Journal of Theoretical Biology 222 (2003) 189–197
On parametric evenness measures
Carlo Ricotta*
Department of Plant Biology, University of Rome ‘‘La Sapienza’’, Piazzale Aldo Moro 5, Rome 00185, Italy
Received 12 February 2002; received in revised form 13 December 2002; accepted 24 December 2002
Abstract
The degree to which abundances are divided equitably among community species or evenness is a basic property of any biological
community. Several evenness indices have thus far been proposed in ecological literature. However, despite its vast potential
applicability in ecological research, none seems to be generally preferred. Furthermore, only very few parametric evenness families
have thus far been proposed. While traditional evenness indices supply point descriptions of community evenness, according to a
parametric evenness family EðaÞ; there is a continuum of possible evenness measures that differ in their sensitivity to changes in the
relative abundances of dominant and rare species as a function of the parameter a: In this review, I first summarize the basic
requirements that a parametric evenness measure should meet to adequately behave in ecological studies. Next, I discuss the major
drawbacks of these requirements and propose some alternative solutions.
r 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Concentration measures; Information theory; Lorenz order; Species independence; Stepwise data transformation
1. Introduction
Biological diversity is a central concept in quantitative
ecology that has been studied extensively for over 50
years (Patil and Taillie, 1982; Magurran, 1988). Intuitively, diversity appears as a simple and unambiguous
notion. However, when we look for a suitable numerical
definition, we find that no single diversity index
adequately summarizes community structure. This observation has led to the well-known comments by
Hurlbert (1971) on the ‘‘non-concept of diversity’’ and
by Poole (1974) that diversity measures are ‘‘answers to
which questions have not yet been found’’. The main
reason for this confusion is that biodiversity measures
combine in non-standard way the two components of
species richness (the number of species in the sample)
and their relative abundance (called variously evenness,
equitability or dominance). High species richness and
evenness, that occurs when species are equal or virtually
equal in abundance, are both equated with high
diversity.
A more complete summarization of community
diversity is possible if, instead of a single index, one
uses a parametric family of diversity indices whose
*Tel.: +39-06-4991-2408; fax: +39-06-4457-540.
E-mail address: [email protected] (C. Ricotta).
members have varying sensitivities to the presence of
rare and abundant species: ‘‘Number of components
(alias: ‘richness’) refers always to zero-ordered diversity,
and extinction of one or more elements must considered
to be a tragic end-result (‘finality’) of a more or less long
stochastic process. If we want to study the process itself,
then, to say the least, we need always vectorial
representations, instead of scalars’’ (Juha! sz-Nagy,
1993). In this paper, after a short overview on
parametric diversity, I show the basic requirements that
a parametric measure of evenness should meet to
adequately behave in ecological studies. Next, I discuss
the major drawbacks of traditional parametric evenness
and propose some alternative solutions.
2. A short overview on parametric diversity
As emphasized by a number of authors (e.g. Hurlbert,
1971; Taillie, 1979; Patil and Taillie, 1982; Rousseau,
1999), both concepts of diversity and evenness are
independent of any single way of measurement. Nonetheless, the few parametric evenness functions proposed
thus far in the ecological literature derive their
conceptual basis from information theory (Shannon,
1948). Also, Buzas and Hayek (1996) proposed an
0022-5193/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0022-5193(03)00026-2
C. Ricotta / Journal of Theoretical Biology 222 (2003) 189–197
190
elegant additive model for partitioning the Shannon
(1948) index into species richness and evenness (see also
Taillie, 1979; Ricotta and Avena, 2002). Thus, information-theoretical measures seem an adequate choice for
providing a statistical base for diversity analysis (He and
! 1993; Ricotta and Avena, 2002).
