Euroean Microbiological Societies

4 4 A. Ramette set. In the latter case, it is still difficult to rovide ecologically meaningful exlanations for these estimates. In any case, the secific handling of missing data should be reorted by the investigator. When dealing with matrices, it is ossible to change a similarity matrix (S) into a dissimilarity matrix (D) by alying the following transformations: D =1 S, D = (1 S), or D = (1 S 2 ). To normalize any D matrix to the interval [ 1], one can comute D/D max, or (D D min )/(D max D min ), where D max and D min reresent the highest and lowest values of D, resectively (Legendre & Legendre, 1998). Exloratory analyses Visualization and exloration of comlex data sets The basic aim of ordination and cluster analysis is to reresent the (dis)similarity between objects (e.g. samles, sites) based on values of multile variables (columns) associated with them, so that similar objects are deicted near from each other and dissimilar objects are found further aart from each other. Exloratory multivariate analyses are thus useful to reveal atterns in large data sets, but they do not directly exlain why those atterns exist. This latter oint is addressed in the third art of the review. Cluster analysis and association coefficients Cluster analysis encomasses several multivariate techniques that are used to grou objects into categories based on their dissimilarities. The aim is both to minimize withingrou variation and maximize between-grou variation in order to reveal well-defined categories of objects, and therefore reduce the dimensionality of the data set to a few grous of rows (James & McCulloch, 199; Legendre & Legendre, 1998). This aroach is thus generally recommended when distinct discontinuities instead of continuous differences (i.e. gradients) are exected between samles (objects) because cluster analysis mostly aims at reresenting artitions in a data set (Legendre & Legendre, 1998). Because distance matrices that are based on differences in DNA or amino acid sequences are commonly used to describe microbial diversity, cluster analysis has become very oular in microbial ecology (Table 1; Fig. 1). This is not surrising because the grouing of organisms based on their henotyic or genotyic similarities in order to infer their taxonomic ositioning is generally and historically based on cluster analysis (or at least based on a tree-like reresentation) and, as such, is central to biology and evolution (Avise, 26). Tyical microbial ecology questions that are addressed by cluster analysis are whether the clustering atterns of molecular sequences reflect samle origin or samling time in order to reveal secific biogeograhical or temoral atterns, resectively (Whitaker et al., 23; Acinas et al., 24). Those factors are generally hyothesized to be of a discontinuous nature, but the rationale of generally reresenting molecular differences as discontinuous clusters in microbial ecology and microbial genomic studies has only started to be questioned (Konstantinidis et al., 26). Another common alication consists of sorting out clones from environmental samles based on secific criteria (e.g. genetic or henotyic markers) because clones or variants are exected to form tight clusters around their arental strains and to be more distinct from other lineages (Acinas et al., 24). In microarray data analysis, cluster analysis has heled identify common exression atterns of grous of genes, which may shed light on functionally related genes or athways (Eisen et al., 1998). Cluster analysis of a data table roceeds in two stes. First, a relevant association coefficient has to be chosen to measure the association (similarity or dissimilarity) among objects or among variables. Second, the calculated association matrix is reresented as a horizontal tree (hierarchical clustering) or as distinct grous of objects (k-means clustering), based on secific rules to aggregate objects. For ecologists, the ower of cluster analysis derives from the existence of different tyes of (dis)similarity coefficients. The choice of aroriate and ecologically meaningful association coefficients is articularly imortant because it directly affects the values that are subsequently used for the categorization of objects. The analysis of similarities among objects (rows) is designated as Q mode analysis, whereas when relationshis among variables (columns) are the focus of the study, this is referred to as R mode analysis (Legendre & Legendre, 1998). Noticeably, the two modes of analysis do not generally use the same association coefficients. Although it is not ossible to give a full review of all association coefficients here, it is useful to known that, for comaring objects (rows) based on their column attributes in Q mode analysis, coefficients may be chosen as a function of data tye (quantitative, qualitative, ordinal, or mixed data, normalized data, resenceabsence), imortance given to rare secies, weight given to each object, and calculation of associated robability levels. For comaring objects in a samle-by-environment table (e.g. water, soil chemistry), selection of aroriate coefficients generally deends on data tye and unit homogeneity of the measured variables. In R mode analyses, in addition to the reviously cited criteria, the choice of a coefficient may also deend on how the variables are related to each other (e.g. linearly, monotonically, qualitatively, ordered), and on how secies absence is handled in the calculations. In most ecological studies, the absence of a secies at two sites being Journal comilation c 27 Federation of Euroean Microbiological Societies

5 Multivariate analyses in microbial ecology 5 comared is not considered as a measure of similarity between those sites. Indeed, a simultaneous secies absence at two sites may be due to different reasons, e.g. the sites offer different hysical chemical conditions and the secies cannot exist under both conditions, and so there is no straightforward conclusion about site similarity that can be drawn in this case. Asymmetric coefficients are coefficients that do not take into account cases of double absences of secies ( double zeros ) in the calculation of airwise similarities among sites. Moreover, in microbial ecology where environmental communities are generally far from being exhaustively samled, a double absence of an OTU has to be regarded more as a lack of information rather than a sign of common structure among samles, and asymmetric coefficients such as Jaccard (191) or Srensen (1948) should be referred. More details about the calculation of association coefficients and their aroriateness can be found, for instance, in Chater 7 of (Legendre & Legendre, 1998). When an association matrix is calculated, the relationshis between