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Phylogenetic diversity measures based

on Hill numbers

Anne Chao1,*, Chun-Huo Chiu1,2 and Lou Jost3

1

Institute of Statistics, National Tsing Hua University, Hsin-Chu, Taiwan 30043

2

Institute of Statistics, National Chiao Tung University, Hsin-Chu, Taiwan 30043

3

Via a Runtun, Ban˜os, Tungurahua Province, Ecuador

We propose a parametric class of phylogenetic diversity (PD) measures that are sensitive to

both species abundance and species taxonomic or phylogenetic distances. This work extends

the conventional parametric species-neutral approach (based on ‘effective number of species’ or

Hill numbers) to take into account species relatedness, and also generalizes the traditional phylo-

genetic approach (based on ‘total phylogenetic length’) to incorporate species abundances. The

proposed measure quantiﬁes ‘the mean effective number of species’ over any time interval of

interest, or the ‘effective number of maximally distinct lineages’ over that time interval. The pro-

duct of the measure and the interval length quantiﬁes the ‘branch diversity’ of the phylogenetic

tree during that interval. The new measures generalize and unify many existing measures and

lead to a natural deﬁnition of taxonomic diversity as a special case. The replication principle

(or doubling property), an important requirement for species-neutral diversity, is generalized to

PD. The widely used Rao’s quadratic entropy and the phylogenetic entropy do not satisfy this

essential property, but a simple transformation converts each to our measures, which do satisfy

the property. The proposed approach is applied to forest data for interpreting the effects of

thinning.

Keywords: doubling property; Hill numbers; phylogenetic diversity; replication principle;

species-neutral diversity; taxonomic diversity

‘We are all blind men (and women) trying to describe a

monstrous elephant of ecological and evolutionary

diversity...’

(Nanney 2004, p. 721)

‘Phylogenetic measures are better indicators of conser-

vation worth than species richness, and measures

using branch-lengths are better than procedures

relying solely on topology...’

(Crozier 1997, p. 243)

1. INTRODUCTION

An enormous number of diversity measures have

been proposed, not only in ecology but also in gen-

etics, economics, information science, linguistics,

physics and social sciences, among others (e.g.

Pielou 1975;Magurran 2004). Until recently, most

of these measures were species-neutral, treating

all species as if they were equally distinct. However,

as Pielou (1975, p. 17) was the ﬁrst to notice,

the concept of diversity could be broadened to con-

sider taxonomic differences between species. Many

biologists (see Vellend et al. 2010 for a review) have

recognized that, all else being equal, an assemblage

of phylogenetically divergent species (say, eagle,

magpie and dunlin) is in an important sense more

diverse than an assemblage consisting of closely

related species (magpie, blue magpie and tree pie).

Since there was never real agreement among

biologists about the simpler base concept of species-

neutral diversity (e.g. Hulbert 1971;Routledge

1979;Patil & Taillie 1982;Purvis & Hector 2000;

Jost 2007;Jost et al. 2010), and since there is even

less agreement about how to incorporate phylogenetic

differentiation (e.g. Crozier 1997;Faith 2002;

Cavender-Bares et al. 2009;Pavoine et al. 2009;

Vellend et al. 2010), we now face a hyperdiverse

and rapidly increasing assemblage of non-neutral

diversity measures.

We show below that most of these non-neutral

measures lack an essential mathematical property

implicit in biological reasoning about diversity. Con-

clusions based on these measures will often be

invalid, especially in conservation applications

(Hardy & Jost 2008). We derive a general class of

measures that take into account both species abun-

dances and species phylogenetic differences, and that

possess all the mathematical properties implicit in

standard biological reasoning about diversity. The

new measures behave more intuitively than previous

measures. Many of the previous measures can be

transformed into this new class.

*Author for correspondence (chao@stat.nthu.edu.tw).

Dedicated to the memory of Ross Crozier, a long-time friend of

Anne Chao.

Electronic supplementary material is available at http://dx.doi.org/

10.1098/rstb.2010.0272 or via http://rstb.royalsocietypublishing.org.

One contribution of 16 to a Discussion Meeting Issue ‘Biological

diversity in a changing world’.

Phil. Trans. R. Soc. B (2010) 365, 3599–3609

doi:10.1098/rstb.2010.0272

3599 This journal is q2010 The Royal Society

2. PREVIOUS NON-NEUTRAL MEASURES

Most of the non-neutral measures that have been pro-

posed are generalization of the classic species-neutral

ecological diversity measures: species richness, the

Shannon entropy and the Gini –Simpson index. The

pioneering work of Vane-Wright et al. (1991) general-

ized species richness to take into account cladistic

diversity (CD), based on the total nodes in a taxo-

nomic tree. Subsequent important work was done by

Faith (1992,1994), Crozier (1992,1997), Weitzman

(1992,1998) and Warwick & Clarke (1995).Faith

(1992) deﬁned the phylogenetic diversity (PD) as the

sum of the branch lengths of the phylogeny connecting

all species in the target community. This concept of

PD is essentially a measure of the total amount of evol-

utionary history embodied in an assemblage since the

time of the most recent common ancestor of the

assemblage. The branch lengths may be proportional

to time of divergence, or they may be proportional to

the number of base changes in a given gene or may

use some other measures of change. If the branch

lengths are proportional to divergence time, all

branch tips are the same distance from the tree base

(the ﬁrst node). Such trees are called ‘ultrametric

trees’ and have particularly simple mathematical

properties.

These generalizations of species richness do not

take into account species relative abundances, because

nearly all early studies were based on a coarse spatial

scale, and data were mostly collected from museum

specimen records; thus, relative abundances could

not be reliably estimated. These measures are still

useful in many conservation purposes or in cases

where species abundances are difﬁcult to count, such

as micro-organisms or clumped plants. However,

species abundances, if available, provide a more com-

plete description of the ecosystem, and it seems

reasonable from the perspective of community ecology

to weigh a lineage by the numerical importance of its

descendants. There is also a strong practical motiv-

ation for using measures that weigh species by their

abundance. In many ecosystems, most species are

rare, and unreasonable or impossible effort is required

to detect them all. Species richness is therefore very

difﬁcult to estimate reliably. By contrast, most abun-

dance-based diversity measures can be reliably

estimated from small samples.

