TESTING FOR PHYLOGENETIC SIGNAL IN BIOLOGICAL TRAITS: THE UBIQUITY OF CROSS-PRODUCT STATISTICS

ORIGINAL ARTICLE

doi:10.1111/j.1558-5646.2012.01823.x

TESTING FOR PHYLOGENETIC SIGNALIN BIOLOGICAL TRAITS: THE UBIQUITYOF CROSS-PRODUCT STATISTICSSandrine Pavoine1,2,3 and Carlo Ricotta4

1Museum national d’Histoire naturelle, Departement Ecologie et Gestion de la Biodiversite, UMR 7204 CNRS UPMC, 55–61

rue Buffon, 75005 Paris, France2Mathematical Ecology Research Group, Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS,

United Kingdom3E-mail: [email protected]

4Department of Environmental Biology, University of Rome ‘La Sapienza’, Piazzale Aldo Moro 5, 00185 Rome, Italy

Received May 8, 2012

Accepted September 18, 2012

To evaluate rates of evolution, to establish tests of correlation between two traits, or to investigate to what degree the phylogeny

of a species assemblage is predictive of a trait value so-called tests for phylogenetic signal are used. Being based on different

approaches, these tests are generally thought to possess quite different statistical performances. In this article, we show that the

Blomberg et al. K and K∗, the Abouheif index, the Moran’s I, and the Mantel correlation are all based on a cross-product statistic,

and are thus all related to each other when they are associated to a permutation test of phylogenetic signal. What changes is only

the way phylogenetic and trait similarities (or dissimilarities) among the tips of a phylogeny are computed. The definitions of the

phylogenetic and trait-based (dis)similarities among tips thus determines the performance of the tests. We shortly discuss the bio-

logical and statistical consequences (in terms of power and type I error of the tests) of the observed relatedness among the statistics

that allow tests for phylogenetic signal. Blomberg et al. K∗ statistic appears as one on the most efficient approaches to test for phy-

logenetic signal. When branch lengths are not available or not accurate, Abouheif’s Cmean statistic is a powerful alternative to K∗.

KEY WORDS: Abouheif test, Blomberg et al. K and K∗, equivalent test statistic, Mantel test, Moran’s I, permutation.

Phylogenetic signal is obtained when phylogenetically related

species tend to have more similar trait values than more distantly

related species. It is tested with different aims: (1) to find models

and rates of evolution for explaining extant species’ traits (e.g.,

Blomberg et al. 2003); (2) to find which test approach should be

used to compare two traits in cross-species analyses and whether

phylogenetic information should be included in these tests (e.g.,

Abouheif 1999; but see Rohlf 2006 and Revell 2010); (3) to

elucidate the processes that underpin patterns in phylogenetic

diversity in ecological studies of communities, species interaction

networks, and ecosystem services (e.g., Mouquet et al. 2012).

Variations in trait states may have various levels of associa-

tion with the species phylogenetic history (Hansen and Martins

1996) and a number of different statistics are widely used to test

for phylogenetic signal (see Revell et al. 2008). Some of these tests

are flexible and might be model-dependent or model-free depend-

ing on how phylogenetic proximities/distances among species are

defined. These include the Mantel (1967) test developed to com-

pare, via a correlation, any kind of dissimilarity matrices (see also

Mantel and Valand 1970), and the Moran (1948) test originally

developed to detect spatial signal in environmental variables and

introduced in a phylogenetic context by Gittleman and Kot (1990).

8 2 8C© 2012 The Author(s). Evolution C© 2012 The Society for the Study of Evolution.Evolution 67-3: 828–840

TESTING FOR PHYLOGENETIC SIGNAL

Abouheif (1999) proposed a model-free test of phylogenetic

signal for a continuous character adapting a diagnostic test for

serial independence originally developed by von Neumann et al.

(1941) in a nonphylogenetic context. Recently, Pavoine et al.

(2008) provided an exact analytic formulation of the Abouheif

test showing that it turns out to be an application of the Moran test

(1948), with a particular definition of the pairwise phylogenetic

proximities between species. In contrast, Blomberg et al. (2003)

proposed two statistics K and K∗ to compare the evolution of

a trait to that expected under a Brownian motion model of trait

evolution.

Being based on different approaches all these methods are

thought to possess different statistical performances in terms of

power and type I error (see, e.g., Harmon and Glor 2010; Hardy

and Pavoine 2012). In this article, we will show that Blomberg

et al. K and K∗, the Abouheif index, the Moran’s I and its gen-

eralizations, and the Mantel correlation are all based on a cross-

product statistic, such that whenever the significance of the tests

is evaluated via permutation procedures, the test procedures are

identical to each other. What changes is the way the phyloge-

netic and trait similarities (or dissimilarities) are computed. Ac-

cordingly, the observed differences in the statistical performances

among the tests are related to differences in how the similar-

ity/dissimilarity matrices are constructed, rather than to the math-

ematical formulation of the tests. We thus compared the ways

phylogenetic (dis)similarity and trait-based (dis)similarity are de-

fined in these statistics of phylogenetic signal and evaluated the

consequences on the performance of the associated tests of phy-

logenetic signal. We end with discussion and recommendation on

which statistic could be usefully preferred in which circumstance.

Matrices of Phylogenetic(Dis)similarityMany tests of phylogenetic signal require the definition of a ma-

trix of phylogenetic similarity or dissimilarity. Hereafter, we will

consider two complementary matrices of phylogenetic similari-

ties among tips: A (Pavoine et al. 2008) and the Brownian co-

variance matrix C. In the next section, we will demonstrate that

the values of C−1, the inverse of C used for instance in Blomberg

et al. (2003) statistics of phylogenetic signal, can be considered

as measures of phylogenetic differences among species.