objects or variables can be reresented following secific aggregation rules. Three general aroaches are commonly used: hierarchical clustering, k-means artitioning, and two-way joining. In hierarchical clustering, a linkage rule to form clusters and the numbers of clusters that best suit the data have to be determined a riori. Clusters, which are nested rather than mutually exclusive here, are either formed by rogressively agglomerating objects from high to low similarity cutoff values (forward clustering), or using the converse strategy, i.e. grouing all cases together and rogressing from low to high cutoff values in order to merge objects and clusters (backward clustering). These two strategies do not necessarily yield the same clusters. The merging of clusters is visualized using a tree format (generally horizontal) and is successful when well-defined clusters are identified in the data set (Sneath & Sokal, 1973). Common linkage rules are, e.g. nearest neighbor (the distance between two clusters is the distance between their closest neighboring oints), furthest neighbor (the distance between two clusters is the distance between their two furthest objects), and the widely used unweighted airgrou method using averages (UPGMA; Sneath & Sokal, 1973), where the distance between two clusters is the average distance between all intercluster airs. When within-cluster homogeneity is desired, Ward s method, which merges clusters only if they increase the within-cluster variation the least, is recommended (Legendre & Legendre, 1998). Finally, equal weight can also be given to clusters that are exected to be of different sizes using the weighted arithmetic average clustering (WPGMA), which consists of giving less weight to the original similarities of the largest grous (Legendre & Legendre, 1998). In k-means clustering, objects are assigned to k clusters (k being defined in advance), based on their nearest Euclidean distance to the mean of the clusters. The mean of the cluster is iteratively recalculated until no more assignments are made and cluster means fall below a redefined cut-off value or until the iteration limit is reached. Different means for each cluster are ideally obtained for each dimension used in the analysis, as indicated by high F-values from the resective analyses of variance. Unlike hierarchical clustering, k-means clustering does not require rior comutation of dissimilarity matrix among objects and is therefore more adated to large data sets (e.g. few thousand objects) where comuting ower is an issue. However, the method is quite sensitive to outliers, which are usually removed before erforming the analyses (Legendre & Legendre, 1998). Two-ste cluster analysis may be useful to grou objects into clusters when one or more of the variables are categorical (not interval or dichotomous). Objects are first groued based on the categories, which are themselves hierarchically clustered as single cases. Because neither a roximity table nor iterative stes are required, the method is articularly suited for the analysis of very large data sets (Eisen et al., 1998). Princial comonent analysis (PCA) PCA has been alied to numerous henotyic and genotyic (e.g. fingerrinting atterns) data sets, and it is one of the most oular exloratory analyses (Table 1), erhas because the technique is generally the first multivariate aroach to be exlained in most data analysis manuals. However, this choice may not always be justified in ecology and recommendations for aroriate alications are rovided at the end of this section and in the Practical considerations art of the resent review. Examles of use in microbial ecology concern the identification of atterns of microbial community change over seasons or geograhic areas (e.g. Merrill & Halverson, 22), or as those atterns relate to different lant comartments at different lant develomental stages (Mougel et al., 26), or the reduction of the comlexity of data sets involving hydrochemistry data, bacterial, and archeal community rofiles in order to visualize and interret comlex multivariate data sets onto two-dimensional geograhic mas of contaminated sites (Mouser et al., 25). The PCA rocedure basically calculates new synthetic variables (rincial comonents), which are linear combinations of the original variables (for instance, the secies of a samle-by-secies table), and that account for as much of the variance of the original data as ossible (Hotelling, 1933). The aim is to reresent the objects (rows) and variables (columns) of the data set in a new system of coordinates (generally on two or three axes or dimensions) Journal comilation c 27 Federation of Euroean Microbiological Societies

6 6 A. Ramette where the maximum amount of variation from the original data set can be deicted. In ractice, PCA is either erformed on a variance covariance matrix or on a correlation matrix. The first aroach is followed when the same units or data tyes are used (e.g. abundance of different secies). The aim is then to reserve and to reresent the relative ositions of the objects and the magnitude of variation between variables in the reduced sace. PCA on a correlation matrix is rather used when descritor variables are measured in different units or on different scales (e.g. different environmental arameters) or when the aim is to dislay the correlations among (standardized) descritor variables. The two aroaches lead to different rincial comonents and different distances between rojected objects in the ordination; hence, the interretation of the relationshis must be made with care (Table 2). Indeed, for correlation matrices, variables are first standardized (i.e. they become indeendent of their original scales), and so distances between objects are also indeendent from the scales of the original variables. All variables thus contribute to the same extent to the ordination of objects, regardless of their original variance. PCA results are generally dislayed as a bilot (Jolicoeur & Mosimann, 196), where the axes corresond to the new system of coordinates, and both samles (dots) and taxa (arrows) are reresented (Fig. 1a). The direction of a secies arrow indicates the greatest change in abundance, whereas its length may be related to a rate of change. Deending on whether a distance or a correlation bilot is chosen, different interretations can be made from the ordination diagram (Table 2). The interretation of the relationshis between samles and secies differs and is directly affected by the scaling chosen, i.e. whether the analysis mainly focuses on intersamle relationshis (scaling 1) or intersecies correlations (scaling 2). For instance, in scaling 1, the distances between objects are an aroximation of their Euclidean distances in the multidimensional sace, but