Diversity measures combining both phylogeny and

abundances have been proposed in the literature

(Rao 1982;Solow et al. 1993;Solow & Polasky

1994;Warwick & Clarke 1995;Izsa

´k & Papp 2000;

Webb 2000; Ricotta & Szeidl 2006,2009;Weikard

et al. 2006;Hardy & Senterre 2007;Hardy & Jost

2008;Allen et al. 2009;Pavoine et al. 2009;Cadotte

et al. 2010). Rao’s quadratic entropy (Rao 1982), a

generalization of the Gini–Simpson index, is the

most well-developed of these. When pairwise differ-

ences between species are speciﬁed, Rao’s Qgives

the mean phylogenetic distance between any two

randomly chosen individuals in the community:

Q¼X

i;j

dij pipj;ð2:1Þ

where d

ij

denotes the phylogenetic distance between

species iand j, and p

i

and p

j

denote species relative

abundance of species iand j.Ricotta & Szeidl (2009)

proposed a transformation of Qif distances are

normalized to the range of [0, 1]:

^

Q¼1

1Q:ð2:2Þ

The advantages of this transformation will become

clear in the following sections.

Allen et al. (2009) generalized the Shannon entropy

to take into account phylogenetic differences. For a

rooted tree, their phylogenetic entropy H

p

is

Hp¼X

i

Liailog ai;ð2:3Þ

where the summation is over all branches, L

i

is the

length of branch i, and a

i

denotes the abundance des-

cending from branch i. This measure includes the

Shannon entropy as a special case.

For ultrametric trees, Pavoine et al. (2009) inte-

grated Faith’s PD, Allen et al.’s H

p

and Rao’s Qinto

a parametric class of measure called I

q

. The parameter

qcorresponds to the order in the Tsallis (1988) gener-

alized entropy. The three named measures correspond

to the orders q¼0, 1 and 2; I

0

¼Faith’s PD minus the

tree height; I

1

¼H

p

and I

2

¼Q.

3. THE REPLICATION PRINCIPLE

While biologists have traditionally used the Shannon

entropy and the Gini– Simpson index to quantify

diversity, this practice is inconsistent with their own

rules of inference about diversity. For example, users

of these measures often judge the compositional simi-

larity of two or more groups by taking the ratio of

mean within-group diversity to total (pooled) diversity.

If the within-group diversity is close to the total diver-

sity, biologists infer that the groups are similar in

composition. Yet, when used with the Shannon

entropy or the Gini –Simpson index, this ratio does

not directly reﬂect compositional similarity. When

within-group diversity is high, the ratio approaches

unity, supposedly indicating that the groups are

nearly identical in composition, even if the groups

are in fact completely distinct (no shared species).

Conservation biologists use diversity measures to

judge the impact of human activities or to design con-

servation strategies. Yet, the Shannon entropy and the

Gini–Simpson index can be very misleading when

judging human impacts, and are logically self-contra-

dictory when used to assess conservation plans ( Jost

2009), because of their nonlinearity with respect to

increasing diversity. We conclude that these measures,

in spite of their popularity, do not capture biologists’

notions of diversity. The forms of reasoning that biol-

ogists apply to diversity lead to invalid conclusions

when used with these measures. These common

forms of reasoning about diversity implicitly assume

that diversity obeys the ‘replication principle’. The

replication principle for species-neutral diversity

states that if we have Nequally large, equally diverse

groups with no species in common, the diversity of

the pooled groups must be Ntimes the diversity of a

3600 A. Chao et al. Phylogenetic diversity measures

Phil. Trans. R. Soc. B (2010)

single group. Many authors (MacArthur 1965,1972;

Whittaker 1972;Peet 1974;Routledge 1979;Jost

2006,2007,2009;Jost et al. 2010) have shown that

only diversity measures that satisfy this replication

principle or ‘doubling property’ (Hill 1973;Jost

2007,2008,2009;Ricotta & Szeidl 2009) give math-

ematically, logically and intuitively correct results.

The replication principle is best known in economics,

where it has long been recognized as an important

property of concentration and diversity measures

(Hannah & Kay 1977).

To see the importance of this property, compare the

behaviour of the Gini –Simpson index 1 Pip2

iwith

that of the inverse Simpson concentration 1=Pip2

i,

which does obey the replication principle. Consider

an archipelago of 20 equally large, equally diverse

islands, each with completely distinctive tree ﬂoras.

There are no shared species among islands. Assume

the tree ﬂoras have the frequency distributions of the

trees of Barro Colorado Island, Panama. To measure

the compositional similarity among the islands, ecolo-

gists will take the mean diversity of the islands (0.95)

and divide it by the diversity of the archipelago as a

whole (0.998). For this example, the ratio is 0.95/

0.998 ¼0.95, near unity, supposedly indicating that

the islands are nearly identical in composition, even

though the islands are actually completely distinct

(no shared species). This ratio does not reﬂect compo-

sitional similarity. Doing the same with the inverse

Simpson concentration gives a ratio of 20.3/406 ¼

1/20, the smallest possible value for a set of 20 equally

large islands, correctly showing that they are comple-

tely distinct in composition. The same problems

apply to the Shannon entropy, and are resolved by

using the exponential of Shannon entropy.

Since the Shannon entropy and the Gini–Simpson

index do not obey the replication principle, neither do

their phylogenetic generalizations—Rao’s quadratic

entropy, Allen et al.’s phylogenetic entropy and

Pavoine’s generalized Tsallis entropy family. Though

each of these has a useful interpretation, they cannot

be applied directly to judge efﬁcacy of conservation

plans, magnitudes of human impacts or compositional

similarity among groups. These considerations motiv-

ate our search for a family of PD measures that are

sensitive to species relative abundances and that obey

the replication principle.

