Vol.3, No.9, 795-801 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.39104
The weighted quadratic index of biodiversity for pairs of
species: a generalization of Rao’s index
Radu Cornel Guiasu1*, Silviu Guiasu2
1Environmental and Health Studies Program, Department of Multidisciplinary Studies, Glendon College, York University, Toronto,
Canada; *Corresponding Author:
2Department of Mathematics and Statistics, York University, Toronto, Canada;
Received 29 July 2011; revised 30 August 2011; accepted 20 September 2011.
ABSTRACT
The distribution of biodiversity at multiple sites
of a region has been traditionally investigated
through the additive partitioning of the regional
biodiversity, called γ-diversity, into the average
within-site biodiversity or α-diversity, and the
biodiversity among sites, or β-diversity. The
standard additive partitioning of diversity re-
quires the use of a measure of diversity which is
a concave function of the relative abundance of
species, like the Shannon entropy or the Gini-
Simpson index, for instance. When a phyloge-
netic distance between species is also taken
into account, Rao’s quadratic index has been
used as a measure of dissimilarity. Rao’s index,
however, is not a concave function of the dis-
tribution of relative abundance of either indivi-
dual species or pairs of species and, conse-
quently, only some nonstandard additive parti-
tionings of diversity have been given using this
index. The objective of this paper is to show that
the weighted quadratic index of biodiversity, a
generalization of the weighted Gini-Simpson in-
dex to the pairs of species, is a concave function
of the joint distribution of the relative abun-
dance of pairs of species and, therefore, may be
used in the standard additive partitioning of di-
versity instead of Rao’s index. The replication
property of this new measure is also discussed.
Keywords: Additive Partitioning of Biodiversity;
Biodiversity Measures; Rao’s Index of Dissimilarity;
Replication Property; Weighted Alpha-, Beta-, and
Gamma-Diversities; Weighted Gini-Simpson Index
1. INTRODUCTION
The amount of turnover among species assemblages is
an important component for the conservation of biodi-
versity. The diversity turnover is called β-diversity while
the regional diversity and the mean of the local diversi-
ties are called γ-diversity and α-diversity, respectively.
Unlike the α-diversity and the γ-diversity, there is no
consensus about how to interpret and calculate the β-
diversity. According to Whittaker [1,2] who introduced
the terminology, β-diversity is the ratio between γ-diver-
sity and α-diversity. This is the multiplicative partition-
ing of diversity. According to MacArthur [3], MacArthur
and Wilson [4], and Lande [5], β-diversity is the diffe-
rence between γ-diversity and α-diversity. This is the
additive partitioning of diversity. Initially, the diversity
measures used in the partitioning of diversity (such as the
classic Shannon’s entropy [6] and the Gini-Simpson in-
dex [7,8]) depended only on the relative abundance of
species. Later, Rao [9] introduced a dissimilarity mea-
sure that takes into account both the relative abundance
of species and an arbitrary distance between species (for
example, the phylogenetic distance). When Rao’s index
is used as a measure in the additive partitioning of bio-
diversity, β-diversity reflects the dissimilarity between
the diversities of the sites of the respective region.
Within the last decade or so, many published studies
attempted to use Rao’s index in the additive partitioning
of biodiversity. Some papers have looked for special
types of distance matrices for which Rao’s index is a
concave function of the distribution of the relative abun-
dance of individual species. Other papers have looked for
nonstandard, particular additive partitioning of diversity
in α-, β-, and γ-diversities. The main difficulty comes
from the fact that, for an arbitrary distance between spe-
cies, Rao’s index is a quadratic, but not concave, func-
tion of the distribution of the relative abundance of indi-
vidual species and a linear, but not quadratic, function of
the joint distribution of the relative abundance of distinct
pairs of species and, as a consequence, it is not suited for
the general standard additive partitioning into α-, β-, and
γ-diversities. The present paper proposes the use of a