The matrix A = (aij) was discovered by Pavoine et al. (2008)

when providing an analytical solution to the test of phylogenetic

signal developed by Abouheif (1999). For a tip i of a phylogenetic

tree, aii is the inverse of the product of the number of branches

descending from each ancestral node of the tip and, for a couple

of tips (i,j), aij is the inverse of the product of the number of

branches descending from the ancestral nodes unshared by tips i

and j and that of their most common ancestor only (nodes located

in the shortest path that connects the two tips). The values aii

have been interpreted as measures of how isolated a tip is in the

phylogenetic tree. A tip is isolated if it descends from lineages

that embed few tips. Extreme isolation is obtained when the tip is

the sole descendent from a branch directly connected to the root.

One of the main characteristics of matrix A is that it avoids the

cost associated with assuming that branch lengths and a model of

evolutionary change are known and accurate (Abouheif 1999).

The matrix C = (cij) is connected with Brownian evolution.

The diagonal value cii is defined as the sum of branch lengths be-

tween tip i and the root of the phylogenetic tree. The off-diagonal

value cij are defined as the sum of branch lengths between the first

common ancestor of tips i and j and the root of the tree (i.e., the

height above the root of the most recent common ancestor of some

pair of tips). C is the basis of the matrix of variance–covariance

V = σ2C (where σ2 is the rate of evolution) among tips’ traits

according to a Brownian motion model. Each diagonal value is

the variance of the trait value at each tip according to a Brownian

evolution from the root of the tree; and each off-diagonal value is

a covariance between the trait values at two tips according to the

same Brownian model. Examples of calculation of the matrix Cof the Brownian motion model and of the phylogenetic proximity

matrix A are given for a simple theoretical tree in Figure 1.

Special Measures of PhylogeneticDifferences: Matrix C−1

Matrices A and C of phylogenetic similarities have been defined

in the previous section. We show below that C−1 is a particular

measure of phylogenetic differences among tips (we use the word

“difference” here instead of “dissimilarity” because the term “dis-

similarity” has usually been associated with non-negative matrices

whereas these values of difference are allowed to be negative).

DEFINITION OF THE VOLUME OF A TREE

Matrix C can be associated with two graphical representations:

a tree and a parallelepiped. Hereafter, we will use the expres-

sion “the volume of a tree” to designate the volume in n di-

mensions of the parallelepiped associated with a tree of n tips

with the following definition. Let (e1 . . . en) be the standard ba-

sis Rn , where e1 = (1, 0, 0, . . . , 0)t is associated with the first

tip, e2 = (0, 1, 0 . . . , 0)t is associated with the second tip, . . . and

en = (0, 0, 0, . . . , 1)t is associated with the last tip. Let us denote

(c1 . . . cn) the columns of C. The coordinates of the vector ck in

the standard basis are the variance of the kth tip on axis ek and

the covariance between the kth tip and each of the other tips on

the other axes. The n vectors c1, . . . , cn, define a parallelepiped

as illustrated in Figure 2 (the set of points whose coordinates

EVOLUTION MARCH 2013 8 2 9

S. PAVOINE AND C. RICOTTA

(a) (b)

(c)

Figure 1. Matrices C and A. (a) Theoretical tree on which Figures 1b, 1c, and 3 are based. The triangle indicates the root node. Squares

indicate interior nodes and circles indicate the tips. (b) Matrices C (Brownian model used by Blomberg et al. 2003) and (c) A (Abouheif

matrix defined by Pavoine et al. 2008) associated to the theoretical tree given in (a).

in the standard basis are in {∑i ti ci |0 ≤ ti ≤ 1}). In the extreme

situations where the matrix C has zero off-diagonal values, the

associated tree is a star phylogeny with all branch lengths de-

scending from the root node and the associated parallelepiped has

right angles and a volume equal to the product of its edge lengths

(i.e., the variances or diagonal values of C; Fig. 2a).

If some of the covariances (off-diagonal values of C) are

positive, then the phylogenetic tree has bifurcating interior nodes

between the tips and the root node and some of the angles in the

parallelepiped are acute. In that case, if three tips are considered,

the three-dimensional parallelepiped looks like a flattened card-

board box (examples are given in Fig. 2). The volume of a tree

thus increases with the variances and decreases with the covari-

ances among tips. For a given number of tips (n), the volume of a

tree is thus a measure of phylogenetic diversity in Rn and it can

be measured by the absolute value of the determinant of matrix C(see File S1 for details).

C−1: MATRIX OF NEGATIVE PHYLOGENETIC

DIFFERENCES

Consider a reference tree from which matrix C is calculated and

C−1 = (δij) is the inverse matrix of C. It can be shown (File

S1) that the diagonal values of C−1 (δii) correspond to the ratio

of the volume in n − 1 dimensions of the new, degraded trees

obtained by removing one tip at a time in the reference tree to

the volume of the reference tree. It can also be shown (File S1)

that the off-diagonal value of C−1, δij, depends on a reduced tree

obtained by dismantling the structure of the tree as follows: (1)

first remove the path that connects the ith to the jth tip; this leads to

several disconnected subtrees; (2) if none of the subtrees contain

the root node of the main tree, δij = 0; otherwise, reconnect the

subtrees by their root, and δij is equivalent to minus the ratio of the

volume of this new reduced tree to the volume of the reference tree

(Fig. 3).

The diagonal value δi i of C−1 is thus positive and it is high

if removing the ith tip hardly change the volume associated to the

reference phylogenetic tree. As the influence of a tip depends on

its variance and covariance with the other tips, this means that δi i

is high if the ith tip is confined in the phylogenetic tree, with lots

of relatives as regards its distance to the root node. Values on the

diagonal of C−1 are thus related to the concept of phylogenetic

originality or distinctiveness of a tip (May 1990; Vane-Wright

et al. 1991; Pavoine et al. 2005, 2008): they increase with de-

creasing phylogenetic distinctiveness.

The value of the off-diagonal δi j entry of C−1 is high if re-

moving the path that connects the ith to the jth tip breaks the

tree into large subtrees. It measures thus some kind of phyloge-

netic difference between tips i and j, which is high if the two tips

compared are far from each other and far from other tips. These

values of difference have the particularity of being negative. A

value close to zero thus means high phylogenetic difference and a

strongly negative value means very low phylogenetic difference.