Species-neutral diversity measures that do obey the

replication principle (the ‘true’ diversity deﬁned by

Jost 2007) include species richness, the exponential

of Shannon entropy and the inverse Simpson concen-

tration. These are special cases of a general class of

measures, known as Hill numbers (Hill 1973):

qD¼X

S

i¼1

pq

i

!

1=ð1qÞ

;ð3:1Þ

where Sis the number of species, p

i

is the relative

abundance of the ith species and the parameter q,

called the ‘order’ of the diversity measure, determines

its sensitivity to species frequencies. The measure

0

D

corresponds to species richness and

2

Dcorresponds

to the inverse Simpson concentration, giving roughly

the number of ‘very abundant’ species in a community

(Hill 1973). The measure is undeﬁned when q¼1,

but the limit as qapproaches unity exists and equals

1D¼lim

q!1

qD¼exp X

S

i¼1

pilog pi

!

;ð3:2Þ

which is the exponential of Shannon entropy. Roughly,

1

Dmeasures the number of ‘common’ (or ‘typical’)

species in a community. Hill numbers provide a

uniﬁed framework for the three most popular groups

of diversity measures, q¼0, 1 and 2.

The Hill numbers are interpreted as the ‘effective

number of species’ or ‘species equivalents’

(MacArthur 1965,1972;Hill 1973;Jost2006,

2007). For any community, if we obtain a value

q

D¼w, then the diversity of this community is the

same as that of a community with wequally abundant

species. Hill numbers will be the basis for our phylo-

genetic generalization. We give the appropriate

phylogenetic generalization of the replication principle

in §6.

4. PHYLOGENETIC DIVERSITY MEASURES

(a)Conceptual framework

To emphasize the conceptual simplicity of our frame-

work, we ﬁrst explain it verbally, and then derive the

corresponding formulae. We start by considering a

phylogenetic tree that uses divergence times to place

the nodes (so that the tree is ultrametric). At any

given moment t, we can ﬁnd the species by slicing

the tree as in ﬁgure 1a. We can ﬁnd their ‘abundances’

by summing the abundances of their descendants in

the present-day assemblage. These abundances are

not estimates of the actual abundances of these ances-

tral species at time t, but rather measures of their

importance for the present-day assemblage. The lin-

eage diversity at time tcan be found by dividing

these abundances by the total abundance at this time

t, and inserting these relative abundances into the

equation for Hill numbers of order q, equation (3.1).

We call this

q

D(t).

We can average the diversities

q

D(t) of the phyloge-

netic tree over any time interval of interest. We will be

interested in the time interval from 2Tyears to the

present time. While previous phylogenetic studies

have focused on Tas the age of the ﬁrst node (root),

we do not make this restriction, because we may

want to compare diversities of systems with different

ages of the ﬁrst node. Also, how diversity varies

with time for any individual tree provides important

information about evolution.

The average diversity of order qover the interval

[2T, 0] incorporates information about the tree’s

branching pattern, its relative branch lengths and the

relative abundances ﬂowing through each of its

branch segments. For a given present-day diversity,

this average will be large when there are many deep

branches, each well represented in the present-day

assemblage. It will be small when all branches

emerge recently and/or when older branches are

poorly represented in the present-day assemblage.

Phylogenetic diversity measures A. Chao et al. 3601

Phil. Trans. R. Soc. B (2010)

It is always less than or equal to the present diversity

of order q.

There are many ways to take this average. If we want

the replication principle to be valid in its strongest

possible form, then we must average the diversities

q

D(t) according to Jost’s (2007) derivation of the for-

mula for the mean (

a

) diversity of a set of equally

weighted assemblages. This mean diversity over the

time interval [2T, 0] will be called q

DðTÞ(mean diver-

sity of order q over T years). With this choice of mean,

when Nmaximally distinct trees with equal mean

diversities (for ﬁxed T) are combined, the mean diver-

sity of the combined tree is Ntimes the mean diversity

of any individual tree. The branching patterns, abun-

dances and richnesses of the Ntrees can all be

different, as long as each of the trees is completely dis-

tinct (all branching off from the earliest point in the

tree, at or before time T). Some choices of averaging

formulae obey weaker versions of the principle, and

these may be useful for some purposes. We discuss

an alternative choice of mean in §8.

We may want to consider not just the mean diversity

but the branch or lineage diversity of the tree as a

whole, over the interval from –Tto present. At any

point within a branch, the abundance or importance

of each branch lineage is the sum of the abundances

of the present-day species descending from that

point, as described above. Then the total diversity of

all the ‘species’ that evolved in the tree during the

time interval [2T, 0] is found by taking the Hill

number of this entire virtual assemblage of ancestral

species. The Hill numbers depend only on the relative

abundances of each species, so we need to divide the

abundances by the total abundance of all the species

in the tree. If each branch is weighted by its corre-

sponding branch length, then we show below that

this diversity depends only on the branching pattern

and on the relative abundances of the species in the

present-day assembly. We call this measure ‘phyloge-

netic diversity of order q through T years ago’or‘branch

diversity’ and denote it by

q

PD(T). This turns out

to be just the product of the interval duration T

and the mean diversity over that interval, q

DðTÞ.For

q¼0 (only species richness is considered), and T¼

the age of the ﬁrst node, this branch diversity is just

Faith’s PD.

Instead of using time as the metric for a phylo-

genetic tree, we often want to use a more direct

measure of evolutionary work, such as the number of

base changes at a selected locus, or the amount of

functional or morphological differentiation from a

common ancestor. The branches of the resulting tree

will then be uneven, so the tree will not be ultrametric.