Orloci,
Imagine a plant community composed of N species,
where pi is the proportional abundance (measured as
number of individuals, dry weight or productivity) of
the
PN ith species ði ¼ 1; 2; y; NÞ such that 0ppi p1 and
i¼1 pi ¼ 1: Working from the observation that traditional diversity indices measure different aspects of the
partition of abundance between species, Hill (1973)
proposed a unifying formulation of diversity, according
to which Re! nyi’s generalized entropy represents the
cornerstone for a continuum of possible diversity
measures. For a relative abundance vector P ¼
ðp1 ; p2 ; y; pN Þ Re! nyi (1970) extended the concept of
Shannon’s statistical information (or entropy) by defining a generalized entropy of order a as
Ha ¼
N
X
1
ln
pa :
1 a i¼1 i
ð1Þ
Ha makes mathematical sense for NpapN:
However, a parameter restriction (aX0) has to be
imposed if Ha should possess certain desirable properties discussed in the remainder that renders it adequate
in ecological research.
Re! nyi proved that Ha satisfies certain axioms that
entitled it to be regarded as a measure of generalized
.
information (Beck and Schlogl,
1993). Mathematically,
the various diversity measures obtained by varying the
parameter a are in fact different moments of the same
basic Re! nyi function. It is easily demonstrated that the
maximum value of Re! nyi’s parametric entropy Hamax ¼
log N is obtained from an equiprobable distribution (i.e.
if pi ¼ pj ¼ 1=N for all species pairs iaj). Conversely,
minimum entropy ðHamin ¼ 0Þ occurs if there is a species
having its relative abundance approaching 1 (the
abundances of all other community species approaching
null).
From an ecological viewpoint, the generalized entropy Ha of a given community is a parametric measure
of uncertainty in predicting the relative abundance of
species. Due to this appealing analogy between entropy
and uncertainty, information-theoretical measures of
community structure appeared in early works on species
diversity (MacArthur, 1955; Pielou, 1966a, b). Notice
that in Re! nyi’s definition the base 2 logarithm is used to
measure information content in bits, while in ecological
applications the natural logarithm is traditionally used.
Notice also that Eq. (1) is not defined
P for a ¼ 1 and its
derivation, H1 ¼ H where H ¼ N
i¼1 pi ln pi is Shannon’s (1948) entropy, is based on l’Hospital’s rule of
calculus.
Hill (1973) showed that the generalized entropies Ha
have many desirable properties as diversity indices. One
particularly convenient property is that a number of
traditional diversity indices popular among ecologists
are special cases of Ha : For a ¼ 0; H0 ¼ ln N; where N is
the total number of community’s species; for a ¼ 1;
H1 ¼ H; for a ¼ 2; H2 ¼ ln 1=D;
D is Simpson’s
P where
2
(1949) dominance index D ¼ N
p
;
and
for a ¼ N;
i¼1 i
HN ¼ ln 1=d ¼ ln 1=pmax ; where d is the dominance
index of Berger and Parker (1970) and pmax is the
proportional abundance of the most frequent community species. While traditional diversity indices supply
point descriptions of community structure, according to
Re! nyi’s formulation, there is a continuum of possible
diversity measures, which differ in their sensitivity to
rare and abundant community species becoming increasingly dominated by the commonest species for
increasing values of the parameter a: In this way, rather
than as a single-point summary statistics, diversity is
seen as a scaling process from community species
richness to its dominance concentration that takes place
not in the real but in the topological data space (Podani,
1992).
The biological explanation for using parametric
diversity measures resides in the observation that the
extent to which plant species affect immediate ecosystem
functions, such as carbon balance or nutrient dynamics,
is largely determined by their contribution to total plant
biomass (Aarsen, 1997; Huston, 1997). This ‘mass-ratio’
effect is dictated by the fact that, for autotrophs such as
plant species, a larger body mass involves major
participation in syntheses, and in inputs to resource
fluxes and degradative processes (Shugart, 1984; Pastor
and Post, 1986; Huston, 1997; Grime, 1998). There is
also increasing evidence that complementary resources
exploitation due to functional differences between
codominants (e.g. in phenology, rooting depth, or
reproductive biology) generally confer beneficial effects
on ecosystem productivity (Hooper and Vitousek, 1997).
