Analyzing or Explaining Beta Diversity

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CONCEPTS & SYNTHESISEMPHASIZING NEW IDEAS TO STIMULATE RESEARCH IN ECOLOGY

Ecology, 87(11), 2006, pp. 26972708 2006 by the Ecological Society of America

ANALYZING OR EXPLAINING BETA DIVERSITY? UNDERSTANDING THETARGETS OF DIFFERENT METHODS OF ANALYSIS

HANNA TUOMISTO1 AND KALLE RUOKOLAINEN

Department of Biology, University of Turku, FI-20014 Turku, Finland

Abstract. It has been actively discussed recently what statistical methods are appropriatewhen one is interested in testing hypotheses about the origin of beta diversity, especiallywhether one should use the raw-data approach (e.g., canonical analysis such as RDA andCCA) or the distance approach (e.g., Mantel test and multiple regression on distancematrices). Most of the confusion seems to stem from uncertainty as to what is the responsevariable in the different approaches. Here our aim is to clarify this issue. We also show that,although both the raw-data approach and the distance approach can often be used to addressthe same ecological hypothesis, they target fundamentally different predictions of thosehypotheses. As the two approaches shed light on different aspects of the ecological hypotheses,they should be viewed as complementary rather than alternative ways of analyzing data.However, in some cases only one of the approaches may be appropriate. We argue that S. P.Hubbells neutral theory can only be tested using the distance approach, because its testablepredictions are stated in terms of distances, not in terms of raw data. In all cases, the decisionon which method is chosen must be based on which addresses the question at hand, it cannotbe based on which provides the highest proportion of explained variance in simulation studies.

Key words: beta diversity; canonical analysis; community composition; ecological hypotheses; Manteltest; multiple regression; multiple regression on distance matrices; spatial variation; species abundances;variation partitioning.

INTRODUCTION

The question of what factors affect community

composition and its variation (beta diversity) has been

of considerable interest to biologists. Although the

concept of beta diversity dates at least to Whittaker

(1960, 1972), interest in it has increased dramatically

since the publication of Hubbells book on the neutral

theory of biodiversity (Hubbell 2001). The neutral

theory challenged the widely held view that environ-

mental factors and ecological-niche differences between

species are the most important factors in determiningwhere species occur and at what abundances. Instead,

the neutral theory proposes that species abundances

fluctuate in a random walk due to random mortality and

stochastic but spatially restricted dispersal.

Multivariate-analysis methods that allow studying

questions related to beta diversity include canonical

analysis (or constrained ordination; e.g., RDA [redun-

dancy analysis] and CCA [canonical correspondence

analysis]) and the Mantel test and its derivatives.

Canonical analysis can be called a raw-data approach,

because there the input data are in the form of raw-data

tables, such as estimates of species abundances at study

sites and measurements of environmental variables at

the same study sites. The Mantel test can be called a

distance approach, because there the input data are in

the form of distance matrices that are based on the raw

data. Both approaches have been extensively used in the

ecological literature, with hundreds of ecological papers

mentioning the Mantel test, redundancy analysis, or

canonical correspondence analysis (easily verified by a

simple search in, e.g., ISI Web of Science or BIOSIS

Previews).

Variation partitioning provides the statistical means

to quantify the relative effects of different groups of

explanatory variables on the response variable of

interest. As proposed by Borcard et al. (1992), variation

partitioning can be based on RDA or CCA to partition

the variation in a species 3 sites raw-data table to

fractions uniquely or jointly explained by variation in

environmental and spatial variables. This kind of

variation partitioning represents the raw-data approach.

Manuscript received 4 January 2006; revised 31 March 2006;accepted 18 April 2006. Corresponding Editor: N. C. Kenkel.

1 E-mail: [email protected]

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More recently, variation partitioning has been extended

to the distance approach by using multiple regression on

distance matrices (Duivenvoorden et al. 2002, Tuomisto

et al. 2003).

The raw-data and distance approaches have tradi-

tionally been used to study similar ecological questions,

and indeed they have been considered alternative

methods for the same purpose (e.g., Legendre 1993).However, recently a discussion has arisen as to which of

the approaches is more appropriate when one is

interested in beta diversity. This question, called the

dilemma between the raw-data approach and the

distance approach, was the main topic of a recent paper

by Legendre et al. (2005).

The main conclusion of Legendre et al. (2005) was

that the proper statistical procedure for testing hypoth-

eses about the origin and maintenance of variation in

community composition among sites is canonical

variation partitioning, and that partitioning on distance

matrices should not be used to study the variation in

community composition among sites. We disagree withthis conclusion, and find that although their paper made

several valuable points that clarified differences between

the analysis approaches, it also confused some impor-

tant concepts. This led to inconsistencies and errors in

their recommendations. We especially disagree with

their suggestion that the raw-data approach is preferable

over the distance approach for the testing of Hubbells

neutral theory.

Most of the confusion in the dilemma between the

raw-data approach and the distance approach seems to

stem from uncertainty as to what is the response variable

in the different analyses. Here we attempt to clarify the

situation by evaluating which ecological and statistical

questions each analysis approach actually targets, and

what should be taken into account when selecting an

analysis method for a particular purpose.