However, we can easily apply the idea of branch diver-

sity to such non-ultrametric trees. The branch lengths

are calculated in the appropriate units, such as base

t = 0

(present time)

p1 + p2 + p3

p2 + p3

p1p2p3p4

T3

t = –T

p4

p4

p1

T

(a)(b)

slice 3

slice 2

slice 1

T2

T1

1

2

4

3.5

p1 = 0.5

p2 = 0.2

p3 = 0.3

p2 + p3 = 0.5

Figure 1. (a) A hypothetical ultrametric rooted phylogenetic tree with four species. Three different slices corresponding to

three different times are shown. For a ﬁxed T(not restricted to the age of the root), the nodes divide the phylogenetic tree

into segments 1, 2 and 3 with duration (length) T

1

,T

2

and T

3

, respectively. In any moment of segment 1, there are

four species (i.e. four branches cut); in segment 2, there are three species; and in segment 3, there are two species. The

mean species richness over the time interval [2T,0]is(T

1

/T)4þ(T

2

/T)3þ(T

3

/T)2. In any moment of segment

1, the species relative abundances (i.e. node abundances correspond to the four branches) are fp1;p2;p3;p4g; in segment

2, the species relative abundances are fg1;g2;g3g¼fp1;p2þp3;p4g; in segment 3, the species relative abundances are

fh1;h2g¼fp1þp2þp3;p4g.(b) A hypothetical non-ultrametric tree. Let

Tbe the weighted (by species

abundance) mean of the distances from root node to each of the terminal branch tips.

T¼40:5þð3:5þ2Þ0:2þð1þ2Þ0:3¼4. Note

Tis also the weighted (by branch length) total node abundance

because

T¼0:54þ0:23:5þ0:31þ0:52¼4. Conceptually, the ‘branch diversity’ is deﬁned for an assemblage

of four branches: each has, respectively, relative abundance 0:5=

T¼0:125, 0:2=

T¼0:05, 0:3=

T¼0:075 and

0:5=

T¼0:125; and each has, respectively, weight (i.e. branch length) 4, 3.5, 1 and 2. This is equivalent to an assemblage

with 10.5 equally weighted ‘branches’: there are 4 branches with relative abundance 0:5=

T¼0:125; 3.5 branches with rela-

tive abundance 0:2=

T¼0:05; 1 branch with relative abundance 0:3=

T¼0:075 and 2 branches with relative abundance

0:5=

T¼0:125.

3602 A. Chao et al. Phylogenetic diversity measures

Phil. Trans. R. Soc. B (2010)

changes. In non-ultrametric cases, the time Tis replaced

by

T, the mean of the distances from root node to each

of the terminal branch tips (i.e. the mean evolutionary

change per species); see ﬁgure 1bfor a numerical

example. Thus, we can obtain the total effective

number of ‘changes’ based on Hill numbers.

(b)Formulae

To make the above discussion precise and derive

formulae from it, we need to introduce some notation.

Assume that for any ﬁxed time Tthe phylogenetic tree

is divided as ksegments with duration T

1

,T

2

,...,T

k

and species richness S

1

,S

2

,...,S

k

as in ﬁgure 1a.

Note that S

1

¼S, the present-day species richness.

Each branching point must form a segment boundary,

so that the species richness in any given segment is a

constant. Our derivation and formulae would be

unchanged by making ﬁner segment divisions. To

obtain the formulae for 0

DðTÞ, assume there are S

i

species (i.e. S

i

branches cut) in the ith segment.

Then, 0

DðTÞ(mean diversity of order 0 over T years)is

0

DðTÞ¼T1

TS1þT2

TS2þþTk

TSk

¼

0PDðTÞ

T:ð4:1Þ

When Tis the time corresponding to the root, then

0

PD(T) is Faith’s PD measure. Our equation (4.1)

connects Faith’s PD to the mean species richness

over the time interval from the terminal tips to

the root.

At each moment within a given segment, the

set of species relative abundances is constant. In

segment 1, the species relative abundances are

fp1;p2;...;pS1g;PS1

i¼1pi¼1. Assume that in

segment 2 the relative abundances are

fg1;g2;...;gS2g;PS2

i¼1gi¼1, ..., and in segment k

the relative abundances are fh1;h2;...;hSkg;

PSk

i¼1hi¼1(ﬁgure 1a). Without loss of generality,

we can assume T

1

,T

2

,...,T

k

are all positive integers,

because the mean diversity q

DðTÞis invariant to

the units of time. Weighing each moment in time

equally, we can conceptually imagine that there

are T

1

assemblages with abundance vector

fp1;p2;...;pS1g,T

2

assemblages with abundance

vector fg1;g2;...;gS2g..., and T

k

assemblages with

abundance vector fh1;h2;...;hSkg. There are a total

of T

1

þT

2

þ þ T

k

¼Tassemblages, and each is

given the same weight 1/T.Jost (2007) showed

that, in the context of calculating alpha diversity for

equally weighted assemblages, the alpha diversity

should be obtained by ﬁrst averaging the sums of

Ppq

i;Pgq

i,..., and Phq

i, and then converting this

average to a ‘true’ diversity by raising it to the power 1/

(1 2q).Weusethissamekindofaveragetoobtainthe

formula for q

DðTÞ(mean diversity of order q over T years)

q

DðTÞ¼ T1

TX

S1

i¼1

pq

iþT2

TX

S2

i¼1

gq

iþþTk

TX

Sk

i¼1

hq

i

()

1=ð1qÞ

:

ð4:2Þ

When q¼0, equation (4.2) reduces to equation (4.1).

Thesameformula(4.2)maybecomputedmore

easily by numbering every branch in the time interval

[2T, 0]. Denote the set of all branches in this time inter-

val by B

T

.Then,q

DðTÞcan be calculated as

q

DðTÞ¼ X

i[BT

Li

Taq

i

()

1=ð1qÞ

¼1

TX

i[BT

Li

ai

T

q

()

1=ð1qÞ

;ð4:3Þ

where L

i

is the length (duration) of branch iin the set B

T

and a

i

is the total abundance descended from branch i.