this aroximation is not valid if scaling 2 is chosen (Table 2). Projecting an object at a right angle on a secies arrow in the ordination diagram aroximates the osition of the Table 2. Interretation of ordination diagrams Linear methods (PCA, RDA) PCA, RDA RDA Scaling 1 Scaling 2 Samles Secies ENV NENV Focus on samle (rows) distance Focus on secies (columns) correlation Euclidean distances among samles Linear correlations among secies Marginal effects of ENV on ordination scores Correlations among ENV Euclidean distance between samle classes Abundance values in secies data Values of ENV in the samles Membershi of samles in the classes Linear correlations between secies and ENV Mean secies abundance within classes of nominal ENV Average of ENV within classes Unimodal methods (CA, CCA) CA, CCA CCA Focus on samle (rows) distance and Hill s scaling Focus on secies (columns) distances Turnover distances among samles w 2 distances between samles - w 2 distances among secies distributions Marginal effects of ENV Correlations among ENV Turnover distances between samle classes w 2 distances between samle classes Relative abundances of the secies table Relative abundances of the secies table Values of ENV in the samles Membershi of samles in the classes Weighted averages the secies otima in resect to articular ENV Relative total abundances in the samle classes ENV averages within samle classes The interretation of ordination diagrams deends on the focus of the study, because samle scores are rescaled as a function of the scaling choice. Aroximate relationshis between and among the different elements reresented in bilots and trilots as secies (reresented as dots or arrows), samles (dots), environmental variables (ENV; arrows), and nominal (qualitative) environmental variables (NENV; dots). A meaningless interretation ( ) haens when the suggested comarison is not otimal because of inaroriate scaling of the ordination scores. Adated from ter Braak (1994); Les & Smilauer (1999); ter Braak & Smilauer (22). Journal comilation c 27 Federation of Euroean Microbiological Societies

7 Multivariate analyses in microbial ecology 7 object along that secies descritor. The length of the secies descritor indicates its contribution to the formation of the ordination sace. For correlation bilots, the length of the orthogonal rojection of a secies arrow on the axes aroximates its SD on the resective axes. Angles between secies arrows reflect their correlations, e.g. utative interactions between secies (scaling 2). An erroneous interretation of the bilot would be to use the roximity of an object oint and the ti of a secies arrow to deduce a relationshi between them. Indeed, only right-angle rojections of samles onto secies arrows are correct to derive aroximated secies abundance in the samles. PCA should generally be used when the objects (sites or samles) cover very short gradients, i.e. when the same secies are mostly identified everywhere in the study area (i.e., when samles mostly differ in secies abundances), and when secies linearly resond to environmental gradients. Because those conditions are often not met in ecological studies, other multivariate aroaches have been rogressively referred over PCA (as also suggested by Table 1) such as corresondence analysis or multidimensional scaling. PCA is successful when most of the variance is accounted for by the largest (generally the first two or three) comonents. The amount of variance accounted for by each rincial comonent is given by its eigenvalue. The mathematical descrition of eigenvalue calculation stes goes beyond the aim of the resent review but can be found in most linear algebra manuals. Eigenvalues derived from a PCA are generally considered to be significant when their values are larger than the average of all eigenvalues (Legendre & Legendre, 1998). The cumulative ercentage of variance accounted for by the largest comonents indicates how much roortion of the total variance is deicted by the actual ordination. High absolute correlation values between the synthetic variables (rincial comonents) and the original variables are useful to identify which variables mainly contribute to the variation in the data set, and this is referred to as the loading of the variables on a given axis. However, because the synthetic and original variables are linearly correlated (i.e. they are not indeendent), standard tests to determine the statistical significance of the correlations between them cannot be used. Princial coordinate analysis (PCoA) The technique is more rarely used by microbial ecologists (Table 1), desite its usefulness at reducing and reresenting atterns resent in distance matrices dislaying dissimilarities among objects (Gower, 1966). Its objectives are very similar to those of PCA in that it uses a linear (Euclidean) maing of the distance or dissimilarities between objects onto the ordination sace (i.e. rojection in a Cartesian sace), and the algorithm attemts to exlain most of the variance in the original data set. In microbial ecology, PCoA has been used, for instance, to test whether virulence rofiles (i.e. resence or absence of secific genes) arising from athogenic strains could differentiate either healthy or contaminated hosts (Chaman et al., 26), or to determine whether PCoA axes could searate grous of Stahylococcus aureus isolates into bovine and human hosts when genetic relationshis among them had been established by random amlified olymorhic DNA-PCR analysis (Reinoso et al., 24). As oosed to PCA, PCoA works with any dissimilarity measure and so secific association coefficients that better deal with the roblem of the resence of many double zeros in data sets can be surmounted. Moreover, PCoA does not rovide a direct link between the comonents and the original variables and so the interretation of variable contribution may be more difficult. This is because PCoA comonents, instead of being linear combinations of the original variables as in PCA, are comlex functions of the original variables deending on the selected dissimilarity measure. Besides, the non-euclidean nature of some distance measures does not allow for a full reresentation of the extracted variation into a Euclidean ordination sace. In that case, the non-euclidean variation cannot be reresented and the ercent of total variance cannot be comuted with exactness. The choice of the dissimilarity measure is thus of great imortance, and subsequent transformation of the data to correct for negative eigenvalues is sometimes necessary (see Legendre & Legendre, 1998, section for how to correct for such negative eigenvalues). Objects are reresented as oints in the ordination sace. Eigenvalues are also used here to measure how much variance is accounted for by the largest synthetic