Consequently, immediate ecosystem functioning is disproportionately influenced by the specific and functional
diversity of the dominant species and is relatively
insensitive to the diversity of subordinates. Further,
although there is increasing evidence that the immediate
functioning of ecosystems is disproportionately influenced by the dominant species, this does not exclude
subordinates from involvement in the determination of
less obvious functions that affect ecosystem sustainability in the long term (i.e. on a biologically more
meaningful temporal scale). Grime (1998) suggests that
the major role of subordinate community species may
consist in acting as a filter influencing regeneration by
dominants following major disturbances. Although
evidence of a filter role of subordinates in the recruitment of dominants remains hypothetical to date, nonetheless a recent experiment of Lyons and Schwartz
C. Ricotta / Journal of Theoretical Biology 222 (2003) 189–197
(2001) demonstrates that rare community species can
significantly influence invasion events by exotic grasses
highlighting the potential role of subordinate species in
the maintenance of long-term ecosystem sustainability.
Obviously, these different ecological functions of
dominant and subordinate species are largely quantified
by the evenness component of diversity measures,
whereas simple species count weights the contribution
of any community species in the same way.
To overcome the logarithmic nature of Re! nyi’s
formulation, Hill (1973) defined another diversity index
family of order a as the antilog of Re! nyi’s generalized
entropy
!1=1a
N
X
a
Na ¼
pi
;
ð2Þ
i¼1
where Na represents the ‘effective number of species’ of
Ha ; that is, the number of species that would give the
same generalized entropy measure if they were all
equally represented (Ludwig and Reynolds, 1988). Hill’s
measurements express therefore diversity in numbers of
species that are biologically more intuitive units than the
‘bits’ of information of Re! nyi’s generalized entropy.
This parametric diversity function was introduced again
by Patil and Taillie (1979, 1982), albeit from a different
mathematical viewpoint. Their notation is
!1=b
N
X
bþ1
Sb ¼
pi
;
ð3Þ
i¼1
where Sb represents the effective number of species of
one more parametric diversity index, the diversity index
of degree b; proposed by Patil and Taillie (1979, 1982)
!
N
X
bþ1
Db ¼ 1 pi
=b:
ð4Þ
i¼1
Although Db ðbX 1Þ has been proposed in the
ecological literature without any formal informationtheoretical interpretation, nonetheless putting a ¼ b þ 1;
the diversity index of degree b is identical
to Tsallis
P
(2002) non-extensive entropy Ha ¼ 1 Sj¼1 paj =a 1;
which is in turn a rediscovery of the generalized entropy
!
of type a proposed by Acze! l and Daroczy
(1975), though
with a different prefactor fitted for base e variables
!
instead of binary variables (Tothm
e! re! sz, 1995; Tsallis,
2002). Also, as shown by Eqs. (2) and (3), putting a ¼
b þ 1; Db shares the effective number of species with
Re! nyi’s generalized entropy.
A desirable property of Eq. (4) is that, unlike Re! nyi’s/
Hill’s diversity functions, Db is decomposable into
species-level patterns.
PNUsing Patilb and Taillie’s
PN (1982)
notation,
Db ¼
p
½ð1
p
Þ=b
¼
i
i¼1 i
i¼1 pi Rðpi Þ
where Rðpi Þ is the rarity measure of the ith species such
that the (weighted) sum of single species rarities gives
the pooled diversity of the species collection. Patil and
191
Taillie (1982) termed this property ‘dichotomy’ because
the diversity of the ith species would be unchanged if the
other species were grouped into a single complementary
category. All this is very classic; for a thorough review
!
on parametric diversity functions, see Tothm
e! re! sz
(1995).
3. Requirements for adequate parametric evenness
measures
As mentioned above, besides species richness, another
basic component of any diversity measure is species
evenness, with indices of evenness being basically
relative diversity measures or normalizations of diversity
measures (Kva( lseth, 1991). Evenness measures quantify
the equality of species abundances in the community,
maximum evenness (1.0) arising for an equiprobable
species distribution, and the more that relative abundances of species differ the lower the evenness is
(Alatalo, 1981). Several evenness indices have been
proposed in the ecological literature. For a review, see
Smith and Wilson (1996) and Ricotta et al. (2001).
However, none seems to be generally preferred.
Furthermore, only very few parametric evenness measures have thus far been proposed, and virtually none
has materialized in practical ecological research.