A value of zero in C is conserved as a zero in C−1. Tips separated

by the root node are thus considered unrelated by this approach

both in C (where zero is the lowest possible phylogenetic similar-

ity) and in C−1 (where zero is the highest possible phylogenetic

difference).

A Comparison between Matrices A,C, and C−1

The main difference between matrices A, C, and C−1 is that matri-

ces A and C measure phylogenetic similarity whereas matrix C−1

8 3 0 EVOLUTION MARCH 2013


Figure 2. Five examples of association between phylogenetic trees and parallelepipeds. The matrix C (of the Brownian model used by

Blomberg et al. 2003) is given in each case. The black vectors of the standard basis give the scale. The red vectors correspond to the

columns of C. Three faces of the parallelograms are colored to help to visualize their shape.

Figure 3. Matrix C−1 (inverse of matrix C given in Fig. 1b) associated with the tree given in Figure 1a designated here by t. V is the

volume associated to a tree (volume of the parallelepiped, see File S1 for a full definition of the parallelepiped associated to a tree).

Dismantled trees are given below the matrix.



measures phylogenetic difference. However, a more subtle prop-

erty is shared by matrices A and C−1; they contain relative (instead

of absolute) measures of phylogenetic (dis)similarity. Usual ma-

trices of pairwise phylogenetic proximities/differences between

tips (e.g., patristic distances) have off-diagonal values that only

depend on the two concerned tips: evolution on all branches is

independent on evolution on all other branches. They are ab-

solute phylogenetic proximities/differences (e.g., Gittleman and

Kot 1990). In contrast, matrices A and C−1 contain relative mea-

sures of phylogenetic proximities and differences, respectively,

that depend on the pool of taxa considered.

In the previous section, we have shown that C−1 measures

pairwise phylogenetic differences among tips influenced by how

confined the pairs of tips are (off-diagonal) and also by pure

phylogenetic confinement of individual tips (diagonal). Confine-

ment means presence in nested species-rich clades (in opposition

to originality that is associated with species-poor clades). This

should be connected to the matrix A. The diagonal elements of

A measure the phylogenetic originality of the tips and the off-

diagonal elements of A measure pairwise phylogenetic proxim-

ities influenced by how confined the pairs of tips are (Fig. 1).

Contrary to classical matrices of phylogenetic distance or prox-

imities (such as patristic distances), values in C−1 and in A thus

depend on the shape of the phylogenetic tree, and not only on the

path that connects the pair of tips.

Unification: Widely Used Tests ofPhylogenetic Signal are Based onthe Cross-Product StatisticThe general form of a cross-product statistic is given by (e.g.,

Getis 1991):

� = c∑n

i=1

∑n

j=1wi j yi j ,

where wij and yij are the elements of two pairwise dissimilarity or

similarity matrices W = (wij), Y = (yij) for objects i and j (i, j =1, 2, . . . , n), and c is a constant that is invariant by permutation.

We will consider hereafter that wij represents some measure of

phylogenetic (dis)similarity between tips i and j of a phylogenetic

tree, whereas yij is a measure of (dis)similarity between trait values

at tips i and j. The cross-product statistic is flexible as wij and yij

can be freely defined. We have suggested above two potential

matrices of phylogenetic similarities (A and C) and a matrix of

phylogenetic differences (C−1).

Because the elements of a (dis)similarity matrix are not in-

dependent, the significance of a cross-product statistic (i.e., the

association between W and Y) is usually tested by randomly

permuting the order of the elements within one matrix (rows

and columns are permuted in tandem) keeping the other matrix

unchanged (Rosenberg and Anderson 2011). P-values are then

computed as the proportion of permutation-derived values that

are as extreme or more extreme than the actual � value.

We demonstrate in Table 1 that several widely used statistics

of phylogenetic signal are applications of the cross-product statis-

tic. As a consequence, the differences between the permutation

tests using these statistics are due to the choice of the matrices

W and Y. We will show in the next section that this choice is

critical to the performance of the test in terms of power and type I

error. We illustrate below that the different cross-product statistics

presented in Table 1 have had very different justifications when

they were first developed and have different levels of flexibility

in the definitions of W and Y.

The first statistic related to the cross-product is Mantel cross-

product (1967, p. 213), where, in a phylogenetic context, Y is

a matrix of pairwise dissimilarity between trait values at tips

and W is a corresponding matrix of phylogenetic dissimilarity.

Mantel test was developed to analyze the correlation between any

two matrices of dissimilarities (first in a context of spatial and

temporal aggregation in disease expansions, Mantel 1967). The

main advantage of this index is its flexibility because the definition

of how to compute the trait- and phylogeny-based dissimilarities

among tips is left completely free to convenience of the user of the

test. Note that using dissimilarities among species in Mantel test

implies that a tip will never be compared to itself (the dissimilarity

between a species and itself is zero: wii = 0 and yii = 0 for all i).

The statistics IR, IN , and IW in Table 1, are all rooted in the

analysis of autocorrelation in time series and spatial data (Cliff and

Ord 1973; Rohlf 2001). They are generalized versions of Moran’s

(1948) I autocorrelation index. They take the classic form of any

autocorrelation coefficient: the numerator term is a measure of

covariance among the trait values at tips and the denominator

term is a measure of variance. The general formula is

I =n

∑n

i=1

∑n

j=1, j �=iwi j

(xi −

∑n

k=1rk xk

)(x j −

∑n

k=1rk xk

)(∑n

i=1

∑n

j=1, j �=iwi j

)(∑n

i=1ri

(xi −

∑n

k=1rk xk

)2) .

Originally, the diagonal values of W = (wij) were set to zero.

Here, we use a more general formula where they are allowed to

be positive. The value xk is the trait value at tip k. In IR and IN ,

rk = 1/n for all k. The difference between IR and IN is that in

IN ,∑n

j=1 wi j = 1 for all i (see Gittleman and Kot 1990 for an

application of IN in a phylogenetic context). With this constraint,

the numerator of IN can be seen as a covariance between the

observed value of i (xi) and the average value of the other tips∑nj=1 wi j x j where each tip is weighted by how closely related

(as measured by wij) it is from tip i (Cliff and Ord, 1981; see

also File S3 where the equations of all statistics are detailed). The

covariance is expected to be high if the value at tip i is close to



Table 1. A comparison of the generalized Moran statistics, Bomberg et al. K, K∗, and the new KW . xi is the trait value for tip i,

x = ∑ni=1

1n xi , ri is the phylogenetic weight for tip i (see main text), R is the diagonal matrix with values ri for all i on the diagonal, 1n is

the unit vector of length n, σ2 is the rate of evolution in the Brownian model.

