SPATIAL COVARIANCE IN PLANT COMMUNITIES: INTEGRATING ORDINATION, GEOSTATISTICS…paulojus:... ·...

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Ecology, 84(4), 2003, pp. 1045–1057q 2003 by the Ecological Society of America

SPATIAL COVARIANCE IN PLANT COMMUNITIES: INTEGRATINGORDINATION, GEOSTATISTICS, AND VARIANCE TESTING

HELENE H. WAGNER1

Department of Biology, Colorado State University, Fort Collins, Colorado 80523 USA, andWSL, Swiss Federal Institute for Forest, Snow, and Landscape Research, 8903 Birmensdorf, Switzerland

Abstract. Spatial structure in plant communities occurs in the forms of (1) single-species aggregation and dispersion patterns, (2) distance-dependent interactions betweenspecies, and (3) the response to the spatial structure of environmental conditions. Differentmethods deal with these components of spatial variation: geostatistical analysis revealsautocorrelation in a spatial sample; the variance of species richness has been used as anindicator for interspecific interactions due to niche limitation; and ordination techniquesdescribe multispecies responses to environmental factors. Based on the mathematical prop-erties of presence–absence data, it is shown how variogram modeling, the testing of in-terspecific associations, and multiscale ordination can be integrated using the same set ofdistance-dependent variance–covariance matrices (variogram matrix). The variogram matrixpartitions the variance of community data into spatial components at the levels of theindividual species, species composition, and species richness. It can be used to factor outthe effects of single-species aggregation patterns, interspecific interactions, or environ-mental heterogeneity. The mathematical integration of traditionally unrelated methods in-creases the interpretability of variograms of plant communities, provides a spatial extensionand an empirical null model for the variance test of species richness, and extends multiscaleordination to nonsystematic spatial samples. Beyond the individual applications, the var-iogram matrix provides a framework for a mathematical unification of geostatistics, mul-tivariate data analysis, and the analysis of variance that may enable ecologists from a broadrange of fields to incorporate spatial effects into their research and to integrate analysesacross different levels of biological organization.

Key words: interspecific associations; multiscale ordination; multivariate geostatistics; nonsta-tionarity; spatial variance; species richness; variance test; variogram matrix.

INTRODUCTION

Differences in species composition between sam-pling units, such as quadrats, are a primary focus ofquantitative vegetation analysis. Ordination is the mainmethod used for analyzing variation in plant commu-nities. Indirect ordination detects intrinsic gradients inspecies composition, while direct gradient analysisidentifies compositional gradients in vegetation as aresponse to measured environmental factors (De’ath1999). Plant communities and environmental factorsoften are spatially structured. Direct and indirect or-dination, however, are both essentially nonspatialmethods.

Due to an increasing awareness of the importance ofspace in ecology and the availability of global posi-tioning systems (GPS), more and more data sets arespatially referenced and lend themselves to spatial anal-ysis. If the data represent a transect or grid of contig-uous quadrats, their spatial structure can be analyzedby block-size variance analysis, also known as pattern

Manuscript received 23 August 2001; revised 25 July 2002;accepted 1 September 2002. Corresponding Editor: D. W.Roberts.

1 Present address: WSL (see above).E-mail: [email protected]

analysis (e.g., Greig-Smith 1952, Hill 1973, Usher1975, Dale 1999). Block-size variance techniques sum-marize the spatial structure of individual species andof pairs of species; they are essentially uni- or bivariatemethods (Fortin 1999, Mistral et al. 2000). The re-spective scales of regular spatial patterns in a com-munity, such as patches and gaps of constant size, areidentified by comparing several uni- or bivariate plotsof variance against block size. Alternatively, patternanalysis can be performed on the scores of an ordi-nation axis (Galiano 1983).

A truly multivariate extension, called multiscale or-dination, was presented by Noy-Meir and Anderson(1971) and further developed by Ver Hoef and Glenn-Lewin (1989). Noy-Meir and Anderson (1971) sug-gested summarizing the spatial structure of a com-munity by calculating a variance–covariance matrix foreach block size. In order to facilitate interpretation, thematrices are added to form a combined covariance ma-trix, which is subjected to principal component analysis(PCA). The scales of spatially overlapping, statisticallyuncorrelated multispecies patterns are identified by par-titioning the variance of each PCA axis by block size.

Increasingly, ecologists are exploring the possibili-ties of geostatistical analysis (e.g., Burrough 1987,Palmer 1988, Legendre and Fortin 1989, Rossi et al.

1046 HELENE H. WAGNER Ecology, Vol. 84, No. 4

1992, Fortin 1999, Koenig 1999). The geostatisticalapproach is based on distance rather than block size,which has the advantage that the quadrats need not becontiguous, nor spaced at regular intervals. The spatialstructure of a dataset is usually described by an em-pirical variogram, which is basically a plot of the var-iance or difference between pairs of observationsagainst their distance in geographic space. A variogramcan be interpreted in a similar way as a plot of varianceagainst block size derived by blocked variance tech-niques (Ver Hoef et al. 1993). In addition to descriptivepurposes, variogram modeling can be used to inter-polate point observations by kriging (Isaaks and Sri-vastava 1989, Cressie 1991, Haining 1997). However,like pattern analysis, geostatistical analysis has beenapplied mostly to single variables such as species rich-ness or the scores of quadrats along a major ordinationaxis (Palmer 1988, Legendre 1993, Jonsson and Moen1998).