This diversity may also be interpreted as the effective

number of maximally distinct lineages (or species)

during the interval [2T, 0]. For maximally distinct

specieswehaveallbranchlengthsequaltoT, and thus

q

DðTÞreduces to Hill numbers

q

Din equation (3.1).

This gives a simple reference tree for a value of

q

DðTÞ¼z, i.e. the observed mean diversity in the time

period [2T, 0] is the same as the mean diversity of a

community consisting of zequally abundant and maxi-

mally distinct species with branch length T.

The effective diversity of the whole tree during the

interval [2T, 0] is the product of the effective

number of lineages during the interval and the duration

of the interval. We denote this measure by

q

PD(T)

(phylogenetic diversity of order q through T years ago):

qPDðTÞ¼Tq

DðTÞ¼TX

i[BT

Li

Taq

i

()

1=ð1qÞ

¼X

i[BT

Li

ai

T

q

()

1=ð1qÞ

:ð4:4Þ

This has dimensions of ‘effective number of lineage

years’. If q¼0, this equals

0

PD(T) as deﬁned above,

regardless of branching pattern or abundances. If all

species are maximally distinct and equally common,

and if Tis the age of the highest node, this equals

Faith’s PD for all q.

For an ultrametric tree, we can express the time

parameter Tas T¼Pi[BTLiai. Therefore, the time

length Tcan also be interpreted as the total abun-

dance (weighted by branch lengths) in the time

interval [2T, 0] and a

i

/Trepresents the relative

abundance of the ith branch. Using this idea,

equation (4.4) suggests that instead of dividing the

tree into several segments and treating the mean

diversity as the alpha diversity of several assemblages,

we could conceptually think of all the branch seg-

ments in the interval [2T, 0] as forming a single

assemblage consisting of relative abundances

fai=T;i[BTg, with each branch weighted by its

corresponding branch length. (Equivalently, we can

also think for each ithat there are L

i

equally weighted

‘branches’ with the relative abundance a

i

/T.) Then

the Hill number of order qfor this assemblage is

exactly the branch diversity

q

PD(T) given in equation

(4.4). Dividing this Hill number by T, we obtain

q

DðTÞgiven in equation (4.3).

For the extension to non-ultrametric trees, let B

T

denote the set of branches connecting all focal species

with mean base change

T. The total node abundance

Phylogenetic diversity measures A. Chao et al. 3603

Phil. Trans. R. Soc. B (2010)

weighted by branch lengths is

T¼Pi[B

TLiai, which

also represents the weighted (by species abundance)

mean evolutionary change per species (ﬁgure 1b). (In

ultrametric trees,

T¼T.) Based on the assemblage

consisting of all branches with relative abundance set

fai=

T;i[B

Tgand under the assumption that each

branch is weighted by its corresponding branch

length (ﬁgure 1b), parallel derivation gives the follow-

ing measures, which are exactly the same as those in

equations (4.3) and (4.4), except that the parameter

Tthere must be replaced by the mean quantity

T:

q

Dð

TÞ¼ X

i[B

T

Li

Taq

i

()

1=ð1qÞ

¼1

TX

i[B

T

Li

ai

T

q

()

1=ð1qÞ

ð4:5Þ

and

qPDð

TÞ¼ X

i[B

T

Li

ai

T

q

()

1=ð1qÞ

:ð4:6Þ

We thus can conclude that the diversity of a non-

ultrametric tree with mean evolutionary change

T

(however this might be measured) is exactly the

same as that of an ultrametric tree with time par-

ameter

T. Therefore, for non-ultametric trees, if

q

Dð

TÞ¼z, then the diversity is the same as the diver-

sity of an ultrametric tree consisting of zequally

abundant and maximally distinct species with

branch length

T.

(c)Relationship with Rao’s Qand phylogenetic

entropy H

p

In the limit as qapproaches unity, the formula q

Dð

TÞ

in equation (4.5) equals

1

Dð

TÞ¼exp X

i[B

T

Li

Tailog ai

"#

:ð4:7Þ

The measure 1

Dð

TÞhas the following simple relation-

ship with the phylogenetic entropy H

p

:

1

Dð

TÞ¼expðHp

TÞor logð1

Dð

TÞÞ ¼ Hp

T:ð4:8Þ

When q¼2, from equation (4.5), we have

2

Dð

TÞ¼ X

i[B

T

Li

Ta2

i

()

1

:ð4:9Þ

After some algebra, we have the relationship between

2

Dð

TÞand Rao’s quadratic entropy Q:

2

Dð

TÞ¼

T

TQ¼1

1Q=

T:ð4:10Þ

Formula (4.10) represents the equivalent number of

completely distinct species (of age

T) for the assem-

blage. Ricotta & Szeidl (2009) derived a similar

formula, given in equation (2.2), for the special case

in which the pairwise distance between any two species

is normalized to the range of [0, 1]. While their

formula is identical to our equation (4.10) for ultra-

metric trees when our time parameter Tis scaled to

1, for non-ultrametric trees, our theory leads to the

conclusion that the equivalent number of species for

Qshould be 1=ð1Q=

TÞ.

We give an example to illustrate this point. Consider

a non-ultrametric tree in which three equally

abundant species are maximally distinct with

branch lengths 1, 1 and 0.2, respectively, from a

divergence point. The pairwise distances between

the three species are d

12

¼1, d

13

¼0.6 and d

23

¼

0.6. We have Rao’s Q¼4.4/9 ¼0.489 and

T¼ð1=3Þð1þ1þ0:2Þ¼2:2=3¼0:733. Based

on our equivalent number of species formula, we

have 1=ð1Q=

TÞ¼3 maximally distinct species

with equal branch lengths of 0.733, and the total

length ¼0.733 3¼2.2, which is Faith’s PD. How-

ever, based on the Ricotta & Szeidl (2009) formula,

we obtain 1=ð1QÞ¼1:957, implying there are

1.957 maximally distinct species with branch length

of 1. The total length is thus 1.957 1¼1.957,

which is not Faith’s PD.