The fact that Hill’s measurements are all expressed in
the same units and differ in their sensitivity to the rarer
species led Hill (1973) to propose that evenness be
measured by a double continuum ratio of diversity
numbers
Ea;b ¼ Na =Nb
ð5Þ
corresponding to all possible pairs of values aab: At the
same time however, Hill (1973) recommended caution in
the use of indices of ‘peculiar’ order, such as for example
N1:5 =N1:414 : ‘‘There is almost unlimited scope for
mathematical generality in relation to measures of
[evenness]. Simple and well-understood indices should
be used’’. Examples of simple special cases of Ea;b
include the index E1;0 (Buzas and Gibson, 1969; Sheldon
1969) and a normalized version of Simpson’s index, E2;0
(Smith and Wilson, 1996).
Many authors (Engen, 1979; Taillie, 1979; Routledge,
1983; Smith and Wilson 1996, Ricotta et al., 2001;
Gosselin, 2001) have proposed a number of basic
criteria that an evenness index should meet to reasonably behave in ecological research. An ecologically
acceptable measure of evenness should be reasonably
simple to compute and applicable to any community
independently of its species-abundance distribution
(Alatalo, 1981; Lande, 1996). Furthermore, it should
possess an appropriate mathematical and statistical
pedigree (Kva( lseth, 1991). Two basic criteria for
an ecologically acceptable evenness index are that
C. Ricotta / Journal of Theoretical Biology 222 (2003) 189–197
192
community evenness decreases both by marginally
reducing the abundance of community rarest species,
and by the addition of a very minor species to the
community. These are Routledge’s (1983) requirements
R1 and R2, respectively. In addition, a meaningful
evenness measure must be independent of species
richness (Hill, 1973; Taillie, 1979). This requirement is
based on the assumption that species diversity can be
partitioned into two components: species richness and
evenness. If the separation is incomplete, so that
evenness is affected by species richness, then differences
in evenness values could result from differences in the
species count rather than any fundamental difference in
community organization (Sheldon, 1969).
To qualify for being independent of species richness
(or, shorter, s-independent), Hill (1973) proposed that
replication should not change the value of community
evenness. Imagine a N-species community characterized
by the relative abundance vector P: It seems reasonable
that replicating the N-species sequence K-times should
give a community with K-times the original species
richness but the same community evenness. Note that
the proposed criteria are part of Taillie’s (1979) more
general requirement that an evenness index maintains
the natural ordering introduced by the Lorenz curves
used by economists to compare wealth distributions.
The Lorenz curve is obtained by plotting the cumulative
species relative abundances as abscissa against corresponding cumulative proportions of species as ordinates
(Taillie, 1979). Arrange the components of the species
relative abundance vector P of a given community in
descending order so that the ranked abundance vector
P# ¼ ðp#1 Xp#2 XyXp#N Þ is obtained, where p#1 p#2 ?p#N :
The Lorenz curve is then defined as the polygonal path
joining the successive points: p0 ¼ ð0; 0Þ; p1 ¼ ðp#1 ; 1=NÞ;
p2 ¼ ðp#1 þ p#2 ; 2=NÞ;y, pN ¼ ðp#1 þ p#2 þ ? þ p#N ;N=NÞ
ð1; 1Þ (Fig. 1). The outcome is very similar to the
intrinsic diversity profile proposed by Patil and Taillie
(1979, 1982) for defining the concept of intrinsic
diversity order: both use as abscissa the cumulative
species relative abundances. However, the intrinsic
diversity profile uses as ordinate the cumulative number
of species, whereas the Lorenz curve uses as ordinate the
cumulative proportion of species. Patil and Taillie (1979,
1982) defined community A to be intrinsically more
diverse than community B without reference to indices,
provided B leads to A by a finite sequence of forward
transfers of abundance from one species to another
strictly less abundant species (for mathematical details,
see Patil and Taillie, 1979, 1982). Following this
definition, the hypothetical community A is intrinsically
more diverse than community B (written as A > B) if
and only if community A has its intrinsic diversity
profile everywhere above that of community B: As a
consequence, each diversity index that conforms to the
intrinsic diversity order, such as Re! nyi’s/Hill’s parametric diversity with 0XaXN or Patil and Taillie’s
index Db with 1XbXN; assumes a smaller value for
community B than for community A: Note that the
intrinsic diversity order is only partial in that if A > B
and B > C; then A > C: Nevertheless, two intrinsic
diversity profiles may intersect. That is, it is not
necessarily true that for every A and B; either A > B or
B > A: In this case, A and B cannot be unambiguously
ordered according to their diversity as different diversity
indices rank them in contradictory ways. This effect
should not be a cause for undue pessimism: Patil and
Taillie (1979) emphasize that ‘‘such inconsistencies are
1
cumulative species relative abundances
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
cumulative proportion of species
Fig. 1. Lorenz curve for an artificial five-species community with relative abundances 0.27, 0.24, 0.21, 0.16, 0.12. The dotted line represents perfect
evenness.