Several authors have proposed ways of plotting somekind of resemblance measure (a multispecies measureof the similarity or dissimilarity of pairs of quadrats)against geographic distance. Examples are the Mantelcorrelogram (Sokal 1986), the method of Nekola andWhite (1999) for determining the rate of distance de-cay, or the ‘‘dissimilogram’’ proposed by Mistral et al.(2000). Such plots provide a very flexible descriptionof the overall multivariate spatial structure of a com-munity. However, generalized variograms cannot beused for interpolation, and their ecological interpre-tation is limited as their behavior is typically unknown.

Spatial structure in plant communities arises from avariety of factors. These factors fall into three broadgroups: (1) morphological factors, such as plant sizeor dispersal mechanism, which influence the spatialaggregation within a population; (2) interspecific in-teractions within a community; and (3) the response toenvironmental factors, which themselves are spatiallystructured (Kershaw 1964, Dale 1999, Koenig 1999).Based on hierarchy theory (Allen and Starr 1982), Le-gendre (1993) postulated that physical processes createenvironmental heterogeneity at broad scales, whilecontagious biotic processes may cause further struc-turing within smaller areas of relative environmentalhomogeneity. This hierarchical view implies that onemay assume local homogeneity within a study areaeven though heterogeneity exists at a larger scale. If,however, there are no independent domains of scale butthe scales of physical and biotic processes overlap, weneed to account for environmental heterogeneity wheninvestigating patterns caused by biotic processes, andvice versa.

Although it is common to use more than one spatialor nonspatial method to analyze the same dataset, thetechniques are usually applied individually rather thanin an integrated way. On a conceptual level, this meansthat we investigate factors determining communitystructure individually, ignoring the contributions of

other factors and their potential interactions. However,one cannot answer the question of why communitiesvary and, ultimately, why species coexist, without un-derstanding how various factors interact and what de-termines their relative importance. For example, testingthe variance of quadrat species richness against a nullmodel has often been used by ecologists as a test ofinterspecific association (for a review, see Palmer andVan der Maarel 1995), but the difficulties of accountingfor spatial autocorrelation, for the distance-dependenceof species associations, and for environmental hetero-geneity severely limit the capacity of this method toprovide evidence for niche limitation (Palmer and vander Maarel 1995, van der Maarel et al. 1995, Wilsonet al. 1995, Roxburgh and Matsuki 1999).

This paper presents a mathematical unification ofgeostatistical analysis, the analysis of interspecific as-sociations, and multiscale ordination. The integrationprovides a framework for partitioning the variance incommunity data into the distance-dependent compo-nents of single-species aggregation patterns, interac-tions between species, and species-specific responsesto environmental gradients. Starting with a mathemat-ical model of species richness as the sum of a set ofbinary species variables, I reexpress the variance ofspecies richness in terms of spatial covariance, whichis conveniently summarized in the variogram matrix.I derive a standardized variogram for binary data thatmakes spatial covariance observed under different en-vironmental conditions directly comparable. I presenta spatial extension of the variance test of species rich-ness that accounts for spatial autocorrelation and thedistance-dependent nature of interspecific associations,and I show how multiscale ordination can be used toremove variance attributed to a larger scale trend. Un-derstory vegetation data from the Oosting Natural Areain North Carolina serve to highlight the ecological ap-plication, and a worked example on artificial data inthe Appendix illustrates the calculations.

MODEL STRUCTURE AND METHODS

A mathematical model of species richness

The following mathematical model reflects the factthat species richness is not a simple quantitative var-iable, but a result of the distributions of individual,interdependent species. Statistically speaking, the oc-currence of a species i in a single quadrat is itself arandom variable xi with a specific probability distri-bution. Therefore, I start with the formal definitions ofspecies occurrence, species composition, and speciesrichness for a single quadrat.

Let xi be a binary variable that takes the value 1 ifspecies i is present in a quadrat, and 0 if it is absent.The defining parameter pi, the mean or probability ofoccurrence of i in the quadrat, will depend on quadratsize and shape:

April 2003 1047SPATIAL COVARIANCE IN PLANT COMMUNITIES

FIG. 1. Hypothetical variograms of the complementarity of species composition, X, and of a single species i. The parameterssill, range, and nugget of the variogram of complementarity are related to the respective parameters of the variograms of theconstituent species.

1 pix 5 i ∈ {1, . . . , s} 0 # p # 1.i i5 60 1 2 pi

(1)

Species composition X is the vector of s binary vari-ables xi that describe the occurrence of s species in thequadrat. Quadrat species richness S is the sum of thevector of species composition X, i.e., the sum of allspecies variables xi:

S 5 X 5 x . (2)O O ii

The expected value of S is the sum of the expectedvalues of the binary variables xi, even if the variablesare not independent. The variance of S is the sum ofthe variance–covariance matrix:

s s i21

Var(S) 5 Var(x ) 1 2 Cov(x , x )O O Oi i ji51 i52 j51

s s

5 Cov(x , x ) i, j ∈ {1, . . . , s}. (3)O O i ji51 j51

If the species variables are independent, the pairwisecovariances between all species i and j are zero andthe variance of species richness S is equal to the var-iance of species composition X, which is the sum ofthe variances of the s species variables xi.

In practice, it is not possible to observe multipleindependent realizations of xi in a single quadrat underexactly the same conditions. Hence, one cannot esti-mate the statistical properties of xi (or of S) directlyfrom empirical data from a single quadrat. One solutionis to observe one realization each of a number of ran-dom variables xia, with a ∈ {1, . . . , N}, from a sampleof N quadrats, assuming the variables xia to be inde-

pendent and identically distributed with the probabilitydistribution function of Eq. 1. Thus the quadrats act asreplicates, and the variances and covariances in Eq. 3are estimated from the replicated data. What happensif the quadrats are not true replicates, either becausethey are not independent due to spatial autocorrelation,or because they are not comparable due to environ-mental heterogeneity, is the primary subject of thispaper.