variables on each PCoA synthetic axis. Although there is no direct, linear relationshi between the comonents and the original variables, it is still ossible to correlate object scores on the main axis (or axes) with the original variables to assess their contribution to the ordination. Corresondence analysis (CA) A basic question that ecologists may want to address when facing a multidimensional table of sites (or samles) by secies is whether certain secies occur at secific sites, as a measure of their ecological references. CA has generally been used in microbial ecology to determine whether atterns in microbial OTU distribution could reflect differentiation in community comosition as a function of seasons, geograhic origin, or habitat structure (Olaade et al., 25; Edwards et al., 26a, b; Kent et al., 27). The overall aim of the method is to comare the corresondence between samles and secies from a table of counted data (or any dimensionally homogenous table) and to reresent Journal comilation c 27 Federation of Euroean Microbiological Societies

8 8 A. Ramette Axis II (PCA) (a) Axis II (CA) (b) Axis I (PCA) Axis I (CA) Fig. 2. Ordination diagrams in two dimensions. (a) In a PCA bilot reresentation, samles are reresented by dots and secies by arrows. The arrows oint in the direction of maximal variation in the secies abundances, and their lengths are roortional to their maximal rate of change. Long arrows corresond to secies contributing more to the data set variation. Right-angle rojection of a samle dot on a secies arrow gives aroximate secies abundance in the samle. (b) In a CA joint lot reresentation focusing on secies distance, both samles and secies are deicted as dots. Secies dots corresond to the center of gravity (inertia) of the samles where they mostly occur. Distances between samle and secies oints give an indication of the robability of secies comosition in samles (see Table 2 for more details about diagram interretation). it in a reduced ordination sace (Hill, 1974). Noticeably, instead of maximizing the amount of variance exlained by the ordination, CA maximizes the corresondence between secies scores and samle scores. Several algorithms exist and the most commonly described one is recirocal averaging, which consists of (1) assigning arbitrary numbers to all secies in the table (these are the initial secies scores), (2) for each samle, a samle score is then determined as a weighted average of all secies scores (this thus takes into account the abundance of each secies at the site and the reviously determined secies scores), (3) for each secies, a new secies score is then calculated as the weighted average of all the samle scores, (4) both secies scores and samle scores are standardized again to obtain a mean of zero and a SD of one, and (5) stes two to four are reeated until secies and site scores converge towards stable solutions in successive iterations (Hill, 1974). The overall table variance (inertia) based on w 2 distances is decomosed into successive comonents that are uncorrelated to each other, as in the PCA or PCoA rocedures. For each axis, the overall corresondence between secies scores and samle scores is summarized by an eigenvalue, and the latter is thus equivalent to a correlation coefficient between secies scores and samle scores (Gauch, 1982). The technique is oular among ecologists because CA is articularly recommended when secies dislay unimodal (bell shaed or Gaussian) relationshis with environmental gradients (ter Braak, 1985), as it haens when a secies favors secific values of a given environmental variable, which is revealed by a eak of abundance or resence when the otimal conditions are met (this can be visualized by lotting secies abundance against the environmental arameter). The unimodal model that suorts the concet of ecological niches has also been shown to be of the right order of comlexity for the ordination of most ecological data (ter Braak & Prentice, 1988). Although examles of unimodal distributions along variables or environmental gradients exist with macroorganisms (ter Braak, 1985), the shae of the distribution of the abundance of microbial secies along environmental arameters or gradients has not been extensively investigated (but see Ramette & Tiedje, 27a, b). This may arise from the fact that, in microbial surveys, environmental samling is mostly erformed blindly in relation to environmental heterogeneity, and the abundance of target secies is generally determined without systematically analyzing associated environmental arameters. Finally, another imortant feature of CA for microbial ecologists is that the recirocal averaging algorithm disregards secies double absences because the relationshis between rows and columns of the table are quantified using the w 2 coefficient that excludes double absences (Legendre & Legendre, 1998). Both samles and taxa are often jointly deicted in the ordination sace (i.e. joint lot; Fig. 2b), where the center of inertia (centroid) of their scores corresonds to the zero for all axes. Deending on the choice of the scaling tye, either the ordination of rows (samles) or the columns (secies) is meaningful, and can be interreted as an aroximation of the w 2 distances between samles or secies, resectively (see Table 2 for more details about interretation). Samle oints that are close to each other are similar with regard to the attern of relative frequencies across secies. It is imortant to remember that in such joint lots, either distances between samle oints or distances between secies oints can be interreted, but not the distances between samle and secies oints. Indeed, these distances are not simle Euclidean distances comuted from the relative row or column frequencies, but rather they are Journal comilation c 27 Federation of Euroean Microbiological Societies