R. C. Guiasu et al. / Natural Science 3 (2011) 795-801 796
weighted quadratic indicator instead of Rao’s index. The
weighted quadratic indicator, a generalization of the
weighted Gini-Simpson index to the pairs of species,
proves to be a concave function of the joint distribution
of the relative abundance of pairs of species and is suit-
able for use in the additive partitioning of biodiversity
induced by the pairs of species when a distance between
species is taken into account. The formula for calculating
the β-diversity, as a measure of dissimilarity among the
diversities of the sites, is given. A numerical example is
presented in order to illustrate how the mathematical
formalism may be applied. In Section 4, a simple alge-
braic transformation is presented, which allows the use
of the weighted quadratic index in the multiplicative par-
titioning of biodiversity, and the corresponding replica-
tion property is discussed.
2. RAO’S INDEX OF DISSIMILARITY
If there are n species, let

1,, n
pp   be a vector
such that:

0,1, ,;1,
ii
i
pi np 


where i p is the relative abundance of species i. The
classic measures of biodiversity are the Shannon en-
tropy ([6]):

ln ,
ii
i
H
pp  

(where 0ln0 0, extending by continuity the function
ln
x
x to be equal to zero at the origin), and the
Gini-Simpson index ([7,8]):

2
1.
i
i
GS p  

Recently, Jost [10,11], and Jost et al. [12] gave some
examples showing that both the Shannon entropy and the
Gini-Simpson index do not behave well when the num-
ber of species n is very large. Guiasu and Guiasu [13]
showed, however, that the Rich-Gini-Simpson index:
 
1
ii
i
RGSn pp  
,
which depends explicitly on the number of species (spe-
cies richness), preserves all the properties of the classic
Gini-Simpson index GS but, unlike GS, behaves very
well when n is large. Another measure of diversity is the
weighted Gini-Simpson index (Guiasu and Guiasu [14]):
 
1
wiii
i
GSw pp  
,
where

1,, n
ww w is the vector whose components
are some nonnegative weights assigned to the species,
such as the conservation values of the respective species,
for instance. Obviously, GSw becomes RGS if i
wn

for each 1,, .in
Let ij
d

 D be a square matrix whose entries are
the distances between species, such as the phylogenetic
distances, for instance. Then, we have:

0,0, ,1,,
ij ii
dd ijn 
Rao’s index [9], also called quadratic entropy or dis-
similarity measure, is:


.
D
ij ij
ij
Rdpp  

Let us assume that in a certain region there are n spe-
cies and m sites. In what follows, the subscripts i and j
refer to species (i, j = 1,,n) and the subscripts k and r
refer to sites, (k, r = 1,,m).
Let


1, ,
,,
kknk
pp   be the vector whose compo-
nents are the relative abundances of the individual spe-
cies within site k, such that:


,,
0,1, ,;1,
ik ik
i
pinp 


for each 1,, .km

 A measure of diversity μ may be
used in the standard additive partioning of diversity in-
duced by individual species if it is a concave function,
which means that it satisfies the inequality:



kkk k
kk

  
 (1)
for arbitrary parameters such that:


0,1, ,,1.
kk
k
km   

 (2)
In such a case, as pointed out by Lande [5], the right-
hand side of (1) is the α-diversity, denoted by α, measur-
ing the average local diversity, and the left-hand side of
(1) is the γ-diversity, denoted by γ, measuring the re-
gional or global diversity. In the additive partitioning of
diversity, the β-diversity is the diference between the
γ-diversity and the α-diversity, ,

  representing
the variation, changes and dissimilarity between the di-
versities of the sites within the given region.
The measures of biodiversity H, GS, RGS, and GSW
are concave functions and satisfy the inequality (1), for
arbitrary parameters (2), and, therefore, are suitable to be
used in the additive partioning of diversity induced by
individual species.
Viewed as a function of the relative abundance of in-
dividual species, RD is a quadratic function of the relative
abundance of individual species. Unfortunately, if the
distance matrix D is arbitrary, Rao’s index RD, taken as a
diversity measure μ, does not satisfy the inequality (1)
for arbitrary parameters (2) and, therefore, cannot be
used in a standard additive partitioning of the diversity.
There is a vast recent literature (Pavoine et al. [15], Ri-
cotta [16], Ricotta and Szeidel [17], Hardy and Senterre
[18], Villéger and Mouillot [19], Hardy and Jost [20],
Ricotta and Szeidel [21], Sherwin [22], De Bello et al.
[23], and Tuomisto [24,25] about how to use Rao’s index
RD in the additive partioning of diversity. Some research