Phylogenetic matrix (W) Trait-based matrix (Y) Constant (c)

IR Any non-negative squaredmatrix

⎡⎣ (xi − x)(x j − x)∑n

i=1

1n (xi − x)2

⎤⎦ 1/1t

nW1n

IN Any non-negative squaredrow-normalized matrix

⎡⎣ (xi − x)(x j − x)∑n

i=1

1n (xi − x)2

⎤⎦ 1/n

IW Any non-negative squaredmatrix

⎡⎢⎢⎣

(xi −

∑n

i=1ri xi

)(x j −

∑n

i=1ri xi

)∑n

i=1ri (xi −

∑n

i=1ri xi )

2

⎤⎥⎥⎦ 1/1t

nW1n

1/K∗ C−1

⎡⎢⎢⎣

(xi −

∑n

i=1ri xi

)(x j −

∑n

i=1ri xi

)∑n

i=1(xi − x)2

⎤⎥⎥⎦

(σ2

n − 1

) (tr [C] − 1t

nC1n

n

)

1/K C−1

⎡⎢⎢⎢⎣

(xi −

∑n

i=1ri xi

)(x j −

∑n

i=1ri xi

)

∑n

i=1

(xi −

∑n

i=1ri xi

)2

⎤⎥⎥⎥⎦

(σ2

n − 1

) (tr [C] − n

1tnC−11n

)

1/KW C−1

⎡⎢⎢⎢⎣

(xi −

∑n

i=1ri xi

)(x j −

∑n

i=1ri xi

)

∑n

i=1ri

(xi −

∑n

i=1ri xi

)2

⎤⎥⎥⎥⎦

(σ2

n − 1

) (tr [RC] − 1

1t C−11

)

the values at its most related tips. The index IW use different ri

values: ri = ∑nj=1 wi j/

∑ni=1

∑nj=1 wi j . This weighting grants a

higher importance to tips having many closely related tips. It was

suggested (Thioulouse et al. 1995) to unify several points of view

on how autocorrelation (spatial autocorrelation in Thioulouse et

al. paper, for us phylogenetic correlation) should be measured and

analyzed including Moran’s (1948) index, Geary’s (1954) index,

the local variance (Lebart 1969), the local principal component

analysis (Le Foll 1982; see Thioulouse et al. 1995 for details).

According to Pavoine et al. (2008, p. 83), Abouheif (1999)’s

Cmean test of phylogenetic signal turns out to be equal to IR(A).

Given that for A = (aij),∑n

j=1 ai j = 1, the weights defined in IW

are simply ri = 1/n, so that Cmean = IR(A) = IW (A) = IN (A).

Finally, the formulas for K, K∗, and KW statistics in Table 1

are connected. The statistic K∗ (Blomberg et al. 2003) was defined

as:

MSE∗

MSE

/ (MSE∗

MSE

)expected with Brownian model

.

The denominator of this ratio is a scaling factor that does not

depend on the traits and that is thus unchanged by permutation. It

is the expected value the numerator would have if the traits were

distributed according to a Brownian model of evolution. K and

K∗ were defined by Blomberg et al. (2003). Consider the model

x = μ1n + ε, where x is the vector of observed trait values at the

tips of a phylogeny, μ is a mean (scalar), 1n is the unit vector of

length n, and ε the vector of residuals. In ordinary least squares

(OLS) the elements of ε are assumed to be independent and iden-

tically distributed according to a normal distribution of mean zero

and variance σ2. The mean square error of the model in OLS is

MSE∗ = 1n−1

∑ni=1 (xi − x)2 where x = ∑n

i=11n xi. In generalized

least squares (GLS), ε is assumed to follow a multivariate normal

distribution of mean 0n (null vector of length n) and of covari-

ance matrix σ2C (of size n × n). The mean square error of the

model in GLS is MSE = 1n−1

∑ni=1

∑nj=1 (C−1)ij(xi − x)(xj − x),

where x = [1t C−11]−1(C−11)t X. In K∗ (see Table 1 for the whole

formula), MSE∗/MSE is thus the ratio of the mean square error

of the model in OLS to the mean square error of the model in

GLS.

K is similar to K∗ except that 1n−1

∑ni=1 (xi − x)2 is replaced

with 1n−1

∑ni=1 (xi − x)2, with x defined above. The justification

for this replacement was that x is the “phylogenetically correct

mean” (Blomberg et al. 2003): the estimated value at the root

node of the phylogenetic tree. Rohlf (2006) considered K∗ but not



K and wrote for the component 1n−1

∑(xi − x)2 of K (Appendix

in Rohlf 2006) that “it is somewhat inconsistent to use deviations

from a GLS mean when computing a MSE representing the re-

sults of using OLS.” Even if K is more widely used in practice

and practically available in statistical softwares (e.g., functions

“Kcalc” and “phylosignal” in picante, Kembel et al. 2010, in R

Development Core Team 2012), K∗ seems to be preferred in the-

oretical, statistical studies (e.g., Ives et al. 2007). On the contrary,

Blomberg et al. (2003) wrote: “we only present results for K,

which we feel has greater heuristic value.” Given that K does not

have a strong theoretical justification, the evaluation of its perfor-

mance in the permutation test, in comparison with K∗, is critical

to determine any recommendation about its future use.