Spatial covariance

Geostatistical methods deal with the question of howvariance and covariance depend on the distance be-tween observations (i.e., quadrats). Spatial autocorre-lation, or distance dependence, is commonly modeledby fitting a variogram function to an empirical vario-gram (Isaaks and Srivastava 1989, Cressie 1991, Hain-ing 1997, Burrough and McDonnell 1998). An empir-ical variogram is a plot of half the squared differencebetween two observations (the semivariance) againsttheir distance in space, averaged for a series of distanceclasses. A simple variogram model is defined by themodel family and the parameters sill (the average halfsquared difference of two independent observations),range (the maximum distance at which pairs of obser-vations will influence each other), and nugget (the var-iance within the sampling unit; Fig. 1).

The most commonly used model families (spherical,exponential, and Gaussian models) assume that thereis no spatial dependence for distances larger than therange. However, ecological communities may be or-ganized in periodic spatial patterns, which result in acyclic pattern of variance plotted against distance. Thiscan be modeled with a hole effect model, a dampening


sine function that is defined by the nugget effect andthe range and eventually stabilizes at the sill (Legendreand Legendre 1998).

Variogram modeling is well developed for metric(ordinary kriging) and binary (probability kriging) uni-and bivariate data, but no standard methods exist forthe multivariate categorical data that are typical forplant community ecology. Hence, I will first developthe empirical variogram description of a set of binaryvariables xi as observed in a sample of N quadrats.

An omnidirectional empirical variogram of a singlevariable xi is constructed by estimating the empiricalsemivariance, i(h), for a range of distance classes hg(Isaaks and Srivastava 1989, Cressie 1991):

12g (h) 5 (x 2 x ) (4)Oi ia ib2n a,b zh øhh ab

where nh is the number of pairs of quadrats a and bseparated by approximately h. Sample size nh decreaseswith large distances h. Distances greater than half themaximum extent of the study area can only be observedfor quadrats outside of the center of the study area.This introduces a bias, as the quadrats from the centerdo not contribute to variance estimates for larger dis-tances. Therefore, interpretation is commonly limitedto distances smaller than an arbitrary maximum dis-tance hmax:

The univariate definition of a variogram can be ex-tended to multivariate data, in which case xa and xb arenot two observations of a single variable x, but vectorsXa and Xb of two observations of s variables xi. Theempirical semivariance (h) becomes half the squaredgEuclidean distance between Xa and Xb and is equal tothe sum of the empirical semivariances i(h) of thegspecies variables xi:

12g (h) 5 \X 2 X \O a b2n a,b zh øhh ab

125 (x 2 x ) 5 g (h). (5)O O Oia ib i2ni a,b zh øh ih ab

In the case of binary variables, (h) equals the meangnumber of species that are present in only one of a pairof observations, regardless of the direction of com-parison. It is a direct measure of species turnover anddescribes the complementarity of the species compo-sition of two quadrats. Therefore, (h) will be referredgto as the variogram of complementarity. The conceptof complementarity is statistically equivalent to the dis-tinctness or dissimilarity of species composition. As aconcept, however, it captures the sense that comple-mentary faunas or floras form parts of a whole, so thatcomplementarity is a positive biodiversity component(Vane-Wright et al. 1991, Colwell and Coddington1994).

In order to obtain a spatial definition of the varianceof species richness, the covariances between species

need to be expressed in terms of cross-variograms. Anempirical cross-variogram ij(h) describes the distance-gdependent covariance between two species, i and j(Isaaks and Srivastava 1989):

1g (h) 5 (x 2 x )(x 2 x ). (6)Oi j ia ib ja jb2n a,b zh øhh ab

Based on Eqs. 3 and 6, the variance of species rich-ness can be expressed as spatial covariance (note thatthe ‘‘semivariance’’ in a variogram is equal to the var-iance of a single species occurrence, complementarity,or species richness):

(x 2 x )(x 2 x )1 ia ib ja jbVar(S) 5 O On 2i j ab

n nh h5 g (h) 5 g (h). (7)O O OS ijn nh h ij

As the worked example in the Appendix shows, Eq.7 provides the empirical variance of species richnessS. In geostatistical analysis, however, only distances.0 are analyzed, so that a quadrat is never comparedto itself. Under this condition that a ± b, N quadratswill provide n 5 N(N 2 1)/2 unique pairs of quadratsa and b, and Eq. 7 results in the unbiased estimator ofthe variance of species richness S.

Eq. 7 includes the definitions of the variance of com-plementarity, pairs of species, and an individual speciesas special cases. The condition i 5 j leads to the var-iance of complementarity; for a given pair of speciesi and j, Eq. 7 results in their cross-variogram; and fora single species i 5 j, it describes the variance of thevariable xi.

The basic element of spatial covariance, ij(a,b), fol-glows from Eq. 7. It is half the product of the observeddifferences between quadrats a and b for two speciesi and j:

1g (a, b) 5 (x 2 x )(x 2 x ). (8)i j ia ib ja jb2

The variogram matrix

The spatial covariance can be summarized in a setof distance-dependent variance–covariance matricesC(h). The matrix elements cij(h) 5 ij(h) are calculatedgfrom Eq. 6. The term variogram matrix of Myers (1997)will be used for such a set of distance-dependent var-iance–covariance matrices C(h). The variogram matrixcan be interpreted in various ways (Fig. 2):

1) A plot of a diagonal element cii(h) against distanceh is the empirical variogram of species i. Its ex-pected value for independent observations in ahomogeneous environment is pi(1 2 pi).