R. C. Guiasu et al. / Natural Science 3 (2011) 795-801 797
focused on finding special kinds of distance matrices D
for which the corresponding Rao’s index RD is a concave
function, such as the matrix D assumed to be Euc-
lidean, for instance. Some other research focused on how
to use Rao’s index for getting a nonstandard additive
partitioning of diversity, which means determining whe-
ther some special parameters (2) could be used in order
to define analog α-, β-, and γ-diversities corresponding to
these particular parameters.
For instance, Hardy and Senterre [18] proposed the
following nonstandard statistical framework for parti-
tioning the phylogenetic γ-diversity into α- and β-com-
ponents using Rao’s index.
Let Aik be the abundance (number of individuals) of
species i within site k. The relative abundance of species
i within site k is:


,i kikik
i
pA A,
and the relative abundance of species i over all sites is
defined as:


,
iik ik
kik
pA A.
Let ij
d be the distance between the species i and j.
Using Rao’s index, Hardy and Senterre [18] defined the
γ-diversity to be:
,,
Tijij
ij
Edpp
measuring the total or regional diversity. The diversity
within-site k is:
,,
,
kijikjk
ij
Edpp.
The average within-site diversity:

1
Sk
k
EmE

is called α-diversity. According to the additive partition-
ing of diversity, β-diversity is defined by them to be:
.
TS
EE

  
Villéger and Mouillot [19] generalized the model pro-
posed by Hardy and Senterre, following an approach
outlined by Ricotta [16]. They kept T E and k E un-
changed but replaced S
E by the new:
Skk
k
EE  ,
where


,,1,,,
kik ik
iik
A
Ak m  
 
which is also a particular case of parameters 1,,
m

 
satisfying (2).
According to Hardy and Jost [20], the approaches
proposed by Hardy and Senterre [18] and by Villégere
and Mouillot [19] are both adequate in their specific
contexts but they provide, however, only a nonstandard
additive partitioning of biodiversity.
3. WEIGHTED QUADRATIC INDEX
Rao’s index is important because it measures diversity
taking into account both the dissimilarity between spe-
cies, as induced by a distance between them, such as a
phylogenetic distance for instance, and the relative abun-
dance of species. Let ij
Dd

 be an arbitrary matrix
whose entries are the distances between the pairs of n
species. Let


1, ,
,,
kknk
pp   be the vector whose
components are the relative abundances of the individual
species within site k, such that:


,,
0,1, ,;1,
ik ik
i
pinp 


for each 1,, .km


As mentioned in the previous section, Rao’s index RD
is a quadratic function of the relative abundances of the
individual species. It, however, may be also viewed as
being a linear function of the joint distribution of the
relative abundance of the pairs of species. Indeed, let us
take the joint distribution of the pairs of species within
site k, induced by the distribution of the relative abun-
dance of the individual species k

within site, namely,
,
π
kijk




, where:

,,,
π,,1, , .
ij ki kj k
pp ijn (3)
In such a case, Rao’s index for site k is:


,, ,
,,
π
D
kijikjkijijk
ij ij
Rdppd 

 (4)
and it is indeed a linear function of the joint distribution
,
π
kijk

 

 of the relative abundance of pairs of spe-
cies.
In dealing with species diversity, a good measure of
the difference or dissimilarity among the sites in a certain
region has to be nonnegative and equal to zero if and
only if there is no such difference. Dealing with pairs of
species, a measure of diversity μ may be used in the
standard additive partitioning of diversity if it is a con-
cave function, which means that it satisfies the inequa-
lity:



kkk k
kk
 


 (5)
for arbitrary parameters 1,,
m

  satisfying (2). In
dealing with the diversity induced by the pairs of species,
the right-hand side of (5) is the α-diversity, denoted by α,
the left-hand side of (5) is the γ-diversity, denoted by γ,
and the β-diversity is



. As
D
R is a linear
function of k

, if we take
D
R

, (5) becomes an
equality. But in such a case the corresponding β-diversity
induced by the pairs of species is equal to zero, for any
distance matrix D and any relative abundance of species.
Therefore, Rao’s index is not suitable for use in the addi-

R. C. Guiasu et al. / Natural Science 3 (2011) 795-801 798
tive partitioning of diversity induced by pairs of species
when a dissimilarity distance between species is taken
into account.
The solution proposed here is to replace Rao’s index
(4) by the weighted Gini-Simpson quadratic index for
pairs of species:


,, ,,
,1
D
kijikjkikjk
ij
GSd pppp 
 (6)
This is a quadratic function of , π
kijk



defined
by (3), which satisfies the inequality (5) for arbitrary
parameters 1,,
m

  for which (2) holds. Therefore,
D
GS is suitable for use in the additive partitioning of
diversity induced by the pairs of species when both a
distance between species and the relative abundance of
species are taken into account.
In order to prove that
D
GS satisfies the inequality (5),
which mathematically means that it is a concave function
of k
, we move to a more general context which, how-
ever, makes the proof simpler and more elegant. Thus, let
,
π
kijk



be an arbitrary joint probability distribution
of the pairs of species, where ,
πij k is the probability of
the pair of species

,ij within site k, such that:

,,
,
π0, ,1,,;π1.
ij kij k
ij
ij n 

 (7)
In a more general context, let ij Ww

 be a square
matrix whose components are arbitrary nonnegative
weights assigned to the pairs of species. Denote by:


,,
,π1π
Wkijijkij k
ij
GS w 
 (8)
the weighted quadratic index of site k.
Remarks: 1) Let us note that if 1
ij
w, for all i and j,
then (8) becomes the classic Gini-Simpson index as-
signed to the pairs of species.
2) If 2
ij
wn, for all i and j, where 2 n is the number
of pairs of n species, then (8) becomes the Rich-Gini-
Simpson index assigned to the pairs of species.
3) If the weights are the distances between species,
ij ij
wd, and the species are independent, ,,, πij ki kj k pp

,
then

Wk
GS  from (8) becomes

D
k
GS  from
(6).
4) Another case of interest is when the species are in-
dependent, namely ,,, π,
ij kikjk
pp and the weights are

1
2
ijijij
wvvd , where ij d is the distance between
the species

,ij and vi is the value, such as the con-
servation value for instance, of species i, in which case
(8) becomes:

 
,,,,,
,
11.
2
vDkij ijikjkikjk
ij
GSvvdp pp p 

If vi = 1 for all i, then

,vDk
GS  becomes

D
k
GS  given by (6).
The weighted quadratic index

Wk
GS  given by (8)
is a concave function of the joint probability distribution
,
π
kijk




 given by (7). Indeed, let:


0,1, ,;1
kk
k
km   


be arbitrary parameters. Taking into account that:




22 22
,, ,
2
111 ,
2
111 ,
ππ 1π
π
π,
kijkk ijkkkijk
kkkmijk
kkkkkkmijk

 
 


 

  


for every 1,km

 we get:



 





,,
,
,,
,
2
,,,
,
22
,,,,
,
2
,,
,
1π
π1π
1π
π2ππ
ππ.
Wkk kWk
kk
ijkij kkij k
ij kk
kijijkijk
kij
ijkkijkkrij kij r
ijkk r
ijkrijkijkij rijr
ijk r
ijkrij kij r
ijk r
GS GS
w
w
w
w
w
 
 

 
 




 








 

 


(9)
We can see that β ≥ 0 and β is equal to zero if and only
if the sites have identical joint relative distribution,
namely, , ππ
ij kij

, for all 1,, .km
If the weight is the distance between species and the
species are independent as far as their relative abundance
is concerned, namely:
,,,
,π,
ijijij kikj k
wd pp


for all ,1,,ijn

 and 1, ,km, where ,ik p is the
relative abundance of species i within site k, then the
corresponding α-, γ- and β-diversities, with respect to the
parameters 1,,
m

 , become:



,, ,,
,
1,
kDk
k
k ijikjk ikjk
kij
GS
dp pp p





 (10)




,, ,,
,,
1,
Dkk
k
ijki kj kki kj k
ij ijk
GS
dpp pp





 (11)
where, according to (9), we have:

2
,, ,,
,ijk rikjkirjr ijk r
dpppp  

 
 (12)
We have β ≥ 0 and β = 0 if and only if the relative
abundance of the pairs of species ,,ikjk pp is the same
for each site k.
Numerical example: Let ik
A
be the abundance (num-
ber of individuals) of species i within site k. Using an
example given by Villéger and Mouillot [19], let us as-
sume that there are three species (n = 3), three sites (m =