9 Multivariate analyses in microbial ecology 9 weighted distances. The roximity between samle and secies oints in the lot can thus be understood as a robability of secies occurrence or of a high abundance in the samles in the vicinity of a secies oint. In scaling 2 (i.e. focus on secies), secies oints found at the center of the ordination sace should be carefully checked with the raw data to clarify whether the secies ordination really corresonds to the otimal abundance or occurrence of the secies, or whether the secies is just badly reresented by the main axes, as it is the case when other axes are more aroriate to reresent the secies. Rare secies contribute little to the total table inertia (i.e. they only lay a minor role in the overall table variance) and are hence ositioned at the edges of the lot, next to the site(s) where they occur. In general, only the secies oints found away from the ordination center and not close to the edges of the ordination have more chances to be related to the ordination axes, i.e. to contribute to the overall variance (Legendre & Legendre, 1998). When the secies comosition of the sites rogressively changes along the environmental gradient, samle ositions may aear in the ordination lot as nonlinear configurations called arch (Gauch, 1982) (or horseshoe in the case of PCA), which may imair further ecological interretation. In CA, the arch effect may be mathematically roduced as a side-effect of the CA rocedure that tries to obtain axes that both maximally searate secies and that are uncorrelated to each other (ter Braak, 1987): when the first axis suffices to correctly order the sites and secies, a second axis (uncorrelated with the former) can be obtained by folding the first axis in the middle and bringing its extremities together, thus resulting in an arch configuration. Further axes can be obtained by further dividing and folding the first axis into segments (Legendre & Legendre, 1998). To remove the arch effect in CA, a mathematical rocedure, detrending, is used to flatten the distribution of the sites along the first CA axis without changing their ordination on that axis. The aroach is then designated as detrended corresondence analysis (DCA). The review of different detrending algorithms such as using segments or olynomials goes beyond the scoe of this review, but more information can be obtained in (ter Braak & Prentice, 1988; Legendre & Legendre, 1998). Some authors have also argued that the arch effect may not be an artifact but an exected feature of the analysis, esecially when secies turnover is high along environmental gradients (James & McCulloch, 199). In that case, if the samles are meaningfully ositioned along the arch, the ordination should be acceted as a valid result. Nonmetric multidimensional scaling (NMDS) NMDS is generally efficient at identifying underlying gradients and at reresenting relationshis based on various tyes of distance measures. Not surrisingly, NMDS has found an increasing number of alications in microbial ecology (Table 1). The technique has been generally alied to identify atterns among multile samles that were subjected to molecular fingerrinting techniques. For instance, NMDS was used to analyze and to comare the reroducibility of various fingerrinting techniques such as ribosomal internal sacer analysis (RISA), terminal fragment length olymorhism (T-RFLP), and denaturing gradient gel electrohoresis (DGGE) between different laboratories when alied to samles chosen from a salinity gradient (Casamayor et al., 22). NMDS was also used to comare diversity atterns of microbial communities (as determined by length heterogeneity-pcr) from samles undergoing different land management ractices (Mills et al., 26). Another examle is the analysis of the bacteriolankton communities of four shallow eutrohic lakes that differed in nutrient load and food web structure using DGGE rofiling, so as to determine the secificity of community signatures in each lake (Van der Gucht et al., 25). The NMDS algorithm ranks distances between objects, and uses these ranks to ma the objects nonlinearly onto a simlified, two-dimensional ordination sace so as to reserve their ranked differences, and not the original distances (Sheard, 1966). The rocedure works as follows: the objects are first laced randomly in the ordination sace (the desired number of dimensions has to be defined a riori), and their distances in this initial configuration are comared by monotonic regression with the distances in the original data matrix based on a stress function (values between and 1). The latter indicates how different the ranks on the ordination configuration are from the ranks in the original distance matrix. Several iterations of the NMDS rocedure are generally imlemented so as to obtain the lowest stress value ossible (i.e. the best goodness of fit) based on different random initial ositions of the objects in the ordination sace. For samle-by-secies tables, simulations have shown that before alying NMDS, a standardization of each secies by its maximum abundance, followed by the comutation of distances between samles based on the Steinhaus or Kulczinski similarity coefficients yielded informative ordination results (Legendre & Legendre, 1998,. 449). In NMDS ordination, the roximity between objects corresonds to their similarity, but the ordination distances do not corresond to the original distances among objects. Because NMDS reserves the order of objects, NMDS ordination axes can be freely rescaled, rotated, or inverted, as needed for a better visualization or interretation. Because of the iterative rocedure, NMDS is more comuter intensive than eigenanalyses such as PCoA, PCA, or CA. However, constant imrovement in comuting ower makes this limitation less of a roblem for small- to medium-sized matrices. Journal comilation c 27 Federation of Euroean Microbiological Societies

10 1 A. Ramette Testing for significant differences between grous In addition to reresenting objects in an ordination lot or as clusters of similar objects, another objective may be to test whether differences between grous of objects (rows) in a multivariate table are significantly different based on the set of their attributes (columns), i.e. to test whether similarities within grous are higher than those between grous. Here, nonarametric multivariate ANOVA (NPMANOVA) and analysis of similarities (ANOSIM), which are commonly found in standard statistical ackages, are briefly reviewed. It is also ossible to use canonical analyses ( Testing for significant differences between grous ) to test for significant differences between grous of objects. These statistical tests, however, must not be used to assess the statistical difference among grous that were derived from a revious cluster analysis on the same variables because, under those conditions, the two aroaches would not be indeendent from each other. Indeed, the grous derived from cluster analysis (which are themselves made to fit the data) would then be used for testing the null hyothesis that there is no difference among the grous. This hyothesis would then not be indeendent of the data used to test it, and would nearly always roduce significant differences between the grous even if it is not the case (Legendre & Legendre, 1998). NPMANOVA The method can be used to test for significant differences between the means of two or more grous of multivariate, quantitative data (Anderson, 21). The null hyothesis of equality of means is tested based on Wilks L (lambda) statistic, which relaces the F-test normally used in univariate ANOVA. When only two grous are comared, Hotelling s T 2 test is more aroriate. The latter test can also be used, as