2) A plot of cij(h) against distance h is the empiricalcross-variogram of species i and j. Its expectedvalue for independent observations in a homo-


FIG. 2. Schematic representation of a variogram matrix.A variogram matrix contains a separate variance–covariancematrix C(h) for each distance class h, up to an arbitrarilydefined maximum distance hmax. Each cell of a matrix C(h)contains an estimate of the variance of one species i [diagonalcells, cii(h)] or of the covariance of a pair of species i and j[off-diagonal cells, cij(h)], based on all pairs of observationsthat are separated by a distance of approximately h. The ar-rows indicate further geostatistical interpretations that are dis-cussed in the text.

geneous environment and independent species iszero.

3) A plot of the sum of the diagonal of C(h), Si cii(h),against distance h is the empirical variogram ofcomplementarity. Its expected value for indepen-dent observations in a homogeneous environmentis Si pi(1 2 pi).

4) A plot of the sum of C(h), Sij cij(h), against dis-tance h is the empirical variogram of quadrat spe-cies richness S. Its expected value for independentobservations in a homogeneous environment andindependent species is Si pi(1 2 pi).

Often, researchers want to test whether the observedautocorrelation is significantly different from random.A randomization test is constructed by randomly per-muting the observations. The easiest way is to permutethe couplets of x and y coordinates (Legendre and Le-gendre 1998). For each of r permutations, a variogrammatrix is derived, providing a reference distribution ofr values for each matrix element under the null hy-pothesis of spatially independent observations.

The standardized variogram

For binary variables such as species presence–ab-sence data, the expected variance is a function of themean. This can be used to make variograms from dif-ferent areas with varying environmental conditionscomparable by scaling the observed spatial covariancegij(a, b) in terms of the expected spatial covariance.The method corresponds to a pairwise relative vario-

gram (Isaaks and Srivastava 1989), and results in astandardized variogram with an expected sill of 1, in-dependent of the mean and variance of a species.

The variance of a binary species variable i with anunderlying mean of pi is pi(1 2 pi). Hence the stan-dardized variogram (h) of species i isg9i

21 (x 2 x )ia ibg9(h) 5 . (9)Oi 2n p (1 2 p )a,b zh øhh i iab

If the mean pia of species i depends on the quadrata, the expected semivariance gij(a,b) of a pair of quad-rats a and b is

p (1 2 p ) 1 p (1 2 p )ia ib ib iaE{g (a, b)} 5 (10)i j 2

and Eq. 9 extends to

21 (x 2 x )ia ibg9(h) 5 . (11)Oi n p (1 2 p ) 1 p (1 2 p )a,b zh øhh ia ib ib iaab

If pi is estimated from the data, the unbiased varianceestimator should be used for the denominator. Thismeans that (h) in Eq. 9 needs to be multiplied byc9ii(N 2 1)/N, where N is the total number of quadrats inthe sample:

1p 5 x (12)Oi iaN a

2N 2 1 1 (x 2 x )ia ibg9(h) 5 . (13)Oi N 2n p (1 2 p )a,b zh øhh i iab

The standardized variogram of complementarity,g9(h), with estimated means pI is

g (h)O iN 2 1 ig9(h) 5 . (14)N p (1 2 p )O i i

i

Under the assumption of stationarity, the autocor-relation function ri(h) of a species i can be expressedin terms of its semivariance gi(h) and its semivariancefor independent samples, gi(`), i.e., its global varianceor sill (Isaaks and Srivastava 1989):

g (`) 2 g (h) g (h)i i ir (h) 5 5 1 2 . (15)i g (`) g (`)i i

Under the assumption that the underlying means pi

or pia are known or correctly estimated, the standard-ized variogram eliminates the effect of nonconstantmeans and variances. This means that the stationaritycriteria are met and the standardized semivariance

(h)provides an estimator for the autocorrelation func-g9ition ri(h):

r (h) 5 1 2 g9(h).i i (16)

Under the same conditions, the standardized cross-covariance (h) of two species i and j is an estimatorg9ijof their cross-correlation function rij(h):


FIG. 3. Grid design of the 1-m2 quadrat data from theOosting Natural Area in Duke Forest, Orange County, NorthCarolina (Palmer and White 1994). The quadrats of the largegrid are located in the lower left corner of each module of16 3 16 m. Each of the three small grids contains 16 3 16contiguous quadrats within a randomly selected module ofthe large grid. The underlying contour plot reflects relativeelevation in intervals of 1 m, ranging from 0–1 m (dark grey)to 10–11 m (white). The contour plot is derived from ele-vation measurements at the corner points of each module.

r (h) 5 g9 (h)i j i j

(x 2 x )(x 2 x )N 2 1 1 ia ib ja jb5 .O

N 2n a,b zh øh Ïp (1 2 p )Ïp (1 2 p )h ab i i j j

(17)

Testing the spatial covariance of species richness

The observed spatial covariance of complementarityprovides an empirical null model of species indepen-dence for the spatial covariance of species richness.The two are affected by spatial aggregation in the in-dividual species variables and species-specific responseto environmental heterogeneity in exactly the sameway. The only difference is in the associations or co-variances between species, which again can be distancedependent. The test statistic is the ratio of the observedand the expected variance, as in the nonspatial variancetest of species richness (Schluter 1984, McCulloch1985, Palmer and Van der Maarel 1995). A varianceratio smaller than 1 is a potential indicator for nichelimitation. The ratio can be derived by first calculatingthe empirical variograms of species richness (‘‘ob-served variance’’) and of complementarity (‘‘expectedvariance’’) before dividing the aggregate values foreach distance class. Alternatively, the observed vari-ance can be divided by the expected one for each pairof observations, as in a pairwise relative variogram(Isaaks and Srivastava 1989). A permutation test forthe null hypothesis of independent species is con-structed by independently permuting each species vec-tor xi.