R. C. Guiasu et al. / Natural Science 3 (2011) 795-801 799
3), and the absolute frequences of the species are:
A11 = 1, A21 = 1, A31 = 2,
A12 = 28, A22 = 1, A32 = 1,
A13 = 1, A23 = 1, A33 = 2.
Therefore, the respective relative frequencies are:
P1,1 = 0.25, P2,1 = 0.25, P3,1 = 0.50,
P1,2 = 0.934, P2,2 = 0.033, P3,2 = 0.033,
P1,3 = 0.25, P2,3 = 0.25, P3,3 = 0.50.
The distance between species is assumed to be defined
by: 121323
1, 2,2.dd d  For the parameters:
123
1/3,

 
the Formulas (10)-(12) give: α = 0.361, γ = 0.678, β =
0.317.
Remark: If the species are independent, namely
πiji j pp, where i p is the relative abundance of spe-
cies i, and the weights are

21
2
ijijij
wn vvd, where
ij
d is the distance between the species


,ij and vi is
the value of species i, such as its conservation value for
instance, in which case the weighted quadratic index:


,π1π
Wijijij
ij
GS w 

becomes the weighted Rich-Gini-Simpson quadratic in-
dex:


2
,,,,,
,1
2
ij
vD kijikjkikjk
ij
vv
RGSndp ppp

 
(13)
a measure of diversity which depends not only on the
distance between species, the relative abundance of the
distinct pairs of species, and the value of the species, but
also on the explicit number of distinct pairs of species
(the richness induced by the pairs of species). If all spe-
cies have the same value,

1,1, ,
i
vi n, then (13)
is the weighted Rich-Gini-Simpson version of the wei-
ghted Gini-Simpson quadratic index (6).
4. REPLICATION PROPERTY
Dealing with the multiplicative partitioning of diver-
sity, Whittaker ([1,2]) suggested the use of the exponen-
tial of the Shannon entropy and the algebraic inverse of
the Gini-Simpson index as measures of biodiversity.
Following this line of thought and taking the exponential
transformation of a general entropy studied by Rényi
[26], Hill [27] introduced a unified index of diversity
suitable to use in the multiplicative partitioning of diver-
sity. Hill [27] and Jost [28] noticed that these multiplica-
tive measures have the so-called doubling property (or
replication property) according to which a measure of
diversity should double when two identically distributed
but distinct communities (with no shared species) are
added together in equal proportions. In a personal corre-
spondence, Professor C. Ricotta asked whether there is a
reasonable index transformation of the weighted Gini-
Simpson index that possesses this doubling property. The
answer is yes.
4.1. Dealing with Individual Species
If in a community we have n species, such that the
distribution of the relative abundance of these species is


1,, n
pp   and the weights assigned to these spe-
cies are


1,, n
ww w, then the corresponding
weighted Gini-Simpson index:

 
1
wiii
i
GSw pp  

,
which can be used in the additive partitioning of diver-
sity induced by individual species, may be transformed
into the measure of diversity:


1
2
1iiwii
ii
wp GSwp 





, (14)
which can be used in the multiplicative partitioning of
diversity induced by the individual species. The measure
(14) has the doubling property. Indeed, if we have two
communities, A and B, having n species each, all being
different but having the same distribution of the relative
abundance


1,, n
pp   and the same qualitative
weights


1,, , n
ww w assigned to the individual
species, then the union C of these two communities will
have 2n species with the distribution of the correspond-
ing relative abundance of these species

12
,,n qq,
where 2
iin i
qq p


, and the corresponding weights


12
,,
n
uu, where iini uuw


.The diversity of C, as
measured by (14), applied to the 2n species of C is:
 
11
22
2,
ii
i
uqwp 


 where (1,,2)n ; i.e.
twice the diversity of A or B.
Chao et al. [29] claim that the doubling property is “an
important requirement for species-neutral diversity”.
This replication property refers, however, to a very sin-
gular situation because, practically, it is almost impossi-
ble to see two communities having the same number of
entirely different species with exactly the same relative
abundance. But there is another problem here. Thus, Jost
([10,28]) states that: “The diversity of a community (say
community C) should double if every species is divided
into two equal groups, say males (community A) and
females (community B) and each group is considered to
be a distinct species…A measure of diversity doubles
when two identically distributed but distinct communi-
ties, with no shared species, (like communities A and B,
respectively) are added together in equal proportions”.
We notice, however, that the simple union of the identi-
cally diverse communities A and B does not reflect the
diversity of community C, because C contains indeed the
diversities of A and B, taken separately, but has now the
additional gender diversity, which is missing in A and in