a ost hoc test, to assess the significance of airwise comarisons statistically between grous, following an overall significant Wilks test. Significance is generally comuted by ermutation of grou membershi, with several thousand relicates, alleviating concerns about multinormality of the data. Because multile airwise comarisons are made, the significance level of the airwise Hotelling s tests needs, however, to be corrected. With the Bonferroni correction, for instance, the P-value usually chosen for significant differences between grous (i.e..5) is relaced by a smaller P-value calculated by dividing the original P-value by the total number of airwise comarisons that are erformed. For instance, for 1 airwise comarisons, the corrected P-value becomes.5. This correction is often judged to be rather conservative as it leads to significance for fewer airwise comarisons (Legendre & Legendre, 1998). ANOSIM This nonarametric rocedure tests for significant difference between two or more grous, based on any distance measure (Clarke, 1993). It comares the ranks of distances between grous with ranks of distances within grous. The means of those two tyes of ranks are comared, and the resulting R test statistic measures whether searation of community structure is found (R = 1), or whether no searation occurs (R = ). R values 4.75 are commonly interreted as well searated, R 4.5 as searated, but overlaing, and R o.25 as barely searable (Clarke & Gorley, 21). The test makes fewer assumtions than MANOVA because it is based on the ranks of distances, and it is often used for samle-by-secies tables, where grous of samles are comared. All grous should have comarable within-grou disersion to avoid finding falsely significant results (Legendre & Legendre, 1998). Alications in microbial ecology include testing for satial differences, temoral changes, or environmental imacts on microbial assemblages. For instance, Kent et al. (27) determined whether bacterial communities from the same lake were more similar in comosition to each other than to communities in different lakes. The bacterial comosition and diversity of samles from different geograhic origins, habitats, and avian hosts were also comared using ANOSIM based on a length heterogeneity (LH)-PCR (Bisson et al., 27). Another examle is the alication of ANOSIM to terminal restriction fragment length olymorhism (T-RFLP)-generated data to determine the imact of B and NaCl on soil microbial community structure in the wheat rhizoshere (Nelson & Mele, 27). Environmental interretation Exloratory analyses may reveal the existence of clusters or grous of objects in a data set. When a sulementary table or matrix of environmental variables is available for those objects, it is then ossible to examine whether the observed atterns are related to environmental gradients. Tyical objectives may be, for instance, to reveal the existence of a relationshi between community structure and habitat heterogeneity, between community structure and satial distance, or to identify the main variables affecting bacterial communities when a large set of environmental variables has been conjointly collected. The significance of the relationshis between secies atterns and environmental variables can generally be assessed by ermutation techniques such as Monte Carlo ermutation tests, which infer statistical roerties from the data themselves. The order of data (generally the rows of one matrix) is ermuted and the relationshis between the observed atterns and environmental variables can be Journal comilation c 27 Federation of Euroean Microbiological Societies

11 Multivariate analyses in microbial ecology 11 assessed for randomness. This aroach is articularly suitable when variables do not follow a normal distribution (which is often the case with environmental or ecological data), as generally required by traditional statistical rocedures (Legendre & Legendre, 1998). Indirect gradient analyses Ordination axes or clusters can be interreted based on additional environmental variables (i.e. variables not used in the ordination or cluster analysis) that rovide ecological knowledge about the studied sites or secies ecological characteristics. When using exloratory ordination aroaches on a samle-by-secies table, samles are dislayed along the axes of main variation in secies comosition. These axes are thus constructed without reference to environmental characteristics, but they can be hyothesized to reresent underlying environmental gradients (e.g. environmental arameters, satial or temoral variables, chemical gradients), which need to be subsequently identified. Such an aroach is designated as indirect, because synthetic variables (i.e. the axes) are first constructed and thereafter related to environmental variation. For instance, the scores of the objects on PCA or CA main comonents (axes) can be further related by standard statistical rocedures (e.g. ANOVA, regression analysis) to environmental variables. Likewise, in PCoA or NMDS, it is ossible to statistically comare the ranks obtained by the objects on each axis and the ranks of those objects on additional environmental variables, using Searman s rank correlation coefficients (Legendre & Legendre, 1998). A convenient method of interretation is to reresent the additional environmental variables as fitted arrows directly on the ordination diagram. These variables are added to the existing ordination by linear regression of their values onto the existing ordination axes. This rocedure is imlemented in various statistical ackages (e.g. CANOCO, R). Hence, it is ossible to assess the direction and magnitude of the most raid change in the environmental variables and to determine whether they corresond to the observed atterns among objects (Oksanen, 27). In cluster analysis, the magnitude of the absolute correlation value between an ordered clustering solution and environmental variables may also rovide clues about utative environmental causes for the observed discontinuities in the data set. Another convenient way of dislaying additional information to hel interret the ordination is to use site symbols whose sizes are roortional to the values of the additional variable. Hence, one can visually assess whether the ordination of objects (samles, sites) matches secific trends in the additional variable. This strategy was, for instance, used on NMDS ordination lots inferred from DGGE rofiles on which the values of five additional environmental variables were individually maed as roortional circles in order to identify the main environmental factors related to the bacterial community structure in four freshwater lakes (Van der Gucht et al., 25). Direct gradient analyses (constrained analyses) In constrained (canonical) ordination