The spatial structure of ordination axes

Multiscale ordination has been proposed for inves-tigating the scale of multispecies patterns, such as acyclic spatial structure of a gradient in species com-position. In multiscale ordination (Noy-Meir and An-derson 1971, Ver Hoef and Glenn-Lewin 1989, Dale1999), blocked-variance techniques are used to cal-culate a variance–covariance matrix C(b) of the speciesfor each of a range of block sizes b. The matrices arethen summed into an overall variance–covariance ma-trix C, which is subjected to principal components anal-ysis (PCA). For each PCA axis, the associated varianceis partitioned by block size and plotted against blocksize.

The method can be adapted to the concept of spatialcovariance and the distance-dependent variance–co-variance matrices C(h) of a variogram matrix. A matrixC(0) for distance class h 5 0 is included and all ma-trices C(h) are weighted with wh 5 nh /n. The summedmatrix ShwhC(h) is equal to the global empirical var-iance–covariance matrix C that is used in PCA. Theeigenvalue lf of PCA axis f is partitioned among dis-tance classes, h, by multiplying its associated eigen-vector uf with each of the matrices C(h) (cf. workedexample in the Appendix):

Tl (h) 5 u C(h)u .f f f (18)

A plot of lf(h) against distance h provides a vario-gram of axis f. It reflects the spatial covariance of com-plementarity explained by PCA axis f, or the differencein species composition due to the intrinsic gradientdescribed by f.

The eigenvalue lf of PCA axis f is a weighted meanof its distance-dependent components lf(h):

nhT Tl 5 u Cu 5 u C(h)uOf f f f fnh

5 w l (h) f ∈ {1, . . . , s}. (19)O h fh

Examples from the Oosting Natural Area

The approach is illustrated with previously publisheddata from a mixed hardwood–pine forest in North Car-olina (Reed et al. 1993, Palmer and White 1994, Palmer1995, Jonsson and Moen 1998). The study area in theOosting Natural Area of the Duke Forest, OrangeCounty, North Carolina, contains several forest com-munities with gradual transitions. The entire datasetdescribes the presence–absence of understory vascularplant species in a nested series of quadrats within asampling grid of 256 modules of 16 3 16 m. For thispaper, I analyzed the 1-m2 quadrats from the main grid,


placed in the lower-left corner of each module, andfrom the three randomly selected modules within whichevery square meter was studied (Fig. 3). In addition toplant community data, I used data on relative elevationand pH measurements obtained at the corners of eachmodule (Reed et al. 1993). For the small grid 3, I es-timated relative elevation at the lower-left corner ofeach 1-m2 quadrat by linear interpolation.

To explore the effects of environmental heteroge-neity, I calculated a variogram of complementarity foreach of the three small grids and for the large grid byaveraging half the squared Euclidean distance per dis-tance class. To make the variograms directly compa-rable, I restricted the analysis to the species that oc-curred at least five times in each of the four data sets.A 95% confidence interval for each variogram and dis-tance class was estimated from the distribution of thebasic elements of spatial covariance (Eq. 8). For everysmall grid, I estimated the mean pi of each of the 25species (Eq. 12) and derived the standardized vario-gram of complementarity (Eq. 14).

Further analysis focused on the small grid 3, usingall of its 72 species variables. I tested the significanceof the departure of the variogram of species richnessfrom the variogram of complementarity by a two-sidedpermutation test (a 5 0.05) with 499 permutations ofthe original observations, permuting each species vec-tor independently. For each permutation, the ratio ofthe variance of species richness to the variance of com-plementarity was calculated for each distance class.The permutations thus provided a reference distributionfor the ratio of each distance class under the assumptionthat the species are independently distributed. For thevariograms of PCA axes, a variogram matrix was con-structed by assembling the variograms and cross-var-iograms calculated from Eqs. 4 and 6 in a set of dis-tance-dependent matrices C(h). The original data fromthe small grid 3 were subjected to PCA. Because a totalof 23 axes had eigenvalues lf . 1, I used a scree test(Cattell 1966) to determine the number of PCA axesto be retained. As l4 lay below the regression line fittedto l2 2 l23, the first three axes were retained for anal-ysis. All axes were partitioned by distance class usingEq. 18. A variogram of complementarity, accountingfor the variance along PCA axis 1, was obtained bysumming the variograms of all other PCA axes by dis-tance class.

All calculations were performed in S-Plus (Beckeret al. 1988).

RESULTS

The effect of environmental heterogeneity onspatial covariance

How strongly does local environmental heteroge-neity influence spatial covariance? The Oosting dataprovide an excellent example, as the four grids weresampled from a total area of only 256 3 256 m. The

study area contained some environmental heterogene-ity (cf. Fig. 3); e.g., the mean relative elevation of thefour corner points of the small grids 1, 2, and 3 was3.7 m, 8.9 m, and 7.1 m, and their respective averagepH was 6.5, 5.7, and 5.9.

Fig. 4 shows the empirical variograms of comple-mentarity for each of the four grids, including only the25 species that occurred at least five times in each grid.Although the four data subsets contained exactly thesame species and originated from the same small studyarea, their variograms of complementarity differedstrongly (Fig. 4). The variogram for the large gridshowed a continuous rise without reaching a sill, in-dicating larger scale heterogeneity. The curves for thethree small grids showed more or less parallel curvesbut approached very different sills.