R. C. Guiasu et al. / Natural Science 3 (2011) 795-801 800
B. The diversity of the union (C) is more than the sum of
the diversities of A and B, taken separately.
4.2. Dealing with Pairs of Species
If there are n species in a community with the joint
distribution of relative abundance πij

 of the
pairs of species and the weights ij Ww

 assigned
to the pairs of species, then the corresponding weighted
Gini-Simpson index :


,π1π,
Wijijij
ij
GS w 

which can be used in the additive partitioning of diver-
sity induced by pairs of species, may be transformed into
the measure of diversity:


1
2
,,
1ππ,
ij ijwij ij
ij ij
wGS w





(15)
which can be used in the multiplicative partitioning of
diversity induced by the pairs of species. The measure
(15) has the quadrupling property. Indeed, assume that
we have two communities, A and B, with n species each,
all being different (only males

1,,n
M
M in A and
only females

1,, n
F
F in B, for instance) with the
same joint distribution  of the relative abundance of
the pairs of species and the same weights W assigned
to the pairs of species. Thus ij w is the same weight as-
signed to the pairs
 
,,,,,
ijijij
M
MMFFM,

,
ij
F
F, which happens, for instance, when the distance
ij
d between species i and j, taken as the weight ij w,
assigned to the pair of species


,ij, does not depend on
sex. Let C be the union of communities A and B. It has
2n species

11
,, ,,,
nn
M
MF F with the joint
22nn distribution of the relative abundance of the
pairs of species from C:


44
,,1,,2,
44
s
s
n






π 
and the 22nn matrix of the weights assigned to the
the pairs of species from C:


,,1,,2
s
WW
s
n
WW

 


Ww .
Then, the diversity of C, as measured by (15), is:



1
1
22
,
4π
ssij ij
sij
w



π

w,
which shows that the community C has a diversity four
times larger than the diversity of A or B. But the same
comment may be made as above: community C, the un-
ion of identical communities A and B, has not only four
times the diversity of A or B (which is the quadrupling
replication property for the pairs of species), but also has
the supplemental gender diversity which is missing in A
or B.
5. CONCLUSIONS
Rao [9] introduced a dissimilarity measure (4) that
takes into account both the relative abundance of species
and an arbitrary distance between species. When Rao’s
index is used in the additive partitioning of diversity, the
β-diversity reflects the dissimilarity between the diversi-
ties of the sites of the respective region.
Rao’s dissimilarity measure takes into account both
the relative abundance of species and an arbitrary dis-
tance between species (for example, the phylogenetic
distance). There is a large number of papers published in
the last decade dealing with different attempts at using
Rao’s index in the additive partitioning of biodiversity.
Some papers have looked for special types of distance
matrices for which Rao’s index is a concave function of
the distribution of the relative abundance of individual
species. Other papers have looked for nonstandard, par-
ticular additive partitioning of diversity into analog α-, β-,
and γ-diversities. The main difficulty comes from the fact
that, for an arbitrary distance between species, Rao’s
index is a quadratic, but not concave, function of the
distribution of the relative abundance of the individual
species and a linear, but not quadratic, function of the
joint distribution of the relative abundance of the distinct
pairs of species and, as a consequence, it is not suitable
for use in the general standard additive partitioning of
diversity into α-, β-, and γ-diversities. The present paper
proposes the use of a weighted quadratic indicator (6)
instead of Rao’s index (4). The weighted quadratic indi-
cator, a generalization of the weighted Gini-Simpson
index to the pairs of species, proves to be a concave
function of the joint distribution of the relative abun-
dance of the pairs of species and is suitable for use in the
additive partitioning of biodiversity induced by the pairs
of species when a distance between species is taken into
account. There is a simple Formula (12) for calculating
the β-diversity, as a measure of dissimilarity among the
diversities of the sites. A simple algebraic transformation
is given which allows the use of the weighted Gini-
Simpson index in the multiplicative partitioning of bio-
diversity induced by the individual species (13) or by the
pairs of species (14), and the corresponding replication
properties are discussed.
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