analyses, only the variation in the secies table that can be exlained by the environmental variables is dislayed and analyzed, and not all the variation in the secies table. Gradients are suosed to be known and reresented by the measured variables or their combinations, while secies abundance or occurrence is considered to be a resonse to those gradients. Constrained ordinations are mostly based on multivariate linear models relating rincial axes to the observed environmental variables, and the different techniques deend on data tyes (matrix or table), and on the hyothesis underlying secies distribution in the gradients (i.e. linear or unimodal). Their aim is to find the best mathematical relationshis between secies comosition and the measured environmental variables, and to assess whether, statistically, such a relationshi could have been roduced due to chance alone using ermutation tests. The resulting ordination diagrams dislay samles, secies, and environmental variables so that fitted secies samles and secies environment relationshis can be derived as easily as ossible from angles between arrows or distances between oints and arrows (Table 2). Redundancy analysis (RDA) In microbial ecology, RDA has been alied, for instance, to test whether the occurrence of biocontrol bacteria with secific carbon source utilization rofiles was related to their origin from different root samles (Folman et al., 23), to determine which environmental factors were the most significant to exlain variation in microbial community comosition in undisturbed native rairies and croed agricultural field (McKinley et al., 25), to examine the effects of samling locations (longitude, latitude, altitude) on genetic diversity of lant athogenic bacteria (Kolliker et al., 26), or to assess the influence of season, farm management, and soil chemical, hysical, and biological roerties on nitrogen fluxes and bacterial community structure (Cookson et al., 26). This method can be considered as an extension of PCA in which the main axes (comonents) are constrained to be linear combinations of the environmental variables (Rao, 1964). Two tables are then necessary: one for the secies data ( deendent variables) and one for the environmental variables ( indeendent variables). Multile linear regressions are used to exlain variation between indeendent Journal comilation c 27 Federation of Euroean Microbiological Societies

12 12 A. Ramette and deendent variables, and these calculations are erformed within the iterative rocedure to find the best ordination of the objects. The interest of such an aroach is to reresent not only the main atterns of secies variation as much as they can be exlained by the measured environmental variables but also to dislay correlation coefficients between each secies and each environmental variable in the data set. When the data set consists of a matrix of distances between objects, distance-based RDA (db-rda; Legendre & Anderson, 1999) can be alied to determine how well additional environmental arameters can exlain the variation among objects in the matrix. The technique first alies a PCoA on the distance matrix to convert it back to a rectangular table containing rows of objects by columns of PCoA coordinates. Those new, uncorrelated coordinates thus corresond to synthetic secies variables that are then related to additional environmental arameters using a classical RDA. For instance, db-rda was successfully used to determine how the variation in matrices of genomic distances among environmental strains could be exlained by factors such as soil arameters, host lant secies, and satial scale, each factor being taken alone or in combination (Ramette & Tiedje, 27b). Most software oututs rovide the total variation in secies comosition as exlained by the environmental axes, the cumulative ercentage of variance of the secies environment relationshi, and the overall statistical significance of the relationshis between the secies and environmental tables. RDA can be reresented by a trilot of samles (dots), secies (arrows), and environmental variables (arrows for quantitative variables and dots for each level of qualitative or nominal variables), or by any combinations thereof (i.e. bilots) (ter Braak, 1994). Deending on the scaling chosen, i.e. whether the analysis mainly focuses on intersamle relationshis or intersecies correlations, the interretation of the relationshis between samles, secies, and environmental variables differs (Table 2). Canonical corresondence analysis (CCA) The aroach is very similar to that of RDA, excet that CCA is based on unimodal secies environment relationshis whereas RDA is based on linear models (ter Braak, 1986). CCA can be considered as the constrained form of CA in which the axes are linear combinations of the environmental variables. CCA uses the unimodal model to model secies resonse to the environmental variation as a mathematical simlification to enable the estimation of a large number of arameters and the identification of a small number of ordination axes. This secies model seems, however, to be robust even when some secies dislay bimodal resonses, unequal ranges, or unequal maxima along environmental gradients, and the technique is thus considered to be the method of choice by many ecologists (ter Braak & Smilauer, 22). It is therefore articularly adated for the environmental interretation of tables of abundance and occurrence of secies, and accommodates well the absence of secies at certain sites in the data set. CCA is sensitive to rare secies that occur in secies-oor samles, and down-weighting of such secies hel reduce the roblem (Legendre & Legendre, 1998). Software oututs are very similar to those of RDA and as for RDA, trilot and bilot reresentations and interretation deend on the choice of the scaling tye (Table 2). The same interretation of the relationshis between samle and secies oints is found in CA and CCA. Right-angle rojection of these oints on the environmental arrows leads to the correct aroximation of the ranking of the oints along environmental variables. CCA has been used in an increasing number of ublications dealing with microbial assemblages in marine and soil ecosystems. Tyical questions that are addressed concern the identification of environmental factors that influence the diversity of bacterial assemblages among large sets of candidate environmental arameters measured for the same samles, when the diversity is determined by cultureindeendent, genetic fingerrinting techniques such as automated ribosomal intergenic sacer analysis (ARISA) (Yannarell & Trilett, 25), DGGE (Salles et al., 24; Sa et al., 27), or T-RFLP (Córdova-Kreylos et al., 26; Klaus et al., 27). Another interest in the technique comes from the ossibility of determining