LEVELS OF ABSTRACTION

The basic concepts

We start by specifying at which levels one can ask

ecological questions that are related to the distribution

of species along environmental and spatial gradients. We

distinguish three levels of abstraction (Fig. 1). The first,

basic level is formed by the raw-data tables, which

consist of the observations of the abundances of one ormore species (A1 to Ap) in more than one study site (s1 to

sn), in which the values of one or more environmental

variables and spatial coordinates (x1 to xm) have also

been measured. The second level of abstraction is

derived from the first level and consists of the variation

in the raw-data tables. The third level of abstraction is

derived from the second level and consists of the

variation in the variation in the raw-data tables; i.e.,

(1) raw data ! (2) variation in the raw data ! (3)

variation in the variation in the raw data.

In the case of the species-data table, the sequence can

equally well be written as follows: (1) community

composition! (2) variation in community composition

! (3) variation in the variation in community compo-

sition. Here the term community composition is used

so that it encompasses both species composition and

species abundances. Variation in community composi-

tion across sites is beta diversity. Overall beta

diversity in the data set can be measured with the sum

of squares (SS) of the raw data. The SS or other beta-diversity indices (which measure the difference in species

composition and species abundances between sites) can

also be computed for all different site pairs and used to

construct a dissimilarity matrix; the mean of the cell

values in this matrix is also a measure of overall beta

diversity (Whittaker 1972, ter Braak 1983, Vellend 2001,

Legendre et al. 2005; Fig. 1). Therefore, the sequence of

the levels of abstraction can also be written as follows:

(1) community composition ! (2) beta diversity ! (3)

variation in beta diversity.

This sequence is conceptually analogous to the

relationship between position, velocity and acceleration

in physics: (1) position (geographical coordinates)! (2)

velocity (variation in position over time) ! (3) accel-

eration (variation in velocity over time, i.e., variation in

variation in position over time).

Legendre et al. (2005) also developed their arguments

from a three-level framework, but their levels were (1)

variation in species identity within communities (alpha

diversity), (2) variation in community composition

among sites (beta diversity), and (3) variation in beta

diversity among groups of sites. However, we think that

this does not provide an optimal framework for

clarifying the concepts, because alpha diversity is not a

logical starting point for deriving beta diversity in the

same way as community composition is. There is no

simple relationship between alpha diversity and either

community composition or beta diversity. If two sites

have exactly the same number of species (in exactly the

same proportions of abundance), their alpha diversities

are identical, but their community compositions can be

anything from identical to completely different, and beta

diversity can hence be anything between 0% (if all

species are shared between the sites in similar abundan-

ces) and 100% (if no species are shared). Therefore, our

level of abstraction 1 is different from that of Legendre

et al. (2005), but the levels of abstraction 2 and 3 are the

same.

Application to ecological questions

Now, let us turn to the kind of ecological questions we

may be interested in studying.

We may want to analyze the abundance of a single

species A, in which case we pay attention to just this

species in the community-composition raw-data table.

Then we are concerned with the questions: Why do some

sites have a higher abundance of species A than others?

i.e., Why is there variation in the abundance of species

A? Can the variation in the abundance of species A be

explained by variation in environmental characteristics

HANNA TUOMISTO AND KALLE RUOKOLAINEN2698 Ecology, Vol. 87, No. 11


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or geographical location of the sites? If so, then we can

predict the abundance of species A at a site if we know

the values of environmental variables at the site and its

geographical position. The response variable y is theabundance of species A at study sites s1 to sn, and the

independent variables x1 to xm are the environmental

variables and geographical coordinates measured at the

sites. Standard methods to test whether the variables are

statistically dependent on each other include correlation

analysis (where no distinction is made between response

and independent variables) and multiple regression

(where the variation in the independent variables is used

to explain the variation in the response variable in a

causal framework). The regression models that are fitted

can be either linear or more complex (e.g., Huisman et

al. 1993, Legendre and Legendre 1998, Oksanen and

Minchin 2002, Karadzic et al. 2003). This is what

Legendre et al. (2005) called the raw-data approach,

which is suitable to answering what they called level-2questions. In terms of our levels of abstraction, in this

approach we analyze data from the level-of-abstraction

1 to test whether we can explain its variation, which is

expressed at level-of-abstraction 2 (Fig. 2).

In the present case we may rather want to analyze the

abundances of all observed species at a time, i.e.,

community composition. Then we are concerned with

the questions: Why do some sites have a different

community composition than others? i.e., Why is there

variation in community composition? Can the variation

in the abundances of species A1 to Ap, which form the

FIG. 1. Three levels of abstraction in studies concerning community composition and beta diversity. The overall amount of betadiversity in the raw-data table (whose cell values are the abundances ofp species in n sites) can be summarized with a single numberat the level of abstraction 2. This measure can be the sum of squares ( SS) of the raw-data table, or the mean of the cell values in adistance matrix. The cell values in the distance matrix can be pairwise SS values or values of any other measure that quantifies thedissimilarity in species composition and abundances (community composition) between the two sites in each site pair. The overallamount of variation in beta diversity in the distance matrix can be summarized with a single number at the level-of-abstraction 3.

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community, be explained by variation in environmental

characteristics or geographical location of a site? If so,

then we can predict the abundances of species A1 to Ap

(community composition) at a site if we know the valuesof environmental variables at the site and its geo-

graphical position. The ecological question is similar to

the previous one, but now there are several response

variables instead of just one. A standard method to test

whether the variation in the independent variables can

explain the variation in the response variables is

canonical analysis, i.e., multiple regression as imple-

mented in RDA (redundancy analysis; used when the

expected response model is linear) and CCA (canonical

correspondence analysis; used when the expected re-

sponse model is unimodal; Legendre and Legendre

1998). Just as in the previous case, this is a raw-data

approach (Fig. 2).