Here, we would like to introduce the new statistic KW , which

could reconcile Blomberg et al.’s (2003) advice of using only the

phylogenetic correct mean in a statistic for phylogenetic signal

with the general idea of having a strong theoretical basis for

any statistic. KW (see Table 1 for the formula) is similar to K

except that it replaces∑n

i=1 (xi − x)2 with∑n

i=1 ri (xi − x)2; with

the notation C−1 = (θij), ri = ∑nj=1 θi j/

∑ni=1

∑nj=1 θi j . With

this definition, it can be shown that KW = λ/IW (C−1) (File S3),

where λ is a scalar (i.e., real value) invariant by permutation. A

permutation test based on KW is thus equivalent to a permutation

test based on IW , which is rooted on a strong statistical framework

that unifies several points of view on the measures and analyses

of autocorrelation as indicated above.

Mantel’s statistic and Moran’s generalized statistics thus be-

longs to the same statistical framework, the cross-product statistic,

as Blomberg et al.’s statistics. Specifically, the observation that

K∗, K, and KW can be considered in the context of cross-product

statistics and are closely related to Moran’s generalized statistics,

is based on our demonstration that C−1 can be interpreted as a

matrix of phylogenetic differences. The characterization of matrix

C−1 was thus critical to unify all these indices. The Mantel test

compares phylogenetic dissimilarities with trait-based dissimilar-

ities and Moran’s tests, when applied to phylogenetic similarities,

compare phylogenetic similarities with trait-based similarities. In

contrast, Blomberg et al.’s K, K∗ and the new KW statistics all

mix phylogenetic differences in C−1 and trait-based similarities.

However, as functions of the inverse of a cross-product, K, K∗,

and KW also increase with phylogenetic signal. If applied to C−1,

any generalized Moran statistic (IR, IN, IW ) decreases with phylo-

genetic signal but can still be used to test for phylogenetic signal

provided one considers that low values of IR, IN, IW in that case

correspond to high phylogenetic signal. To evaluate the perfor-

mance of these statistics in testing for phylogenetic signal, we

provide in the next sections a comparison between the general-

ized Moran statistics, K, K∗, and KW statistics and between the

definitions of phylogenetic (dis)similarity in terms of power and

type I error of tests for phylogenetic signal.

Table 2. Statistics used in the simulations. Note that when the

diagonal values of A are not set to zero, Abouheif (1999), Cmean =IR(A) = IN(A) = IW (A), but when the diagonal values of A are set to

zero, IR(A) �= IN(A) �= IW (A). In addition, when the diagonal values

of C−1 are not set to zero, the statistical performance of KW and

IW (C−1) are equal.

Statistics compared, which have an intrinsicMatrix W definition of Y (see Table 1)

W = A IR(A), IN(A), IW (A)W = C IR(C), IN(C), IW (C)W = C−1 IR(C−1), IN(C−1), IW (C−1), K, K∗

Type I Error and PowerRecommendations that increase the power of the Mantel’s test,

and a comparison between the power of the Mantel’s and

Blomberg et al.’s (2003) test can be found in Hardy and Pavoine

(2012). We focus here only on the different generalizations of

the Moran test (IR, IN , IW ), and the statistics K and K∗. The

performances of KW are equivalent to those of IW (C−1) and the

performances of Abouheif’s test are equivalent to those of IW (A)

(knowing that IW (A) = IR(A) = IN(A) see section Unification:

Widely Used Tests of Phylogenetic Signal are Based on the Cross-

Product Statistic). We have simulated data to evaluate type I error

and power (1-type II error) associated with each coefficient of

phylogenetic signal (IR, IN , IW , K, and K∗) when permutation

tests are used. The indices IR, IN , IW were applied with matrices

A, C, C−1 first with the diagonal values given by their definitions

and then by artificially setting zero on their diagonal to evaluate

the role of diagonal values in power and type I error (see also

File S4). We have shown in the previous sections that the indices

IR, IN , IW , K, and K∗ are all part of the same statistical framework,

the cross-product statistic. The differences between these indices

concern mostly the way the trait-based similarities among tips are

computed (column Y in Table 1), but also some restrictions on

the way phylogenetic (dis)similarities are computed (column Win Table 1) such as the row normalization in IN . The restriction

on W in K and K∗ is stronger as they were developed only to

be used with matrix C−1 of phylogenetic differences among tips.

We summarize in Table 2 the statistics of phylogenetic signal

compared.

The calculus of the coefficients and the randomizations were

done with functions “gearymoran” of package ade4 (Dray and

Dufour 2007) of R and “Kcalc” of picante (Kembel et al. 2010)

and with personal R scripts. Matrix A was computed with func-

tion proxTips of package adephylo of R (Jombart and Dray

2010), Matrix C with function “vcv” of the package ape (Paradis

et al. 2004) and the inverse of C was obtained by function “ginv”

of package MASS (Venables and Ripley 2002) and checked for

congruence with function “solve” of the basis of R.



We simulated a series of trees as follows. We first simulated

pure birth trees (with birth rate of 0.1 as in Harmon and Glor 2010)

leading to relatively well-balanced trees (function “sim.bd.taxa”

in the package Treesim, Stadler 2011, of R, R Development Core

Team 2012). Next, we analyzed the effect of the strength of covari-

ance among tips by transforming the previous trees first moving

back most speciation events near the root (low expected covari-

ance among tips) and then, inversely, moving forward most spe-

ciation events near the tips (low expected covariance among tips)

using package geiger of R (function “deltaTree” with δ = 10 and

0.1, respectively, Harmon et al. 2009; Hardy and Pavoine 2012).

We then obtained asymmetric trees by applying UPGMA on val-

ues randomly drawn from a log-normal distribution (Euclidean

distance; mean and standard deviation of the distribution on the

log scale equal 0 and 1, respectively). Finally, we also analyzed

the power of the tests on nonultrametric trees (where the distance

from tips to root is not a constant) by simulating trees where the

topology is generated by splitting randomly the edges (function

“rtree” of package ape of R, Paradis et al. 2004). We generated

the branch lengths with a uniform distribution (bound between

0 and 1), next with a log-normal distribution (mean and stan-

dard deviation of the distribution on the log scale equal 0 and 1,

respectively). Nonultrametric trees can represent unequal evolu-

tionary rates in different parts of the phylogeny. Power analyses

were based on 1000 trees per model and type I error on 10,000

trees to have a better precision on the deviation from the nominal

α = 5% level. We simulated trees with 23 = 8, 25 = 32 and 27 =128 tips.