5. TAXONOMIC DIVERSITY

Rather than using time or the number of base

changes at a locus as our measure of evolutionary

work, we might want to use a more holistic measure

of evolutionary work, such as a phylogenetic tree

based on the classical Linnaean taxonomic categories.

Consider the special case in which each Linnaean

taxonomic category is given unit length, and assume

all species are classiﬁed in all levels. Our formulae

above can be easily applied to this ultrametric tree,

with Treplaced by an integer representing the

number of taxonomic categories needed to character-

ize the assemblage. We thus change the continuous

time parameter Tto an integer parameter L(level )

to distinguish taxonomic diversity from the general

PD measures q

DðTÞand

q

PD(T). If we use species

and genus, then L¼2; if we use species, genus and

family, then L¼3. Additional intermediate levels,

such as subgenus or subfamily, may be appropriate

depending on the group. Notice that in a taxonomic

tree, the total length is identical to the total number

of nodes. Setting all the segment lengths L

i

to unity

in equations (4.3) and (4.4), we have the following

mean diversity of order q for L taxonomic levels,q

DðLÞ,

q

DðLÞ¼ Piaq

i

L

1=ð1qÞ

¼1

LX

i

ai

L

q

()

1=ð1qÞ

;ð5:1Þ

where iis over all nodes in the Llevels taxonomy

tree. The measure q

DðLÞquantiﬁes ‘the mean effec-

tive number of cladistic nodes per level in a

taxonomic tree of Llevels’. The diversity of a taxon-

omy tree with q

DðLÞ¼zis the same as the diversity

of a community consisting of zequally abundant

species, with each species classiﬁed in its own genus

and family, so that there are zspecies, zgenera and

zfamilies.

3604 A. Chao et al. Phylogenetic diversity measures

Phil. Trans. R. Soc. B (2010)

The taxonomic diversity of order q for L levels,

q

TD

(L), is the product of q

DðLÞand the level L. This

measure quantiﬁes ‘the effective number of total

cladistic nodes in a taxonomic tree of Llevels’ and

has the formula

qTDðLÞ¼Lq

DðLÞ¼ X

i

ai

L

q

()

1=ð1qÞ

:ð5:2Þ

In the special case L¼1, the measure q

DðLÞ¼qD.

When q¼0,

0

TD(L)¼total number of nodes,

which is Vane-Wright’s CD. Equations (4.8) and

(4.10) reduce to the following transformations:

1

DðLÞ¼expðHp=LÞand 2

DðLÞ¼1=½1ðQ=LÞ; see

table 1 for a summary of all proposed measures

and their relationships with conventional measures.

The decomposition of taxonomic diversity into

diversity of each level is provided in the electronic

supplementary material.

6. REPLICATION PRINCIPLE FOR

PHYLOGENETIC DIVERSITY

Some basic properties of our proposed measures

(table 1) are summarized in the electronic supplemen-

tary material; details of the proofs are provided in

Chiu (2010) and Jost & Chao (in preparation).Inthis

section, we only reﬁne the concept of the replication

principle for phylogenetic trees, and prove its validity

for the most general case (i.e. non-ultrametric case),

implying that it is valid for all measures in table 1.

Suppose we have Ncompletely distinct assemblages

(no shared lineages), all with the same mean branch

length

T(hence same Tin the case of ultrametric

trees) and the same mean PD q

Dð

TÞ¼X. Then we

can prove the following strong replication principle:

if these assemblages are pooled in equal proportions,

the pooled assemblages have mean PD NX.

Proof. Suppose in tree k, the branch set is B

T;k

(we omit

Tin the subscript and just use B

k

in the fol-

lowing proof for notational simplicity) with branch

lengths fLik;i[Bkgand the corresponding

nodes abundances faik;i[Bkg,k¼1, 2, ...,N.

The Ntrees have the same mean diversity X, implying

Pi[BkðLik=

TÞaq

ik ¼X1qfor all k¼1, 2, ...,N. When

the Ntrees are pooled with equal weight for each tree,

each node abundance a

ik

in the pooled tree becomes

a

ik

/N. Then, the q

Dð

TÞmeasure for the pooled tree

becomes

X

N

k¼1X

i[Bk

Lik

T

aik

N

q

()

1=ð1qÞ

¼fN1qX1qg1=ð1qÞ

¼NX:

In our proof of this replication principle, the Nassem-

blages must have the same average quantity

T, but may

have different numbers of species if q.0, and the tree

structures of the Nassemblages can be totally

different.

7. EXAMPLES

To show the general behaviour of our proposed

measures, we give two simple hypothetical examples

in the electronic supplementary material. Here, we

apply the proposed q

DðTÞand

q

PD(T) measures to

the real forest data discussed by Shimatani (2001),

who collected data from the over-storey tree species

in the Fred Russ experimental forest in Michigan.

For illustrative purpose, we only consider the abun-

dance data of block 4 in his paper for two sites: CT

(thinned site) and CU (un-thinned site). Both sites

were 28-year-old (in 1999) secondary forests. The

two sites were dominated by oak trees. No thinning

was conducted for the CU site after clear cutting in

1971, while thinning was done for non-oak species

in the site CT in 1982 and 1996.

Shimatani (2001) proposed a four-level (species,

genus, family, subclass) taxonomic measure based on

the Simpson index, and concluded that the traditional

diversity indices and the taxonomic diversity consider-

ing species relatedness give different conclusions about

the effect of thinning. We constructed the phylogeny

trees for species in each site by using the software

PHYLOMATIC (from http://www.phylodiversity.net/phy-

lomatic;Webb & Donoghue 2004). The phylogenetic

tree for the species in the two sites, and the two sets

of species relative abundances, are shown in ﬁgure 2.