We may also want to analyze variation in community

composition, i.e., beta diversity. Although beta diversity

can be computed for groups of sites that consist of any

number of sites !2, the analytical methods are best

developed for the special case where each group of sites

consists of exactly two sites. Using a fixed number of

sites in all groups simplifies the analyses, and using the

smallest possible number of sites per group maximizes

the power of the statistical tests, because each group of

sites is one data point in the analyses. For simplicity, we

therefore limit our present discussion to this situation.

Then we are concerned with the questions: Why are

some site pairs more different in community composi-

tion than others? i.e., Why is there variation in beta

diversity? Can the variation in the difference incommunity composition between two sites be explained

by variation in difference in environmental character-

istics or geographical location? i.e., Can variation in beta

diversity be explained by variation in environmental

difference or geographical distance? If so, then we can

predict the degree of beta diversity between two sites if

we know how different their environments are and how

far apart they are situated geographically. The response

variable Y is a distance matrix consisting of the n(n 1)/

2 pairwise differences in community composition (i.e.,

floristic or faunistic distances) between all possible pairs

of the study sites s1 to sn. This distance matrix can be

based on any of the various resemblance measures that

have been designed for species data, e.g., Jaccard, Bray-

Curtis, and Hellinger indices, or any other measure that

quantifies variation in community composition (such as

SS). The independent variables X1 to Xm are matrices of

geographical distances and the differences between sites

in environmental variables at the sites. Standard

methods to test whether the variables are statistically

dependent on each other include the Mantel test (to test

for linear or monotonic correlation between two

distance matrices), multiple regression on distance

matrices (which fits a linear regression), and generalized

FIG. 2. An analysis of level-1 data (the raw-data matrix) focuses on modeling what factors explain level-2 data (variation in theraw-data matrix). This is a level-2 question, which can be addressed using the raw-data approach. An analysis of level-2 data (thedistance matrix) focuses on modeling what factors explain level-3 data (variation in the distance matrix). This is a level-3 question,which can be addressed using the distance approach.



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distance modeling (which can fit more complex regres-

sion models; Legendre and Legendre 1998, Ferrier et al.

2002). This is what Legendre et al. (2005) called the

distance approach, which is suitable to answering

what they called level-3 questions. In terms of our levels

of abstraction, in this approach we analyze data from

the level of abstraction 2 to test whether we can explain

its variation, which is expressed at level of abstraction 3(Fig. 2).

The distance approach includes also spatial autocor-

relation analysis using correlograms or variograms,

where an autocorrelation coefficient (which can be

interpreted as a similarity measure) or semi-variance

(which can be interpreted as a dissimilarity measure) is

plotted against inter-site geographical distance (Legen-

dre and Legendre 1998).

THE DIFFERENCE BETWEEN ANALYZING

AND EXPLAINING BETA DIVERSITY

In the previous section, we consistently used the

convention that analyzing X refers to an analysis

where X is the response variable, and that such an

analysis explains the variation in X with the variation

in the independent (or explanatory) variables. This is

how these words are commonly used in a statistical

context; e.g., a regression line resulting from a regression

analysis can be said to explain a certain proportion of

the variance in the response variable (e.g., Legendre and

Legendre 1998).

Under this convention, explaining beta diversity

and analyzing beta diversity are clearly different

things. Explaining beta diversity is a level-2 question:

the response variable is community composition, and

what gets explained is variation in community compo-

sition (i.e., beta diversity). In contrast, analyzing beta

diversity is a level-3 question: the response variable is

beta diversity (i.e., variation in community composi-

tion), and what gets explained is variation in beta

diversity (Fig. 2). Consequently, if one aims at explain-

ing beta diversity then using the raw-data approach is

indicated, whereas if one aims at analyzing beta

diversity then using the distance approach is indicated.

These differences may be clarified by our physics

example. Analyzing community composition at different

points in space is like analyzing the position of an object

(say, a kite) at different points in time. Analyzing betadiversity is like analyzing the velocity of the kite (Fig. 2).

Say we are interested in understanding the causes of the

velocity of a kite. If we run an analysis using the raw-

data approach (where position is the response variable),

we learn how much of the kites observed overall

velocity is due to movement in the updown, leftright,

and forwardbackward directions. But this does not tell

us why the kite had this particular overall velocity rather

than some other velocity; to answer this question, we

need to run an analysis using the distance approach

(where velocity is the response variable). Say we are

interested in understanding the causes of beta diversity

in a region. If we run an analysis using the raw-data

approach (where community composition is the re-

sponse variable), we learn how much of the observed

overall beta diversity in the region can be explained by

environmental factors and spatial coordinates. But this

does not tell us why the region had this particular overall

beta diversity rather than some other beta diversity; to

answer this question, we need to run an analysis usingthe distance approach (where beta diversity is the

response variable).

Against this background, it can be observed that

Legendre et al. (2005) accused several studies of having

misused the Mantel test, when in fact its use had been

entirely appropriate. Legendre et al. (2005:438439)

wrote: Here are examples from the recent literature in

which authors used a Mantel approach... although they

declared that the purpose of their study was the analysis

of the variation in community composition among

sites. Following the above convention, analysis of

the variation in community composition among sites

(i.e., analysis of beta diversity) is a level-3 question.