For power analyses, trait values were simulated per tree based

on Brownian (BM) and Ornstein–Ulenbeck (OU) models with

σ2 = 1, θ = 0, and α = 2, 4, 6, 8, or 10 (scaling the maximum

distance from a tip to the root of the tree equal to unity, see

Pavoine et al. 2008 for details). We analyzed type I error by four

models: (1) trait values drawn from a normal distribution with

mean θ = 0 and variance σ2 = 1; (2) trait values drawn from

a log-normal distribution with mean θ = 0 and variance σ2 =1 on the log scale; (3) for n values simulated, n−1 were drawn

from the normal distribution and an extreme value was added as

max(n−1 values) + range(n−1 values) × 10; (3) for n values

simulated, n−1 were drawn from the normal distribution and

an extreme value was added as max(n−1 values) + range(n−1

values) × 100.

With the random normal and the log-normal distributions of

trait values, all type I errors were correctly close to 5% (results

of the type I and power analyses are detailed in File S5). The dis-

tribution of the 10,000 simulated P-values per model was always

uniform (even) from 0 to 1. When the trees were ultrametric and

relatively well balanced (i.e., here with the pure birth model), the

distribution of P-values was still close to even from 0 to 1 when an

extreme value was added to the trait dataset. When the trees were

ultrametric but unbalanced and the number of tips was high (32 or

128) the type I error was inflated by an extreme trait value in which

case the distribution of P-values was right-skewed with many low

P-values and a few large P-values (typically when the diagonal

values of the phylogenetic proximity/differences matrices were

included) or of U-shape with high number of P-values close to 0

and 1 and low number of intermediate values (typically when the

diagonal values of the phylogenetic proximity/differences matri-

ces were not included). In contrast, when the number of tips was

low (eight tips only), the type I error decreased below 5% leading

to too conservative tests.

When the trees were nonultrametric and whatever the branch-

length model, the type I error was correct for IR(C−1) with diag-

onal values and always near to correct (except with eight tips) for

K∗. It was inflated in all other cases except two: (1) with IR(C)

with diagonal values the type I error was lower than 5% and the

distribution of P-values was skewed to the left with 32 or 128 tips

and correctly close to 5% with eight tips; (2) with IN(C−1) and

the most extreme value we considered the type I error was lower

than 5% and the distribution of P-values was skewed to the left.

Powers of tests increased with the number of tips in the

phylogenetic trees as previously shown for instance by Pavoine

et al. (2008) and Hardy and Pavoine (2012). We present the results

of the power analyses for 128 tips in Figure 4. Results for eight

and 32 tips can be found in File S5. Whatever the number of tips,

we obtained the following main results (Fig. 4):

Result 1: Power is impacted by the shape of the phylogenetic

tree, the matrix of phylogenetic proximity/differences and the

way trait-based proximities are computed in the different statistics

based on Moran (1948) and Blomberg et al. (2003) and summa-

rized in Table 1.

Result 2: When A is used to describe phylogenetic prox-

imities, considering the positive diagonal values of matrix A as

defined in Pavoine et al. (2008) instead of arbitrarily setting them

equal to zeros can increase the power of the generalized Moran

tests.

Result 3: When C is used to describe phylogenetic proximi-

ties, the row normalization used in index IN decreases the power

of the test for phylogenetic signal.

Result 4: When C−1 is used to describe phylogenetic dif-

ferences, the differences in power among the generalized Moran

tests are strongly dependent on whether the phylogenetic tree is

ultrametric. With ultrametric trees, indices IR and IN used with

zero values on the diagonal of C−1 decrease power; the index IW

slightly decreases power in comparison with K, K∗, and IR with

positive diagonal values for C−1. With nonultrametric trees, K∗

and IR with positive diagonal values for C−1 have the highest

powers.

Result 5: The highest powers are associated with coefficients

that use C−1.



Figure 4. Power analyses for trees with 128 tips. Results are given per tree shape and per phylogenetic matrix used (A, C, or C−1). A

symbol legend is given at the bottom of the panel. All numerical values can be found in File S5. Results for trees with eight and 32 tips

can be found in File S5.

Result 6: The use of C to describe phylogenetic proximities

generally decreases power in comparison with A and C−1.

Because Pavoine et al. (2008) suggested that contrasting val-

ues in phylogenetic (dis)similarity matrices could increase the

power of the tests, we calculated the coefficient of variation (CV)

of matrices A, C, and C−1. Detailed results are given in File S6.

On average over all simulated trees, the CV increased from C(1.645 for the off-diagonal values and 0.145 for the diagonal val-

ues), through A (3.152 for the off-diagonal values and 1.637 for

the diagonal values), to C−1 (12.304 for the off-diagonal values,

in absolute value, and 1.342 for the diagonal values). The CV in-

creased with the number of tips for A, C−1, and the off-diagonal

values of C but not for the diagonal values of C.

DiscussionIn this article, we showed that all tests for phylogenetic signal we

considered (Mantel, Moran, Abouheif, and Blomberg et al.) are

connected to each other being related to a cross-product statistic.

This means that the observed differences in their statistical perfor-

mances cannot depend on the tests themselves; the only relevant

change among the different approaches is the way the pairwise

phylogenetic and trait (dis)similarities are computed. A posteriori,

this seemingly counterintuitive result is not completely surpris-

ing. For instance, all tests for phylogenetic signal are aimed at

comparing two datasets that are usually produced in two different

formats: a phylogenetic tree and a vector (or a matrix) of traits.

Accordingly, all these methods explicitly or implicitly convert

trait values and phylogeny into pairwise (dis)similarities to calcu-

late the test statistic shifting the attention from the mathematical

formulation of the tests to how the (dis)similarity matrices are ob-

tained. This observation has a number of valuable consequences.