The expected variance estimated from the speciesmeans in each grid predicted the different sills fairlywell. This means that between-grid heterogeneity isresponsible for the observed difference in the sills. Thestandardized variograms of complementarity in Fig. 5(note the different scaling of the x axis as compared toFig. 4) support this interpretation, as the curves for thethree grids coalesced after accounting for variable spe-cies means.

Distance dependence of the variance ofspecies richness

Does the spatial version of the variance test of spe-cies richness provide more or better information thanthe global test? In the example of the small grid 3, theglobal variance of species richness was only slightlylower than expected under the hypothesis of indepen-dent observations and independently distributed spe-cies. Based on the global variance test, the differencewas clearly not statistically significant (chi-square 5248.76, df 5 256, P 5 0.384). The plot against dis-tance, however, revealed a systematic change from pre-dominantly negative covariances between species atsmall distances to mainly positive covariances at largerdistances (Fig. 6). The permutation test showed sig-nificant departures of the two curves for the smallestand the largest distance classes. Hence the spatial var-iance test of species richness was able to detect a var-iance deficit at small distances undetected by the globaltest.

Spatial structure of intrinsic gradients

Indirect ordination methods such as PCA describevegetation as the sum of overlapping, statistically un-correlated intrinsic gradients. What does multiscale or-dination tell us about the spatial structure of these gra-dients? Fig. 7 shows the variograms of PCA axes 1–3for the small grid 3. The variance along the first axisincreased strongly and continually with distance, in-dicating the presence of larger scale trend. A strongcorrelation between axis scores and the interpolatedrelative elevation (Pearson correlation, r 5 0.73, P ,


FIG. 4. Empirical variograms of complementarity for the large grid and for each of the three small grids, calculatedindependently for each grid. The point symbols represent the average observed semivariance per distance class, based onthe 25 plant species that were present in at least five cells of each of the four grids. Error bars mark the 95% confidenceinterval for the mean of each distance class. The solid lines indicate the expected variance based on the species means withineach grid.

0.001) suggests that the variance along PCA axis 1 islargely due to environmental heterogeneity. All otherfactors showed only a modest increase with distance(e.g., axis 2), or even suggested slightly cyclic patterns(e.g., axis 3) as one would expect to result from com-munity-level processes.

How important are the factors contributing to thevariance of species richness? Fig. 8 illustrates for thesmall grid 3 how multi-scale ordination can be used topartition the variance in a dataset into distance-depen-dent components of interspecific interactions, largerscale trend as caused by environmental heterogeneity,and single-species aggregation patterns. The strong in-crease of the variance of species richness with distancewas largely due to interspecific interactions and thetrend reflected in PCA axis 1. Only a small portion ofthe remaining variance could be explained by single-species aggregation patterns, peaking at ;7 m. Theobserved variance of complementarity without axis 1appeared to oscillate around its global variance, indi-cating that all larger scale trend had been accountedfor by removing PCA axis 1.

DISCUSSION

Spatial analysis of plant communities in aheterogeneous environment

Can variograms from different communities be com-pared? The Oosting example illustrates the drastic ef-fect that within-site environmental heterogeneity canhave on spatial analysis. Researchers often assume ho-

mogeneity over relatively small study areas. However,heterogeneity may occur at all scales, leading to dif-ferences in the mean and variance of species variablesacross space. The empirical variogram of the large griddid not reach a sill, indicating a violation of the second-order stationarity assumption (e.g., Bellehumeur andLegendre 1998). The three intensively sampled mod-ules of 16 3 16 m, separated by 150–250 m, differedin their species composition and in the frequency ofoccurrence of the more abundant and ubiquitous spe-cies. Even when the analysis was restricted to the lattergroup, the resulting variograms of complementaritydiffered strongly in their sills due to the differences inspecies means between the grids.

The spatial description by a generalized variogramor ‘‘dissimilogram’’ (e.g., Mistral et al. 2000) wouldnormally stop here. The mathematical approach de-veloped here, however, allowed the different sills to bepredicted from the observed species means so that thevariograms could be made comparable by standardi-zation. The standardized variograms of the three smallgrids were relatively well behaved, although all ap-peared to reach sills slightly .1, indicating some un-accounted internal heterogeneity. In the standardizedform, the three curves coalesced and thus provided ageneral description of small-scale autocorrelation inspecies composition within the study area, independentof environmental conditions.

The variogram matrix provides two ways for dealingwith larger scale heterogeneity. Eq. 11 can be used to


FIG. 5. Standardized variograms of comple-mentarity (cf. Eq. 14), derived independentlyfor each of the small grids. Each point repre-sents, for a distance class, the average of theobserved squared Euclidean distance betweenobservations divided by the expected variancebased on the mean of each species in the grid.The expected sill of a standardized variogramis 1.

FIG. 6. Variograms of complementarity andof species richness for the small grid 3, basedon all 72 species observed in that grid. Signif-icance of the departure of the variogram of spe-cies richness from the variogram of comple-mentarity was determined by a two-sided per-mutation test (see Model structure and methods:Examples from the Oosting Natural Area). Theobserved global variance (solid line) is the non-spatial variance of quadrat species richness, andthe expected global variance (dashed line) is thesum of the (nonspatial) species variances, asused in the global version of the variance testof species richness.

predict and account for the expected spatial covariancebased on any given trend model, which may be obtainedby local interpolation or by modeling species responseto known environmental gradients. Multiscale ordina-tion provides a direct way of separating variance at-tributed to a ‘‘true gradient’’ from the variance of‘‘false gradients,’’ as determined by the variograms ofordination axes. This approach needs to be extendedto direct gradient analysis with Redundancy Analysis(RDA; Rao 1964, Legendre and Legendre 1998), wherePCA axes are linear combinations of observed envi-ronmental variables.