Since level-3 questions need to be addressed using the

distance approach, the Mantel test is a justified choice.

We have seen many papers that are inconsistent or non-

explicit about whether they are addressing level-2

questions or level-3 questions (including our own earlier

work), and we hereby urge ecologists to become more

aware of the levels of abstraction in ecological questions.

Failure to do so easily leads to misinterpretation of the

results.

THE TARGETS OF THE RAW-DATA

AND DISTANCE APPROACHES

The questions of interest in the raw-data approach

concern the relationships among the raw-data variables

that were measured in the field (level of abstraction 1),

and the analyses are based on quantifying to what extent

the variation in one group of raw data variables can be

explained by the variation in another group of raw-data

variables (level-of-abstraction 2; hence the term level-2

question). The questions of interest in the distance

approach concern the relationships among distances

based on the raw data (level-of-abstraction 2), and the

analyses are based on quantifying to what extent the

variation in one group of distances can be explained by

the variation in another group of distances (level-of-abstraction 3; hence the term level-3 question).

One crucial difference between the two approaches is

that when the focus is on distances, neither the species

identities, the actual geographical locations of the study

sites, nor the actual values of the environmental

variables are relevant; we are interested only in how

big the differences in them are. In contrast, the raw-data

approach explicitly models the abundances of specific

species as a function of specific spatial coordinates and

specific values of environmental variables, which leads

to important differences in how the results of the two

approaches should be interpreted.



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Generally, when one expects a monotonic relationship

between the raw-data variables, one can also expect a

monotonic relationship between the distances based on

these raw data. For example, if the abundance of a given

species increases or decreases monotonically along a

given environmental gradient (e.g., soil nitrogen con-

tent) or a spatial gradient (e.g., longitude), then the

abundance of the species is more similar in sites that areclose to each other along that gradient than in sites that

are far apart. No matter whether the abundance of the

species increases or decreases along the gradient (e.g.,

whether the correlation between species abundance and

soil nitrogen content is positive or negative), the

expectation in terms of distances is the same: environ-

mentally more similar (or geographically more prox-

imate) sites have more similar species abundances than

more dissimilar/distant sites.

However, if the relationship between the raw-data

variables is not monotonic, then a monotonic relation-

ship cannot be expected in the distances either. For

example, if species abundance has a unimodal response

to an environmental gradient (abundance is first zero,

then increases to a maximum and decreases back to

zero), difference in abundance will be very small both

when sites are environmentally very similar and when

they are very different. But even if the environmental

gradient is so long that it exceeds the range of a single

species, community composition as a whole may still

have a monotonic relationship with the environmental

gradient; this happens if different species replace each

other along the gradient.

Data at higher levels of abstraction are derived from

the data at the lower levels of abstraction. Therefore, a

given behavior of the level-1 data predicts a given

behavior of the level-2 data. However, the opposite is

not true. When distances are computed, the information

on the identities and abundances of individual species is

lost, as is the information on the values of the

environmental variables and the geographical locations

of the study sites. This information cannot be recovered

from the distance data, because the same distance matrix

can be derived from an unlimited number of different

raw-data tables. For example, adding any constant to all

values in a raw-data table makes no difference to the

Euclidean distances derived from the table.

Consequently, a process that is defined in terms ofhow level-2 data behave cannot be used to predict how

level-1 data should behave. If we know that the velocity

of a kite decreases when it flies against the wind, we can

use this information to model the velocity of the kite at

any point in time on the basis of the strength of the wind

and the velocity of the kite at some other point in time.

But this information cannot be used to model the

position of the kite, because when velocities were

computed, information on position was lost. Because

the same velocity can be obtained from an unlimited

number of starting positions, absolute position (level-1

data) cannot be recovered from velocity (level-2 data).

Similarly, spatial autocorrelation is a phenomenon that

is independent of absolute position. Spatial autocorrela-

tion causes nearby sites to be more similar than faraway

sites, at least over some distance interval, irrespective of

the actual spatial locations of the sites (Legendre and

Legendre 1998). Information on the strength of spatial

autocorrelation and the geographical distance between

two sites makes it possible to predict how different thetwo sites are in community composition. However, since

information on both absolute position of the sites and the

identities and abundances of individual species was lost

when the distances were computed, this information

cannot be used to model which species should be present

in the community at any particular site, or how

community composition should change towards any

particular direction. We return to this in the next section.

ECOLOGICAL VS. STATISTICAL HYPOTHESES

Three ecological hypotheses

Three hypotheses on the organization of communitycomposition have been actively discussed recently, and

testing them was a central issue in the paper by Legendre

et al. (2005). In brief, the hypotheses are as follows: (1)

Species composition is uniform and the same dominant

species are found over large areas (e.g., Pitman et al.

2001); (2) Species composition fluctuates in a random,

autocorrelated way (e.g., Hubbell 2001); and (3) Species

composition is related to environmental conditions

(numerous authors).

Before moving on, it is important to notice that species

composition is not an entity that has ecological behavior

of its own, but it is a result of how individuals belonging

to different species behave. Therefore, these threehypotheses should be seen as logical consequences of

the following more fundamental ecological hypotheses:

(A) Individuals of all species are able to grow equally

well at all sites and in all ecological conditions present in

the area of interest. Species differ in competitive ability,

and the best competitors become dominant at all sites,

whereas less good competitors remain rare at all sites.