the secific secies or OTUs that resond to articular environmental variables, and as such that can be identified as candidate indicator secies. Those secies can then be subjected to further exeriments so as to confirm their status of indicator secies. For instance, the relationshis, as determined by CCA, between bacterial community comosition and 11 environmental variables for 3 lakes in Wisconsin, revealed that atterns in bacterial communities were best exlained by regional- and landscae-level factors, as well as by secific seasons, H, and water clarity (Yannarell & Trilett, 25). CCA was also successfully used to demonstrate that former land use management affected the comosition of the targeted soil microbial community (Burkholderia) to a larger extent than did lant secies (Salles et al., 24). Another interesting examle in the marine ecosystem is the study of the interactions between various abiotic arameters and hytolankton community data (biotic arameter) to exlain bacteriolankton dynamics in the North Sea and the subsequent identification of the bacterial hylotyes resonding more secifically to the factors (Sa et al., 27). Another examle of using CCA to identify some microbial communities as ollution indicators can be found in (Córdova-Kreylos et al., 26). Journal comilation c 27 Federation of Euroean Microbiological Societies

13 Multivariate analyses in microbial ecology 13 (a) Fig. 3. Partitioning biological variation into the effects of two factors. The large rectangle reresents the total variation in the biological data table, which is artitioned among two sets of exlanatory variables (a, b). Fraction 4 shows the unexlained art of the biological variation. Fractions 1 and 3 are obtained by artial constrained ordination or artial regression, and can be tested for significance. For instance, fraction 1 corresonds to the amount of biological variation that can be exclusively exlained by (a) effects when (b) effects are taken into consideration (i.e., when b is considered as a covariable). Fraction 2 [i.e., variation indifferently attributed to (a) and (b) or a covariation of (a) and (b)] is obtained by subtracting fractions 1 and 3 from the total exlained variance, and cannot be tested for statistical significance. Partial ordination, variation artitioning When the effects of a articular environmental variable need to be tested after elimination of ossible effects due to other (environmental) variables, artial ordination may be used (e.g. artial CCA, artial RDA). Such an aroach is also referred to as artialling out or controlling for the effects of secific variables, which are secified as covariables in the constrained analysis. For instance, in a study dealing with the effects of environmental and ollutant variables on microbial communities, Córdova-Kreylos et al. (26) observed that variation in microbial communities was more due to satial variation than to ollutants. The use of artial CCA to account for satial variation in the biological data set revealed that metals had a greater effect on microbial community comosition than organic ollutants. This idea of controlling for the effects of secific variables can be extended to evaluate the effects of all the different sets (factors) of environmental variables resent in a study so as to determine the relative contribution (amount of variation exlained) and significance of each variable set on the total biological variance. The so-called variation artitioning rocedure (Borcard et al., 1992) artitions the total variance of the secies table into the resective contribution of each set of environmental variables and into their covariations using both standard and artial constrained ordinations (Fig. 3). Two methods have traditionally been used to artition the variation of community comosition data, i.e. canonical artitioning and regression on distance matrices based on Mantel tests (Legendre & Legendre, 1998). The canonical aroach has been shown to be more aroriate to artition the b diversity correctly among sites and to test (b) 4 hyotheses about the origin and maintenance of its variation (Legendre et al., 25). Alications of variation artitioning in microbial ecology include, for instance, the study by Ramette & Tiedje (27b), which alied the technique in the context of RDA to disentangle the effects of sace, environmental soil arameters, and lant secies on Burkholderia community abundance and diversity. By quantifying the amount of biological variation that is left unexlained when all environmental variables had been considered, the study suggested that much less of the biological variation could be redicted at the intrasecific level comared with higher taxonomic levels. Another interesting examle is the study of seasonal changes in bacterial community comosition in shallow eutrohic lakes, in which to-down regulation (grazers) of bacterial community comosition was examined after accounting for bottom-u regulation (resources) (Muylaert et al., 22). Linear discriminant analysis (LDA) When grous or clusters of objects have been obtained by exloratory analyses for instance, LDA can be used to identify linear combinations of additional environmental variables that best discriminate those grous. In that resect, LDA can be seen as an extension of MANOVA for two or more grous, in which environmental variables that secifically exlain the grouing of objects are identified. Another alication consists of assigning new objects to reviously defined grous for rediction or classification uroses based on the calculated discriminant function. For instance, Fuhrman et al. (26) used the technique to evidence the existence of reeatable temoral atterns in the community comosition of marine bacteriolankton over 4.5 years. The technique is mostly recommended for multinormal data for which attribute data are linearly related and for which variances and covariances of the variables are good summary statistics. A visual reresentation of LDA can be erformed, and in the resulting ordination, the axes are then the discriminant functions. The distances between objects, which corresond to Mahalanobis distances that take into account the correlations among descritors (Mahalanobis, 1936), are indeendent of the scale of measurement of the various descritors and are mostly used to comare grous of sites or objects with each other (Legendre & Legendre, 1998). Selection of variables in regression models In the revious constrained methods where linear combinations of environmental (exlanatory) variables are used, the inclusion of too many exlanatory variables to describe secies distribution may lead to difficult ecological interretations and to lower redictability of the models, due to Journal comilation c 27 Federation of Euroean Microbiological Societies