Here we analyzed three matrices of phylogenetic

(dis)similarities. We discovered C−1 as a matrix of phylogenetic

differences among tips where the phylogenetic difference depend

on volumes of subtrees degraded by the loss of one or two tips

in a phylogenetic tree. This definition of C−1 allowed us to unify

Mantel’s and Moran’s statistics with Blomberg et al. statistics and

thus to compare their performance within this unique statistical

framework of the cross-product. Power analyses for phylogenetic

signal tests were performed in Pavoine et al. (2008) and Hardy

and Pavoine (2012). In Pavoine et al. (2008) only IW was ana-

lyzed and it was shown that there may be a strong effect of the

choice of the phylogenetic similarity matrix in Moran’s permuta-

tion test. In Pavoine and Hardy (2012), the focus was put on Man-

tel test and K and it was shown that the power of the Mantel test

might in certain circumstances exceed the power associated with

K. Compared to these previous studies here we analyzed several



generalizations of Moran’s index, K, K∗, and KW , which altogether

encompassed a range of different ways of measuring phylogenetic

(dis)similarities and trait-based (dis)similarities among the tips of

a phylogenetic tree. We also analyzed the impact of the positive

diagonal of matrices A, C, C−1 of phylogenetic (dis)similarities

among tips on the performance of the tests.

Harmon and Glor (2010) stated that converting raw data into

matrices of pairwise differences among species is an inefficient

process that reduces power of tests. They also stated that, unlike

the Mantel test, data are not converted into pairwise distances

to calculate K. To the contrary, although this is not immedi-

ately evident, we have seen that data are actually converted into

(negative) measures of phylogenetic differences between tips. In

fact, Blomberg et al.’s K uses a particular phylogenetic matrix

of differences among tips that are positive on the diagonal and

negative elsewhere. In our simulations, this matrix C−1 of phylo-

genetic differences was often associated with the highest powers

of test for phylogenetic signal. When the trees were ultramet-

ric, K and IW (C−1) gave close but slightly lower powers than K∗

and IR(C−1). Pavoine et al. (2008) found that IW (A) had a higher

power than IW (C) whatever the tree shape except for comb-like

trees with which IW (C) had approximately the same but slightly

higher power than IW (A). A similar result was obtained here when

speciation events were driven toward the tips of the tree (see also

Hardy and Pavoine 2012 for a similar result with Mantel test).

However even in that case, our simulations showed that K∗ and

IR(C−1) had higher power than all statistics applied to matrices

A and C. When the trees were not ultrametric, K∗ and IR(C−1)

clearly reached much higher powers and correct type I errors

even in presence of outliers in trait values. IR(C−1) is very closely

related to Mantel tests, which confirms that Mantel test power

might be high with appropriate definitions of phylogenetic and

trait differences among tips (Hardy and Pavoine 2012). K∗ dif-

fers from the scheme of the Mantel test by the use of both a

phylogenetically weighted mean for the tips’ trait values and an

unweighted mean. Despite these differences we obtained similar

results in both type I and power analyses for K∗ and IR(C−1). More

generally, in our simulations, the diagonal values of phylogenetic

proximity/difference matrices tended to increase the power of the

tests. Contrary to what is done in spatial analysis, we thus recom-

mend their use in phylogenetic analysis. Finally, despite not based

on branch lengths, matrix A provided surprisingly high powers as

observed already by Pavoine et al. (2008). Matrix A with positive

diagonal values satisfies IW (A) = IR(A) = IN(A). A with any gen-

eralized I index is thus particularly recommended in all situations

when branch lengths are unavailable.

In addition to power analyses, we have provided a thorough

analysis of type I error. The type I error of tests for phyloge-

netic signals has generally been studied with normal traits and

the sensitivity to asymmetric distributions and to outliers was

neglected. Asymmetric distributions of trait values might result

from independent evolution along a phylogenetic tree where a tip

is greatly separated to the other tips. However real data might also

contain outliers or might have intrinsic asymmetric distributions

even if this asymmetry is not the result of an evolution along a

long branch in a phylogenetic tree. Our simulations show that

most generalized Moran tests are impacted by outliers but not by

asymmetric distributions (here log-normal distributions). Despite

K∗ and IR(C−1) were robust to outliers with nonultrametric phy-

logenetic trees, they were associated with inflated type I error in

presence of outliers as all other statistics with ultrametric phylo-

genetic trees. We thus advocate that transformations are used on

data to limit the impact of outliers in all tests for phylogenetic

signal. For instance, body weight, one of the most studied traits,

can be transformed by cubic root and logarithm to reduce the

effect of potential outliers.

Given the lack of theoretical justification for K and the lower

power and less stable type I error compared to K∗, we suggest

abandoning this statistic K. Compared to KW , K∗ was found to

be more powerful with correct type I error in our simulations and

should thus be preferred when used in permutation tests at least in

the situations covered by our simulations. Overall, based on our

results, we recommend the use of K∗ or IR(C−1) (with positive

diagonal values for C−1) to test for phylogenetic signal whatever

the tree shape when branch lengths are available and accurate.

Nevertheless, although branch lengths were assumed to be known

in our simulations, in many real datasets they are expected to be

estimated. Incorrect branch lengths estimate could decrease the

power of K∗ and IR(C−1) to detect phylogenetic signal. To avoid

the unknown cost associated with assuming that the branch lengths

are known, Abouheif’s Cmean statistic can be recommended as it

provided nearly as high power as K∗ and IR(C−1) whatever the

tree shape. It should be recalled that all statistics assume that the

topology is known and accurate.

To explain differences in power of the statistics of

phylogenetic signal when different matrices of phylogenetic

(dis)similarities were used, Pavoine et al. (2008) suggested that

the high power associated with A could be due to more contrast-

ing values in A than in C. To demonstrate that, they calculated the

average CV of the off-diagonal values of A and C, and obtained

a higher CV with A. With our data, the CV of the three matrices

of phylogenetic (dis)similarity increased from C, through A, to

C−1. We also found that in the only situation where higher power

is associated with C than with A (when most speciation events

are moved back near the root of the phylogenetic tree), the CV

of C is higher than that of A. Pavoine et al. (2008) suggestion is

thus supported by our article. Pavoine et al. (2008) also suggested

that the higher power associated with A could be due to the fact

that A never considers that tips connected only at the root node

of the tree are not related. However, this hypothesis is refuted by



our present article as, despite C−1 considers that tips connected

only at the root node of the tree are not related, it is associated to

the highest power.