(B) Individuals of all species are able to grow equally

well at all sites and in all ecological conditions present in

the area of interest. All species are competitively equal,

and their abundances fluctuate in a random walk due to

random mortality and random but spatially autocorre-

lated dispersal.(C) Individuals of all species are not able to grow

equally well at all sites and in all ecological conditions

present in the area of interest. Species abundances vary

between sites in response to how suitable the environ-

mental conditions are for each species. All species are

not competitively equal, and competitive ranking may

change in response to changes in environmental

conditions. Species may also be excluded from some

sites because they are physiologically not able to grow in

the environmental conditions present.

Each of the hypotheses AC describes an ecologists

view on how species behave, but before they can be



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formally tested, it is necessary to derive statistical

hypotheses (null and alternative hypotheses) from them.

As outlined in the previous sections, such statistical

hypotheses can be formulated either for each species

separately or for all species at the same time, and either

using the raw-data approach or the distance approach.

A part of the confusion in the paper by Legendre et al.

(2005) seems to stem from a failure to make a distinction

between ecological and statistical hypotheses. When

testing the ecological hypotheses AC, one first needs to

think about what ecological predictions they imply, and

how these predictions can be translated into testable

statistical hypotheses. Only thereafter can one decide

which statistical method is appropriate. In the following,

we briefly describe such statistical hypotheses about

community composition and beta diversity that can be

derived from the ecological hypotheses AC.

Testing ecological hypothesis A

The ecological hypothesis A means that the samecompetitively superior species are always most abun-

dant, so community composition is uniform over the

landscape and all sites have the same species in the same

(species-specific) abundances. The expected abundance

of any given species at any given site is equal to the mean

abundance of that species over all study sites, and any

deviations from the mean are due to sampling error.

From the ecological hypothesis A it follows that the

abundance of any given species should not vary much

over the study sites, and that community composition

should not vary much either; any variation found at

level-of-abstraction 2 should be within the limits of

random variation and not explainable by variation inenvironmental variables or spatial location. This pre-

diction is testable with statistical methods that use the

raw-data approach. An example of a statistical hypoth-

esis that can be derived from this prediction is H0: when

community composition is regressed on soil nitrogen

content in CCA, the regression coefficient equals zero.

From the ecological hypothesis A it also follows that

beta diversity is small and similar over different pairs of

study sites; any variation found at level-of-abstraction 3

should be random and not explainable by variation in

environmental differences or geographical distance. This

prediction is testable with statistical methods that use

the distance approach. An example of a statistical

hypothesis that can be derived from this prediction is

H0: the Mantel correlation coefficient between floristic

distances (as measured with the Bray-Curtis index) and

differences in soil nitrogen content (as measured with the

Euclidean distance) is equal to or smaller than zero.

It should be noted here that Pitman et al. (2001) may

not have meant the hypothesis to be interpreted as

strictly as this, but if competitive ability is allowed to

vary in response to environmental conditions, then the

ecological hypothesis A becomes indistinguishable from

ecological hypothesis C.

Testing ecological hypothesis B

The ecological hypothesis B means that any species

can become abundant or rare at any site by chance,

because all species are competitively equal. Species

composition at any one site is not constant but fluctuates

randomly, so there is no way to predict for a given point

in time which species occur at which sites, and at what

abundances they occur at those sites where they do

occur. However, the fluctuations are spatially autocor-

related due to spatially limited dispersal. Sites may lose

or gain different species by chance, but the closer two

sites are to each other, the stronger the homogenizing

effect of dispersal between them (Hubbell 2001, Condit

et al. 2002).

Community composition is heterogeneous over the

landscape at all spatial scales as a result of the

cumulative effects of spatially autocorrelated random

walk in species abundances. This spatial structure is

entirely due to autocorrelation, and spatial dependence

on underlying environmental variables is not present.No directional forces are operating, so space is assumed

isotropic, i.e., the change in community composition per

unit geographical distance is the same to all directions.

From ecological hypothesis B it follows that two

nearby sites should share more species in more similar

abundances than two sites further apart, but differences

in environment are irrelevant. Consequently, variation

in beta diversity at level-of-abstraction 3 should be

explainable by variation in geographical but not

environmental distances. This prediction is testable with

the distance approach.

According to ecological hypothesis B, species abun-

dances fluctuate randomly and are therefore inherentlyunpredictable. As we saw above (see The targets of the

raw-data and distance approaches), the presence of

spatial autocorrelation does not help in predicting how

species abundances (level-1 data) should behave. Even

though spatial autocorrelation may give rise to spatial

structure in community composition, such structure is

random by definition, so it is not possible to predict a

priori where specific species should attain high abun-

dances, or where a specific community composition

should occur. An existing spatial pattern in community

composition can be described a posteriori, especially by

such powerful methods as PCNM (principal coordinates

of neighbor matrices) analysis (Borcard and Legendre2002). However, doing so does not test the neutral

model, because the neutral model did not predict that

this was the particular spatial pattern that was expected

to emerge in this particular case. Any specific spatial

pattern in community composition is just as much in

accordance with the neutral model as any other, as long

as the degree of spatial autocorrelation is similar. And

since spatial autocorrelation is defined in terms of

distances, its presence and strength can only be tested

using the distance approach.