On purely empirical grounds, however, it is not pos-sible to distinguish between trend, or systematic var-iation of the mean, and autocorrelation, or local de-pendence of the departure from the mean. The ap-pearance of any spatial pattern depends highly on thegrain and extent of a study. The same spatial structuremay appear as trend in a fine-scale study and as patternof a specific scale if a broader extent is considered.

Advantages of a spatial variance test ofspecies richness

The example in Fig. 5 clearly illustrates the distance-dependent nature of interspecific interactions in a com-munity and their effect on the variance of species rich-ness. A global, nonspatial variance test is prone to missthe systematic departure of the variance of species rich-ness from its expected value if negative covariances atsmall distances are cancelled out by positive covari-ances at larger distances. The different results of thetwo tests do not merely reflect their statistical power,i.e., their ability to detect a given effect with a sampleof a certain size. Here, the effect itself, the global sumof the associations, increased with increasing distancebetween observations. In such a situation, the chancesof detecting a variance deficit with the global test arelikely to decrease with increasing sample size, as largerdistances are included.

The negative covariance at short distances may in-dicate niche limitation, but it may also be an effect of


FIG. 7. Variograms of PCA axes for thesmall grid 3. The lines connect the point esti-mates of the variance along the first, second,and third axis, respectively, as partitioned bydistance using Eq. 18. The percentage of thetotal variance explained by the first three axesis indicated in parentheses.

FIG. 8. Partitioning of the variance of species richness of the small grid 3 into the additive contributions of interspecificinteractions, environmental heterogeneity as reflected in PCA axis 1, and single-species aggregation patterns. The pointsymbols represent, for each distance class, the variance of species richness, complementarity, and the variance of comple-mentarity after accounting for PCA axis 1 as described in the text.

cell size, or of rarefaction (Økland 1994), the physicallimitation of the number of individual plants and thusthe number of species within relatively small areas. Theincrease in positive covariance with distance is prob-ably due to increasing environmental heterogeneity,which is one of the factors determining niche limita-tion.

The variogram of complementarity provides a nullmodel that already includes the autocorrelation due tosingle-species aggregation patterns. Hence, it elimi-nates the need for keeping the spatial pattern of each

species constant while permuting observations. Whilethis statement is based on mathematical considerations,it remains to be verified by a thorough empirical com-parison to published permutation tests (Palmer and Vander Maarel 1995, Roxburgh and Matsuki 1999).

The spatial variance test of species richness elimi-nates two former impediments of observational studiesof the variance deficit in species richness, namely theconfounding of negative associations at short distanceswith positive interactions at larger distances, and thespatial autocorrelation in the distributions of the in-


dividual species. Hence, it opens the door for a sub-stantially new approach to the research on niche lim-itation.

A geostatistical perspective on multiscale ordination

This paper proposes two important deviations fromearlier presentations of multiscale ordination. First,Noy-Meir and Anderson (1971) suggested a simple ad-dition of the variance matrices for each block size toobtain a global matrix that is subjected to PCA. Dale(1999) recommended weighting by the expected inten-sity, or amplitude of a cyclic variance, as a function ofblock size. Here, I suggest weighting the distance-de-pendent variance–covariance matrices by the numberof pairs of observations in each distance class. Thisrecreates the global, nonspatial variance–covariancematrix, which is used not only in PCA but in manymultivariate methods (see worked example in the Ap-pendix). This compatibility may lead to further inte-gration of geostatistical modeling with nonspatial mul-tivariate methods.

Second, in a geostatistical framework, the block sizeof the original definition of multiscale ordination cor-responds to distance. While block sizes are defined bythe analytic method, distances are a property of thesampling design and may take any value. This is a greatadvantage as it extends multiscale ordination to non-systematic spatial samples, although the sampling de-sign will still determine whether a specific pattern inan ecological community is detected by the method.

The often arbitrary choice of distance classes may,however, affect the shape of a variogram. The geostatis-tical solution is to analyze the variogram cloud insteadof the empirical variogram defined by distance classes.The variogram cloud is a scatter plot where each dotrepresents the semivariance of a single pair of obser-vations. Such a plot can easily be constructed by plot-ting the basic elements of spatial covariance (Eq. 8).

From a geostatistical perspective, there is anotherimportant extension. Spatial autocorrelation is oftenanisotropic, that is, it depends on the geographic di-rection in which it is measured. If the sampling designis two dimensional and the sample is large enough,directional variance–covariance matrices C(h,d ) can becalculated, e.g., for four directional sectors d. However,I would expect that most cases of anisotropy in eco-logical data sets are due to environmental heteroge-neity, so that a removal of larger scale trend wouldusually eliminate the need for anisotropic models.

Assumptions and limitations

Many readers may feel uncomfortable about theheavy reliance on the Euclidean distance. A main con-cern is that the Euclidean distance does not reflect thespecial nature of the value zero in biotic data. Althoughthe Euclidean distance is especially unsuitable for cov-er or abundance data that include zeros (Legendre andLegendre 1998), it is more robust for presence–absence

data, as illustrated in the following example: Assumethat the respective abundance of species i in three quad-rats a, b, and c is 4, 1, and 0. Based on the squaredEuclidean distance, D2, b and c are more similar (D2

5 1) than are a and b (D2 5 9), although a and b sharespecies i, which is absent from c (Legendre and Le-gendre 1998). However, converted to presence–absencedata, b and c are more different (D2 5 1) than a andb (D2 5 0).