C. Ricotta / Journal of Theoretical Biology 222 (2003) 189–197
inevitable whenever one attempts to reduce a multidimensional concept to a single number [y]. For
example, the mean and median are inconsistent measures
of
central
tendency;
likewise,
the
standard
deviation, mean absolute deviation, and range are
inconsistent measures of spread’’ (Patil and Taillie, 1979).
Similarly, community A is intrinsically more even
than community B if and only if community A has its
Lorenz curve everywhere above that of community B
(Taillie, 1979). Consequently, a measure of evenness E
maintains the Lorenz ordering provided that E is at the
same time invariant under species replication and
consistent with the intrinsic diversity order when
restricted to communities with the same number of
species (Taillie, 1979). For instance, when diversity
comparisons are restricted to communities with the
same number of species, since there is no fundamental
difference between diversity and evenness when species
richness is held constant, the intrinsic diversity order is
identical to the corresponding Lorenz order. Note
that twice the area
under the Lorenz curve
PN # #
I ¼ 2 i¼1 i pi 1 =N; where i# denotes the rank
of the ith species within the ranked abundance vector
P# ; is generally known as the Gini evenness index.
Within this framework, it is easily shown that the
function Da ¼ ln N Ha maintains the Lorenz order in
the inverse sense. That is, if community A has its Lorenz
curve everywhere above that of community B; then
Da ðAÞoDa ðBÞ: Since from Eq. (1) it follows that, for a
given relative abundance vector P; Ha > Ha ’ for any a0 >
a; the function Da can be interpreted as a parametric
measure of dominance concentration or ‘unevenness’
that quantifies the loss of statistical information
obtained by selecting a moment Ha ða > 0Þ of Re! nyi’s
generalized information instead of simple species richness H0 ¼ ln N for summarizing community diversity.
In other words, Da summarizes the information loss
obtained by weighting abundant species more than rarer
ones in the formulation of the diversity index.
Nonetheless, despite its appealing information-theoretical meaning, Da has one disadvantage with respect to
other parametric measures for summarizing dominance
or unevenness, namely, it has no upper bound. This
makes comparisons between different communities
somewhat more difficult. Yet, composing it with a
strictly decreasing function with range [0,1] yields a
normalized parametric evenness measure that conforms
to the Lorenz order. The most natural way to perform
this transformation is by the arc tan function (Njissen
et al., 1998). Consequently, the function
2
1 arctan Da
p
ð6Þ
results in a good parametric evenness measure that takes
values between zero and one. A simple unbounded
193
special case of Eq. (6) was suggested by Njissen et al.
(1998) as 1=ðln N HÞ where H is Shannon’s entropy.
Another unfamiliar way of normalizing Da consists in
computing the antilog of Da : The resulting function
expðDa Þ ¼ Ea;0
ð7Þ
yields another good parametric evenness measure.
Taillie (1979) demonstrated that Hill’s generalized
evenness (sub)family Ea;0 ða > 0Þ conforms to the Lorenz
order. Conversely, for ba0; the resulting evenness
figures Ea;b do not conform to the Lorenz order and
are therefore inadequate for summarizing community
evenness. For a discussion on the relation between Ea;0
and Kullback’s (1959) divergence, see Ricotta and
Avena (2002).