Consequently, from ecological hypothesis B follow no

testable predictions about the expected behavior of the



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variation in the raw data at level-of-abstraction 2. The

raw-data approach is concerned with quantifying the

effect of location on the abundances of specific species,

rather than quantifying the effect of distance between

locations on beta diversity. Therefore, the raw-data

approach fails to address the neutral model in a relevant

way, and is unable either to falsify the neutral hypothesis

or to quantify its relative contribution to the observedspatial pattern.

Testing ecological hypothesis C

The ecological hypothesis C means that species

abundances vary between sites in response to variation

in environmental conditions. Different species reach

high abundances in different parts of the environmental

gradient, and species may be restricted to just a portion

of the gradient. Accordingly, a gradual turnover in

community composition is observed along the environ-

mental gradient.

Community composition is heterogeneous over the

landscape, and its spatial structure is determined by

spatial dependence on underlying environmental varia-

bles, which themselves may be spatially structured and/

or autocorrelated. The spatial pattern in community

composition may be very complex, especially if different

environmental variables show different spatial patterns.

From ecological hypothesis C it follows that the

variation in species abundances and community com-

position at level-of-abstraction 2 should be explainable

by variation in environmental variables. This prediction

is testable with the raw-data approach.

From ecological hypothesis C it also follows that two

sites with similar environmental conditions (i.e., with

small environmental distance) should have more similar

community compositions (i.e., smaller degree of beta

diversity) than two sites with more different environ-

ments; variation at level-of-abstraction 3 should be

explainable by variation in differences in environmental

conditions. This prediction is testable with the distance

approach.

Summary of ecological-hypothesis testing

It is important to notice that the level-2 statistical

hypotheses and the level-3 statistical hypotheses are

independent from each other in the sense that one is not

derived from the other. Instead, both are derived directlyfrom predictions of an ecological hypothesis. A level-2

prediction is stated in terms of the raw data, and leads to

a statistical hypothesis that can be tested with the raw-

data approach (where community composition is the

response variable). In contrast, a level-3 prediction is

stated in terms of distances, and leads to a statistical

hypothesis that can be tested with the distance approach

(where beta diversity is the response variable).

The above considerations lead to the conclusion that

all three ecological hypotheses can be tested with the

distance approach, but only hypotheses A and C can be

tested with the raw-data approach.

When the three ecological hypotheses are tested using

the distance approach, hypothesis A is indicated when

neither environmental nor geographical distances pro-

vide a significant regression model of beta diversity.

Hypothesis B is indicated when geographical distances

but not environmental distances provide a significant

model, and hypothesis C is indicated when environ-

mental distances do provide a significant model. Differ-entiating between hypotheses B and C is difficult,

because this necessitates differentiating between spatial

autocorrelation and spatial dependence. This is espe-

cially difficult when the measured environmental varia-

bles are autocorrelated, as they often are. And even if

the analyses indicate that some of the variation in beta

diversity be entirely due to variation in geographical

distances, and hence may be an expression of spatial

autocorrelation, the possibility exists that there actually

is spatial dependence on an unmeasured, spatially

autocorrelated environmental variable.

THE TARGET OF VARIATION PARTITIONING

Variation partitioning aims to quantify the relative

effects of different groups of explanatory variables on

the response variable(s) of interest. Variation partition-

ing in the context of community ecology was originally

based on redundancy analysis (RDA) or canonical

correspondence analysis (CCA), and its aim was to

partition the variation in the raw community-composi-

tion data table to fractions explainable by variation in

environmental variables and spatial location (Borcard et

al. 1992). This form of variation partitioning is a raw-

data approach (Figs. 1 and 2). A more recent extension

of variation partitioning is based on multiple regression

on distance matrices, and its aim is to partition the

variation in floristic distances, i.e., the variation in beta

diversity, to fractions explainable by variation in

geographical distances and environmental differences

(Duivenvoorden et al. 2002, Tuomisto et al. 2003). This

form of variation partitioning is a distance approach

(Figs. 1 and 2).

Variation partitioning has become a popular method

to address questions related to the ecological hypotheses

A, B, and C mentioned above (see Ecological vs.

statistical hypotheses). Examples of studies that have

used the RDA/CCA-based variation-partitioning meth-

od are Duivenvoorden (1995), Gilbert and Lechowitch(2004), Svenning et al. (2004), and Cottenie (2005).

Examples of studies that have used the distance-based

variation partitioning are Duivenvoorden et al. (2002),

Tuomisto et al. (2003), Vormisto et al. (2004), and Jones

et al. (2006).

Legendre et al. (2005) discuss the difference between

the two approaches at length, and show that the

variance of the dissimilarity matrix is not the same as

the variance of the raw-data table, and that there is no

simple relationship between the two variances. To this

extent, we agree with them. If the raw-data table (level-

of-abstraction 1) has nonzero variance (level-of-abstrac-



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tion 2), this indicates that there is nonzero beta diversity

(analogous with nonzero velocity of a kite). Even if this

is the case, the variance at level-of-abstraction 3 may still

be zero; this happens when all s ites within the

community are equally different from each other (no

variation in beta diversity; analogous with a kite moving

with a constant velocity). Consequently, it is true by

definition that a decomposition of the variance of thedissimilarity matrix does not decompose the variance of

the raw-data matrix.