Table 1. A summary of species-neutral and phylogenetic diversity measures and their interpretations; all satisfy the

replication principle. CD, cladistic diversity (total number of nodes) by Vane-Wright et al. (1991); PD, phylogenetic diversity

(sum of branch lengths) by Faith (1992);Q, quadratic entropy, equation (2.1); H

p

, phylogenetic entropy, equation (2.3).

diversity types

species-neutral

diversity

taxonomic classiﬁcation

(Llevels)

ultrametric phylogenetic

tree

non-ultrametric phylogenetic

tree

diversity or

mean

diversity of

general

order q

q

D: equation (3.1),

Hill numbers

(effective number

of species)

q

DðLÞ: equation (5.1),

mean effective

number of cladistic

nodes per level

q

DðTÞ: equation (4.3),

mean effective number

of species (or lineages)

over Tyears

q

Dð

TÞ: equation (4.5), mean

effective number of

species (or lineages) over

Tmean base changes

q¼0 species richness CD/LPD/TPD/

T

q¼1 exp(entropy) exp(H

p

/L) exp(H

p

/T) expðHp=

TÞ

q¼2 1/Simpson 1/[1 2(Q/L)] 1/[1 2(Q/T)] 1=½1ðQ=

TÞ

branch (or

lineage)

diversity

q

DðLÞL: equation

(5.2), effective

number of cladistic

nodes for Llevels

q

DðTÞT: equation

(4.4), effective number

of lineage lengths over

Tyears

q

Dð

TÞ

T: equation (4.6),

effective number of base

changes over

Tmean base

changes

Phylogenetic diversity measures A. Chao et al. 3605

Phil. Trans. R. Soc. B (2010)

We calculated three types of diversities: (i) the mean

diversity q

DðTÞand the phylogenetic diversity

q

PD(T) based on the phylogeny trees and the relative

abundances in ﬁgure 2, (ii) the taxonomic diversity

q

DðLÞbased on taxonomic classiﬁcation in ﬁg. 1 of

Shimatani (2001), and (iii) the species-neutral diver-

sity based on Hill numbers (

q

D) in equation (3.1) for

q¼0, 1 and 2.

In ﬁgure 3, the proﬁle of q

DðTÞand

q

PD(T) when

0,T,150 is shown for q¼0, 1 and 2. For ultra-

metric trees, the two measures give consistent

comparison as clearly seen in ﬁgure 3. We focus on

comparing the measure q

DðTÞ, which gives the mean

effective number of species as a function of evolution-

ary time T. Based on species richness (q¼0), the

diversity q

DðTÞof the thinned site CT dominates

that of un-thinned site CU for all values of T. But for

the common species (q¼1) and very abundant species

(q¼2), we have the reverse conclusion. When abun-

dance is taken into account, the un-thinned CU site

is more diverse than the thinned CT site for all

values of T, except for a very small interval in the

case of q¼2.

Table 2 shows the three types of diversity

(q

DðTÞ;q

DðLÞand

q

D) for three orders of q(0, 1

and 2). All these three measures are in the same

units of species. The q

DðTÞmeasure is only shown

for T¼142.3, which is the age of the root in the

pooled phylogenetic tree. The taxonomic measure

q

DðLÞis computed for L¼4 level classiﬁcations. For

any ﬁxed order q, we had proved that

q

Dis always

greater than or equal to q

DðTÞand q

DðLÞ, and this is

seen numerically in table 2.

Based on table 2, we conﬁrm the ﬁnding of

Shimatani (2001) that the traditional Simpson diver-

sity measure

2

Dimplies that the thinned site is less

diverse. A similar implication is also valid for the

1

D

measure, whereas species richness

0

Dshows that the

thinned site is more diverse. Based on q

DðLÞ, the taxo-

nomic diversity of the thinned site for all three orders is

greater, but the difference is not large. Shimatani thus

concluded that the thinning operation contributed to

an increase in taxonomic diversity.

In contrast to Shimatani’s conclusion, our results

based on q

DðTÞfor q¼1 and 2 imply the opposite

conclusion, as shown in ﬁgure 3, and our results are

consistent with those based on the species-neutral

diversity. Our conclusion may be understood intui-

tively by noting that thinning concentrates the

abundance into a few species of intermediate phyloge-

netic distinctiveness (ﬁgure 2), while in the un-thinned

site, abundance is spread more equitably throughout

the phylogenetic tree. The plots in ﬁgure 3 provide

additional insights about the thinning effect when

both evolutionary history and species abundances

(q¼1 and 2) are considered.

8. CONCLUDING REMARKS AND DISCUSSION

(a)Advantages of the new measures

We have proposed a uniﬁed class of PD measures that

are based on Hill numbers and that obey the replica-

tion principle (§§3 and 6). Most previous PD

measures that take into account species abundances,

such as Rao’s (1982) quadratic entropy Q,Allen

et al. (2009) phylogenetic entropy H

p

and Pavoine

et al. (2009) generalized phylogenetic entropy I

q

,do

not obey the replication principle.

Measures that do not obey the replication principle

give self-contradictory results in conservation analyses

(Jost 2009). Furthermore, for such measures, the

commonly used ratio of within-group to total ‘diver-

sity’ does not reﬂect the compositional similarity of

the groups, since it always approaches unity when

diversity is high (§3 and Hardy & Jost 2008). Finally,

it is difﬁcult to use such measures to judge the magni-

tude of human or natural impacts on the environment.

The problem with these measures is their nonlinearity

with species addition. A numerical example is pro-

vided in the electronic supplementary material. Our

measures solve these problems.

If a dendrogram can be constructed from a trait-

based distance matrix using a clustering scheme

(Petchey & Gaston 2002), then we can apply our pro-

posed measures to quantify functional diversity; see

Chao & Jost (2011) for interpretation. Our proposed

approach can also be extended to the case of multiple

communities. The formulations of phylogenetic alpha,

beta and gamma diversities as well as the construction

of similarity (or differentiation) measures are devel-

oped in Chiu (2010). These results will be reported

in forthcoming papers.