To the best of my knowledge, Ea;0 and 1 2=
p arctan Da are the only parametric evenness functions
proposed thus far in ecological literature that maintain
the Lorenz ordering. Ea;0 is also a particularly adequate
evenness measure in its relation to diversity. As pointed
out by Molinari (1989) and Bulla (1994), since diversity
can be informally partitioned into richness and evenness, an ecologically meaningful evenness measure
should be such that, when multiplied by the number of
community species N; it will produce an index of species
diversity. Generally, no such formal relation can be
derived. However, Ea;0 is formally related to Hill’s
generalized diversity Na by the simple expression
Na ¼ Ea;0 N:
ð8Þ
Two artificial communities with the following relative
abundance vectors are used to illustrate the performance
of Hill’s generalized evenness Ea;0 in quantifying
community structure:
A ¼ ð0:27; 0:24; 0:21; 0:16; 0:12Þ;
B ¼ ð0:37; 0:23; 0:16; 0:09; 0:07; 0:05; 0:03Þ:
Species richness N is 5 for community A and 7 for
community B: Re! nyi’s generalized diversity and Hill’s
evenness profiles of both communities extending from
a ¼ 0 up to 10 are shown in Figs. 2 and 3, respectively.
In Fig. 2 where Re! nyi’s diversity profiles cross, ordering
both communities by Re! nyi’s parametric entropy Ha
reveals that A and B are ranked differently by a ¼ 1 and
a ¼ 2: For the two communities the values of H1 and
H2 are:
H1 ðAÞ ¼ 1:571o1:657 ¼ H1 ðBÞ;
H2 ðAÞ ¼ 1:539 > 1:462 ¼ H1 ðBÞ:
In this situation, for a ¼ 1; B represents the more
diverse community, whereas for a ¼ 2 Re! nyi’s entropy
ranks community diversity in the opposite way. Conversely, in Fig. 3, once the influence of species richness is
eliminated from the computation of the evenness
C. Ricotta / Journal of Theoretical Biology 222 (2003) 189–197
194
2
Community A
Community B
1.9
1.8
1.7
Hα
1.6
1.5
1.4
1.3
1.2
1.1
1
0
1
2
3
4
5
6
7
8
9
10
α
Fig. 2. The relation between R!enyi’s generalized entropies Ha and their order a for the artificial communities A and B extending from a ¼ 0 up to 10:
Since the two diversity profiles cross, A and B are said to be non-comparable.
1.2
Community A
Community B
1
Eα ,0
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10
α
Fig. 3. The relation between Hill’s entropy figures Ea;0 and their order a for the artificial communities A and B extending from a ¼ 0 up to 10: Apart
from the trivial case a ¼ 0; community A is the most even one (i.e. Ea;0 ðAÞ > Ea;0 ðBÞ for any a > 0).
profiles, community A results as the most even one (i.e.
Ea;0 ðAÞ > Ea;0 ðBÞ for any a > 0). Note that, for a ¼ 0;
Hill’s generalized evenness of both communities reduces
to the trivial case Ea;0 ¼ 1: In other words, by tuning the
parameter a of Hill’s generalized evenness Ea;0 we can
focus on different aspects of the partition of relative
abundances between community species independently
of species richness. Within this context, Hill’s generalized evenness presents particularly good results because
of the formal relationship between species richness,
evenness and diversity expressed by Eq. (8).
4. On the major drawbacks of parametric evenness and
alternative solutions
The major drawback of parametric evenness indices
that conform to the Lorenz order such as Ea;0 or
C. Ricotta / Journal of Theoretical Biology 222 (2003) 189–197
1 2=p arctan Da is that evenness does not depend
continuously on the proportional abundance of any
species. This is Routledge’s (1983) requirement R3 (see
also Engen, 1979). This requirement implies that an
arbitrarily small error in the estimated abundance of a
given species should not induce a substantial error in the
estimated evenness. In this view, any evenness index that
conforms to the Lorenz partial order cannot be
continuous at e-0: For a mathematical proof, see
Routledge (1983). This in particular means that for
example community (0.49, 0.49, 0.02) has an equitability
that is much less than that of community (0.05, 0.05). To
the contrary, following requirement R3, the community
(0:5 e; 0:5 e; 2e) should have an evenness that tends
to the maximum when e tends to zero. In other words,
for evenness measures that conform to the Lorenz order,
it makes a big difference whether a species is detected
with a very small abundance or the same species is not
detected. In that sense, the Lorenz partial order is
dependent on species richness, and especially on the
scale and the intensity of the ecological releve! (Gosselin,
2001). ‘‘Such a strategy is acceptable when one wishes
only to study the evenness of the collection of organisms
actually sampled, but not to draw any inference about
some larger community’’ (Routledge, 1983). Conversely,
if we wish to respect the continuity of evenness measures
when a very minor species is added to the community,
we are closer to the notion of ‘dominance equitability
measures’ sensu Engen (1979).