Legendre et al. (2005:441, 447) conclude from this that

a decomposition of the variance of the dissimilarity

matrix among two or several explanatory tables

represented by dissimilarity matrices cannot help us

understand the causes of the variation of community

composition (beta diversity) across an area and

partitioning on distance matrices should not be used

to study the variation in community composition among

sites. Here we disagree with them. The words study

and understand do not have established statistical

meanings, so they can be interpreted to imply either

explaining or analyzing. Legendre et al. (2005)

argue that only the explaining beta diversity inter-

pretation is correct; we argue that also analyzing beta

diversity is appropriate (see The difference between

analyzing and explaining beta diversity, above).

The RDA/CCA-based variation partitioning is ap-

propriate when one is interested in explaining beta

diversity (level-2 question). This method models species

abundances, and can tell us that in a particular case,

variation in geographical location alone explained x%,

variation in the measured environmental variables alone

explained y%, and variation in the two jointly explained

z% of the overall beta diversity (i.e., of the variation in

species abundances). The distance-based variation par-

titioning is appropriate when one is interested in

analyzing beta diversity (level-3 question). This method

models beta diversity, and can tell us that in a particular

case, variation in geographical distances alone explained

a%, variation in the differences in the measured

environmental variables explained b%, and variation in

the two jointly explained c% of the variation in beta

diversity.

MOVING BETWEEN RAW DATA AND DISTANCES

The raw data and distance worlds are intimatelyinterlinked, and it is quite easy to move between them.

Any raw-data table can be used to compute a distance

matrix, which in turn can be used in principal

coordinates analysis (PCoA) to reconstruct raw-data-

like principal coordinates (Fig. 3). The reconstructed

raw data consist of the coordinates of the sites in the

ordination space rather than actual spatial coordinates

or species abundances, but the information on interplot

relationships remains the same. If Euclidean distances

are computed from the principal coordinates, a distance

matrix identical to the original is obtained. However, if

the originally chosen distance measure is not metric,

only the Euclidean part of the distances is reconstructed

(see Legendre and Legendre [1998] or Legendre and

Anderson [1999] for details).

In spite of this close connection between the raw-data

table and the distance matrix, it is important to know

which of them is used in data analysis.

If the final analysis is based on raw data (recon-

structed or original; level-1 data), it addresses a level-2question, even if some earlier phase of the process

involved also distance matrices (level-2 data). Remem-

bering this is relevant, for example, when one considers

the use of distance-based redundancy analysis (db-RDA;

Legendre and Anderson 1999). In db-RDA, a distance

matrix is first computed from the species abundance

data (using any appropriate dissimilarity measure), and

then PCoA is used to obtain principal coordinates (as in

Fig. 3), which are finally used in RDA. In spite of its

name, distance-based RDA is not a distance approach

sensu Legendre et al. (2005). Like normal RDA, db-

RDA is a raw-data approach; it analyzes data taken

from level-of-abstraction 1 in order to explain its

variation at level-of-abstraction 2. The very purpose of

PCoA in the process is to reconstruct a raw data table

from a distance matrix, because the latter is a level-2

data set and hence cannot be used in RDA.

Another raw-data approach that is called distance

based because it involves a distance matrix in an

intermediate phase is nonlinear canonical analysis of

principal coordinates (NCAP; Millar et al. 2005).

Whereas db-RDA is restricted to linear-regression

models, NCAP can incorporate more complex models,

but the two methods are similar in that both analyze

level-1 data.

CHOOSING AMONG STATISTICAL METHODS

Since ecological hypotheses may yield more than one

testable prediction, more than one kind of statistical

approach is often possible when testing them. This is the

case with ecological hypotheses A (uniformity) and C

(environmental control): both can be tested either with

the raw-data approach or with the distance approach.

So even if one confuses the two analysis approaches, the

results one gets are still relevant to the ecological

hypothesis of interest.

However, the same is not true of ecological hypothesis

B (the neutral theory): as we saw in Testing ecologicalhypothesis B (above), its testable predictions are stated in

terms of distances, not in terms of raw data, so it can

only be tested with the distance approach. Therefore,

attempting to test this ecological hypothesis using the

raw-data approach may give quite misleading results.

At least two recent studies (Cottenie 2005, Legendre et

al. 2005) have attempted to test hypothesis B using the

raw-data approach. They assumed that the presence of

significant spatial patchiness (different from random) in

the distributions of species supports the neutral theory.

However, the presence of spatial autocorrelation (the

hallmark of hypothesis B) cannot be tested using the



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raw-data approach, but instead it is entirely possible to

find significant patchiness that is in conflict with thepredictions of the neutral theory. For example, a

significant effect of the northsouth coordinates but

not of the eastwest coordinates implies that space is not

isotropic, contrary to the assumption of the theory.

Similarly, if a significant term is found that includes the

square of one of the coordinates, this may indicate that

sites at the extremes of the study area are more similar to

each other in community composition than to sites in

the central parts of the study area, which conflicts with

the prediction that similarity in community composition

decreases with increasing geographical distance.