(b)Interpretation of the new measures

For ultrametric trees, the mean diversity (in unit of

species) q

DðTÞ, deﬁned in equation (4.3), quantiﬁes

‘the mean effective number of species from the present

to Ttime units ago’. Here the parameter qdetermines

the diversity’s sensitivity to node (or branch segment)

abundances; high values of qemphasize those nodes

with high relative abundances. The product of q

DðTÞ

and Tis the phylogenetic diversity measure

q

PD(T),

deﬁned in equation (4.4), and quantiﬁes the ‘effective

branch diversity’ of the phylogenetic tree. For a

non-ultrametric tree, the only difference is in the

replacement of Tby the mean evolutionary change

T

(the mean of the distances from root node to each of

Sassafras albidum

CT

0 3.4

7.6 4.1

0.8 2.1

30.3 1.4

1.5 2.1

1.5 17.2

0.8 0

37.9 34.5

13.6 12.4

4.5 22.8

1.5 0

CU

Populus grandidentata

Ostrya virginiana

Quercus rubra

Ulmus americana

Ulmus rubra

Celtis occidentalis

Prunus serotina

Acer rubrum

Acer saccharum

Fraxinus american

Figure 2. The combined phylogenetic tree for the species in

the site CT (grey line) and site CU (black line). The age of

the root for the CT site is 116.6 units and 142.3 units for the

CU site and the pooled site. The species relative abundance

(%) in the two sites CT and CU are shown in the last two

columns (abundance data are from Shimatani 2001).

3606 A. Chao et al. Phylogenetic diversity measures

Phil. Trans. R. Soc. B (2010)

the terminal branch tips.). See table 1 for a summary

of the proposed measures.

For ultrametric trees, the most complete picture of

PD is provided by graphing it as a function of T.We

recommend three proﬁles for q¼0, 1 and 2

(ﬁgure 3) and a range of time between 0 and a maxi-

mum value T(such as the age of the ﬁrst node or

the age at which the group of interest diverges from

other groups or the time of origin of life). Proﬁles

may also be taken for ﬁxed T(using the Tvalues just

described, for example) as a function of q. Such pro-

ﬁles will show the effect of taking abundances into

account (q¼0 gives no abundance accounting, while

high qtakes into account only the most abundant

species). For non-ultrametric trees, similar recommen-

dations can be made based on the mean base change.

Species-neutral diversity measures discussed in this

paper are featured in the program SPADE (Chao &

Shen 2010), which can be freely downloaded from

the website http://chao.stat.nthu.edu.tw/softwareCE.

html. The new PD measures will be featured in the pro-

gram PhD (phylogenetic diversity) in the same website.

(c)Alternative formulation

We have developed our new measures to obey the

strongest possible version of the replication

principle, facilitating decomposition into independent

within- and between-group components. This was

accomplished by taking the average of

q

D(t) over the

time interval Tusing the mean derived by Jost

(2007). However, some other kinds of means may

also yield useful results. If we had used the ordinary

mean of

q

D(t) over the time interval T, we would

obtain the expectation value of

q

D(t) over the interval

T. Multiplying this by Twould give a measure of the

amount of evolutionary history embodied by the

tree in this interval, or the amount of evolutionary

work done on the assemblage during this interval.

This product would be monotonically increasing in T,

an advantage over the formulation we have developed

above. However, this alternative mean does not obey

the strong version of the replication principle, but only

the following weaker one: when Nmaximally distinct

trees with equal diversities at each time t, and equal

total abundances, are combined, the mean diversity of

the combined trees is Ntimes the mean diversity of

any individual tree. When this weaker version of the

replication principle is deemed sufﬁcient, the alternative

formulation may be useful in some applications.

This paper is dedicated to Ross Crozier, a pioneer in the

phylogenetic research and in the study of genetics in social

insects. Ross unfortunately passed away in November

q = 0

mean diversity

q = 1

mean diversity

q = 2

q = 0 q = 1 q = 2

mean diversity

time, T

PD(T)

PD(T)

0 50 100 150

time, T

0 50 100 150

time, T

0 50 100 150

4

5

6

7

8

9

10

2

3

4

5

6

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0

200

400

600

800

0

50

100

150

200

250

300

0

100

200

300

400

PD(T)

Figure 3. The proﬁle of q

DðTÞand

q

PD(T) measures for 0 ,T,150 and q¼0, 1 and 2. Solid line, CT site; dashed line,

CU site.

Table 2. Comparison of three types of diversities (q

DðTÞ,q

DðLÞand

q

D) for q¼0, 1 and 2. The q

DðTÞvalue is computed at

T¼142.3, which is the age of the root of the pooled tree. See ﬁgure 3 for other values of q

DðTÞ.

order q

site CT (thinned site) site CU (un-thinned site)

q

DðT¼142:3Þq

DðL¼4ÞqDq

DðT¼142:3Þq

DðL¼4ÞqD

q¼0 5.402 7.25 10 5.338 6.750 9

q¼1 2.660 3.951 4.967 2.797 3.904 5.664

q¼2 1.940 3.187 3.809 2.054 3.012 4.548

Phylogenetic diversity measures A. Chao et al. 3607

Phil. Trans. R. Soc. B (2010)

2009. A.C. sincerely thanks Ross for his friendship of many

years and for his inspiring and encouraging discussion and

guidance to the ﬁeld of phylogenetic diversity. We also

acknowledge helpful discussions with Carlo Ricotta, Olivier

Hardy and Bruno Senterre. The valuable comments and

suggestions from Nick Gotelli and an anonymous reviewer

helped substantially improve the paper. A.C. and C.C.

were supported by Taiwan National Science Council. L.J.

is grateful for support by a grant from John Moore to the

Population Biology Foundation.

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