A zero may be stochastic (the sampling unit repre-sents suitable habitat, but is unoccupied) or it may bestructural (unsuitable habitat, where the species cannotoccur). Structural zeros, which can inflate observedspecies associations, occur if the sample includes alarge degree of environmental heterogeneity. In such asituation, local means should be accounted for in themodeling of variograms (Eq. 11). If a multiscale or-dination is performed, this adjustment may not beenough. PCA assumes that the individual species arelinearly related to the main gradients in species com-position. If there is a considerable degree of beta di-versity, as a result of environmental heterogeneity, spe-cies may appear and disappear along a compositionalgradient. In order to accommodate such a unimodalresponse or optimum curve, the variogram matrix ap-proach needs to be generalized and extended to a chi-square measure of distance that is compatible with cor-respondence analysis (CA; cf. Legendre and Legendre[1998] for an overview of chi-square based ordinationmethods). An approach based on the chi-square dis-tance would also accommodate abundance data.

The Euclidean distance has some distinct advantag-es. First, it leads to a direct mathematical extension ofthe variogram and is compatible with multivariategeostatistical methods used in other fields for quanti-tative variables (e.g., Goovaerts [1994] in soil science,Myers [1997] in remote sensing). Second, it mathe-matically links the spatial structure of individual spe-cies variables, of complementarity, and of species rich-ness. A similar partitioning may be possible for otherresemblance measures, such as the chi-square distance.The direct ecological interpretations, however, and spe-cifically the explicit decomposition of the variance ofspecies richness, are unique to the Euclidean distance.

The biggest advantage of the Euclidean distance isits general (though often implicit) use in statistics,which makes the variogram matrix approach highlygeneralizable. Any variable that is the sum of othervariables can be defined and interpreted as the sum ofa variance–covariance matrix. If variance is calculatedfrom pairwise differences instead of individual differ-ences from the mean, any variance or variance–co-variance matrix can be partitioned by distance.

A last concern is that the observed spatial covariancewill strongly depend on quadrat size, as Palmer andWhite (1994) illustrated with additional data collectedat eight nested quadrat sizes from the Oosting studyarea. Scale dependence has many faces and aspects,


which can be related to the two basic components ofscale, grain, and extent (Wiens 1989). While mostblock-size variance techniques manipulate grain, orquadrat size, geostatistical analysis focuses on extent,or distance between observations. A parallel geosta-tistical analysis at different cell sizes may help to un-tangle the effects of grain and extent on ecologicalpatterns and processes.

Conclusions

Variance in ecological communities and systems isspatially structured. This is true for abiotic and bioticfactors, for populations, communities, and biodiversity.The ecological literature most often treats spatial struc-ture as a problem rather than as information, stressingproblems for significance tests due to spatial autocor-relation instead of the additional insights that can begained from spatial analysis (cf. Legendre 1993). Whenecologists analyze variation in plant communities, theyeither focus on plant–environment interactions usingordination methods that ignore spatial structure, or theydescribe the pattern in one or two species without tak-ing into account the spatial structure caused by envi-ronmental factors or community-level processes.

The composition and diversity of a plant communityresults from processes at different levels of biologicalorganization operating in a spatially structured envi-ronment. As the Oosting example illustrated, the pat-terns created by abiotic and biotic processes overlapto form a complex spatial covariance structure that isdifficult to compare or interpret. However, the jointanalysis of single-species, interspecific, and abiotic ef-fects revealed component patterns that were highly in-terpretable. This shows that an integrated methodolog-ical approach is needed to understand what determinescommunity structure, and ultimately, why species co-exist.

The variogram matrix is a geostatistical extension ofmultiscale ordination. It partitions the variance in eco-logical communities into spatial components on thelevels of individual populations, species composition,and species richness, and can be used to factor out theeffects of single-species aggregation patterns, inter-specific interactions, or environmental heterogeneity.By integrating three traditionally unrelated methods, itincreases the interpretability of variograms of plantcommunities, provides a spatial extension and an em-pirical null model for the variance test of species rich-ness, and extends multiscale ordination to nonsystem-atic spatial samples.

Beyond the individual applications, the variogrammatrix provides a framework for the mathematical uni-fication of geostatistics, multivariate data analysis, andthe analysis of variance. An integration of geostatisticswith general multivariate statistical methods may en-able ecologists from a broad range of fields to incor-porate spatial structure and processes into their re-

search and to integrate analyses across different levelsof biological organization.

ACKNOWLEDGMENTS

This research is part of a project funded by the Environ-mental Protection Agency (EPA) under STAR grant No.R826764-01-0. Additional funding came from the Swiss Na-tional Science Foundation (SNF), NCCR Plant Survival, grantNo. 5005-65748. Mike Palmer, Oklahoma State University,provided the Oosting data sets and many valuable comments.Robert Peet, University of North Carolina at Chapel Hill,provided the Oosting elevation data. Jennifer Hoeting, JohnWiens, Bea Van Horne, and Jonathan Bossenbroek, all atColorado State University, P. A. Burrough at University ofUtrecht, and Sucharita Ghosh at WSL Swiss Federal ResearchInstitute, provided important critique and support. I wouldlike to thank David Roberts, Bengt Gunnar Jonsson, and ananonymous reviewer for their valuable comments.

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APPENDIX

A worked example of spatial covariance in plant communities is available in ESA’s Electronic Data Archive: EcologicalArchives E084-023-A1.

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