Future researches now need to take account, in all these co-

efficients of phylogenetic signal (Mantel r, generalized Moran

IR, IN , and IW , Blomberg et al. K, K∗, and the new KW ), of (1)

within-species variations or measurement errors; (2) groups of

traits; (3) traits of different statistical types. Within-species vari-

ations and measurement errors have already been considered for

K∗ (Ives et al. 2007). This consideration increased the power

of the permutation test based on K∗ (Hardy and Pavoine 2012).

In addition, when approaches that incorporate measurement er-

rors were used, phylogenetic signal could be detected even with

small phylogenies with low number of tips (Zheng et al. 2009).

Adapting these developments for the other indices is needed to

complete the evaluation of their performance. Regarding the use

of groups of traits, tests of phylogenetic signal based on combined

traits have been developed by Zheng et al. (2009). Jombart et al.

(2010) used IN with several traits at the same time to describe

lineage-dependent phylogenetic signals in combinations of traits.

These developments are particularly needed in ecology as many

ecological processes involve a combination of traits rather than a

single trait. From a biological viewpoint, the species ecological

behavior, for example, is expected to be driven by complex in-

teractions among functional traits that are not fully independent

from each other (see for instance Milla et al. 2009). Therefore,

in some cases the researcher may be more interested in testing

for phylogenetic signal in a combination of traits, rather than in a

single trait only, and this is easily done if single trait differences

between species are combined into a multivariate pairwise dis-

similarity matrix (Pavoine et al. 2009, Ricotta and Moretti 2010).

Regarding the statistical type of traits, while tests of phyloge-

netic signal are easily performed on quantitative variables or on

ordinal variables transformed to ranks, it is unclear how to deal

with nominal traits. The usual approach consists in coding the

information in nominal variables by as many independent binary

variables as the number of categories. Then, each binary variable

is tested separately for phylogenetic signal. However, the cate-

gories of a nominal variable may often be nonexclusive. That is,

a species may be characterized by the simultaneous presence of

two or more character states. These nonexclusive states may be

binary coded as is done for the exclusive states, or they may be

fuzzy coded (Chevenet et al. 1994). In this case, each character

state receives a positive score in the range of 0–1 that describes

the affinity of a species for that state. For example, diet habits

in animals or Grime’s (1979) CSR (Competitor-Stress tolerator-

Ruderal) strategies in plants are typically coded as fuzzy variables.

Nonetheless, irrespective of how nonexclusive nominal variables

are coded, a multivariate measure of dissimilarity or disagreement

between pairs of species seems the most straightforward solution

for summarizing interspecies functional differences (Podani and

Schmera 2007). The cross-product can thus be adapted easily to

deal with any type and number of traits.

We analyzed here a limited number of indices that can be

derived from the cross-product statistic. Overall the power of the

test is influenced by the choice of the phylogenetic and functional

interspecies (dis)similarities wij and yij. Although the choice of an

appropriate functional (dis)similarity depends on the number and

type (i.e., quantitative, ordinal, nominal, etc.) of the selected traits

(Legendre and Legendre 1998; Pavoine et al. 2009), the choice

of an appropriate phylogenetic (dis)similarity matrix primarily

depends on whether branch lengths are known and whether a par-

ticular model of evolution is assumed (e.g., Brownian model). In

this view, the general formulation of equation (1), which is at the

core of the tests for phylogenetic signal analyzed in this article

offers very high flexibility in the calculation of the phylogenetic

distance matrix W = [wij], depending on the available phylogeny

and on the problem under scrutiny, that is to say the reasons why

the test of phylogenetic signal is performed. In Abouheif’s test,

the matrix of phylogenetic proximity is obtained solely from the

tree topology assuming all branch lengths to be equal. However,

as stressed by Crozier (1997, p. 243): “[Phylogenetic] measures

using branch-lengths are better than procedures relying solely

on topology.” Despite that, we showed that matrix A maintain

high power of tests for phylogenetic signal especially when it

is used with its positive diagonal and matrix A was found often

highly associated with matrix C−1. These two matrices have the

particularity of being relative measures of phylogenetic proximi-

ties/differences, as the value of proximity/difference between two

tips is not only dependent on the shared history of the taxa at these

tips but also depend on the level of shared history with the other

tips of the tree. Such relative, integrative measures of phylogenetic

proximities/differences appear to enhance the power of tests.

To conclude, methods for testing for phylogenetic signal

in functional traits should be possibly designed to deal with the

particular question asked. The cross-product statistic offers a

common statistical framework for widely used tests for phylo-

genetic signal. It also offers large flexibility into well-known

tests for phylogenetic signal, which provides the opportunity for

adapting tests to the taxa at hand, the quality of the phylogeny and

trait measurements and to the particular question asked. By this

it opens new directions for analyzing multivariate phylogenetic

signals in biological traits ensuring correct type I error and high

power.

ACKNOWLEDGMENTSThe authors would like to thank the editor and anonymous referees fortheir useful comments on our article.



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Associate Editor: E. Abouheif



Supporting InformationAdditional Supporting Information may be found in the online version of this article at the publisher’s website:

File S1. Blomberg et al. (2003) K and K∗ are based on a cross-product.

File S2. Resemblance between the matrix C−1 used in Blomberg et al. (2003) and matrix A implicitly used in Abouheif (1999)

and defined by Pavoine et al. (2008).

File S3. Details on the statistics of phylogenetic signal.

File S4. Positive diagonal values in matrices of phylogenetic proximities/differences.

File S5. Power and type I error analyses—all values.

File S6. Coefficient of variation of matrices A, C, and C−1—all values.

File S7. R scripts.

File S8. R functions (see file S7 for explanations).


Documents

TESTING FOR PHYLOGENETIC SIGNAL IN BIOLOGICAL TRAITS: THE UBIQUITY OF CROSS-PRODUCT STATISTICS