An analysis using the raw-data approach is concerned

about modeling community composition (species abun-dances) at different points in space (compare with

modeling the position of a kite at different points in

time). A successful model can accurately (with a high R2

value) predict community composition at a given site on

the basis of its position along the environmental and

geographical gradients and the community compositions

of other sites whose positions along the same gradients

are also known. An analysis using the distance

approach, in contrast, is concerned about modeling

beta diversity (the difference in community composition)

between site pairs (compare with modeling the velocity

FIG. 3. The derivation of distances from raw data and the reconstruction of raw data like principal coordinates from distances.Both the raw data and the reconstructed raw data are at level-of-abstraction 1, the distances are at level-of-abstraction 2 (PCoA,principal coordinates analysis). In this example, the raw data consist of UTM coordinates of study sites, so Euclidean distancesindicate the distances between sites in kilometers. The original coordinates can be visualized in a scatterplot; because the UTMcoordinates include information of absolute location, the orientation of the scatterplot in relation to compass bearings is known,and its location can be related to external landmarks. The information on absolute location is discarded when distances arecomputed, so the reconstructed raw data include the original information on the positions of the study sites in relation to eachother, but no information on their positions in relation to external landmarks. With species-abundance data the situation is similar:species identities are lost when the distances are computed. If a distance measure other than the Euclidean distance is used to obtainthe distance matrix from the raw data (as is usually the case with species-abundance data), then the relationships among plots willdiffer in the visualizations based on the raw data vs. the reconstructed raw data.



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of a kite between different points in time). A successful

model can accurately (with a high R2 value) predict the

degree of beta diversity between two sites on the basis of

their environmental and geographical distance and the

degree of beta diversity between other site pairs whose

environmental and geographical distances are also

known. In the latter analysis, both geographical position

and community composition are irrelevant and thereforenot included in the model. A given change in geo-

graphical distance has the same effect on beta diversity

no matter which species are actually involved, and no

matter whether the sites of interest are actually situated

east or west of the Greenwich meridian (compare with a

kite hitting a tree: the impact on velocity is the same no

matter what the position of the tree is).

Legendre et al. (2005:438) used simulated data to

compare the statistical power of the two procedures

[RDA and Mantel test] for partitioning the variation of

raw data. They found that RDA (redundancy analysis)

yielded higher R2 values and more powerful significance

tests, and recommended that it be used instead of the

Mantel test.

We find this a very problematic comparison. The raw-

data approach (RDA) and the distance approach

(Mantel test) have fundamentally different null hypoth-

eses, and only one of them (RDA) actually targets the

stated question of interest. The simulations did not show

that one method is better than the other, because the two

methods are not alternative ways of analyzing the same

statistical question. The only thing that can be

concluded from the simulation results is that if you

ask a different question, you may get a different answer.

In this case, the environmental and spatial model, as

implemented in RDA, explained a higher proportion of

the variance in the raw species-data table (level-2 data)

than the model of environmental and geographical

distances, as implemented in Mantel tests, explained of

the variance in the floristic distance matrix (level-3 data).

A data set might be simulated where the distance

approach yields a higher R2 value than the raw-data

approach, but this would not be a valid argument to

recommend the use of the distance approach to analyze

level-2 questions.

Another problem in trying to rank the performance of

different methods according to their R2 values or the

power of their significance tests is that these measuresdepend heavily on several details in the analysis

methods. Often the particular details that are relevant

in the raw-data approach lack counterpart in the

distance approach and vice versa. For example, in the

raw-data approach both P values and R2 values will

change depending on whether the regression that relates

species abundances to environmental variables models a

linear, unimodal symmetric or unimodal skewed re-

sponse, and whether space is modeled using just the x

and y coordinates, also their polynomial terms, or

PCNM (principal coordinates of neighboring matrices)

variables. In the distance approach, none of these

choices is relevant, because it is not the raw data that

are regressed against each other. Instead, both P values

and R2 values will change depending on which dissim-

ilarity measures are used, whether all environmental

variables are combined into a single distance matrix or

used in separate matrices, whether the geographical

distances are ln-transformed or not, and whether the

distance matrices are related to each other using a

monotonic, linear, or more complex function.

If one needs a model to predict the position of a kite,

one does not compare a model built to predict position

with a model built to predict velocity, and choose the

one that happens to yield higher R2 values in a

simulation study. One of the models can, and should,

be discarded from the outset because it does not model

the variable of interest. The situation is no different in

the community-composition case. The first criterion for

choosing a method of analysis is whether it is

appropriate for testing the question at hand or not,

and for making that decision it is irrelevant what R

2

values the available methods have obtained in simu-

lation studies.

CONCLUSIONS

Throughout this paper we have emphasized that

analyses based on the distance approach ask different

statistical questions than analyses based on the raw-data

approach. Therefore, they should be used for different

purposes. When researchers evaluate the answers they

get through statistical analysis, it is essential to under-

stand what questions those analyses ask. One can only

make justified statements about those predictions of the

relevant ecological hypotheses that have actually beentested, so one should not claim to have tested one

prediction when in fact the analysis method tested

another. But if an ecological hypothesis yields predic-

tions at different levels of abstraction, all of these can

fruitfully be tested, if this can computationally be done.

There is no reason to claim that testing one prediction is

more valid than testing another prediction; rather, the

different approaches should be viewed as complement-

ing each other.

ACKNOWLEDGMENTS

Many of the ideas presented in this paper were developed

during inspiring discussions with Pierre Legendre, DanielBorcard, and Pedro Peres-Neto. We thank all of them, as wellas Rune kland and two anonymous reviewers, for construc-tive comments on the manuscript. Financial support wasobtained from the Academy of Finland.

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