To remedy the discontinuity of traditional evenness
indices when the relative abundance of a very minor
species tends to zero, Engen (1979) proposes to look at
the variance of the rarity function of dichotomic
diversity indices. For a given community we have
(Engen, 1979).
"
#2
N
N
X
X
2
VarR ¼
pi R ðpi Þ pi Rðpi Þ :
ð9Þ
i¼1
i¼1
Thus, combining Eq. (9) with Patil and Taillie’s (1979,
1982) parametric diversity function Db ; we obtain
"
#2
N
N
X
X
b
b
2
Varb ¼
pi ½ð1 pi Þ=b pi ½ð1 pi Þ=b ; ð10Þ
i¼1
i¼1
where Varb is a continuous, s-dependent parametric
generalization of Engen’s (1977) index of variability
P
2
P
N
2
n0 ¼ N
: Varb equals zero
i¼1 pi ðln pi Þ i¼1 pi ln pi
for equiprobable distributions and is otherwise positive.
Therefore, combining Varb with the arc tan function
results in an adequate continuous evenness measure 1 2=p arctan Varb that takes values between zero and one.
Alternatively, if instead of using the (a priori
unknown) number of community species N; evenness
is computed for a fixed number K of categories, such as
Raunkiaer’s (1934) life forms or, in landscape ecological
195
applications, the land cover types of a given land cover
classification (e.g. Anderson et al., 1976; Anon, 1992),
we can substitute concentration indices for traditional
evenness indices. Among others, an adequate continuous, s-dependent parametric concentration measure is
Tha ¼ ln K N
X
1
pa ¼ ln K Ha ;
ln
1 a i¼1 i
ð11Þ
where Tha is the generalization ofP Theil’s (1967)
concentration index Th ¼ ln K þ N
to
i¼1 pi ln pi
Re! nyi’s parametric entropy Ha : For a review on
concentration measures, see Izsa! k (1992) and Izsa! k
and Papp (1998). It is easily shown that, like Da ; Tha has
no upper bound. Nonetheless, its normalized expressions 1 2=p arctan Tha and expðTha Þ both take
values between zero and one.
Note that the property of continuity is related to the
sensitivity of evenness measures to rare species; for
instance, discontinuous measures are more sensitive to
rare species than discontinuous ones. Note also that,
since continuous evenness indices are not too much
affected by very rare species, such measures may violate
the intuitive notion of evenness. For instance, using a
continuous measure, the two-species community (0.001,
0.999) may have higher evenness than (0.1, 0.9) (Engen,
1979).
An additional general drawback of parametric evenness indices is that there is no clear quantitative
interpretation of the relative contribution of rare and
abundant community species as a function of the
selected parameter a ðbÞ: In other words, at a fixed
value of a ðbÞ; the weight attributed to rare and
abundant species in the formulation of the indices will
be different for different communities (Dennis et al.,
.
1979; Kroel-Dulay,
1998). As a consequence, the
statistical and biological effects of varying the parameter
a for computing different moments of any parametric
evenness measure cannot be easily followed in detail
(Ricotta and Avena, 2001). Therefore, field ecologists
generally prefer theoretically less desirable but practically more easily interpretable alternatives based on a
stepwise decomposition of community diversity by
dropping out species sequentially (Rajczy and Padisa! k,
.
1983; Kroel-Dulay,
1998). For example, in a similar
.
context, Kroel-Dulay
(1998) proposes a stepwise data
transformation procedure where a series of cover
thresholds is applied to sequentially delete species with
low cover values from the sampled data set. If after
each step of deletion, a (punctual) evenness index is
computed, a ‘sequential’ parametric evenness index is
obtained that summarizes the abundance distribution of
the remaining species as a function of the selected cover
threshold. Similarly, Gosselin (2001) proposes three
alternative families of evenness indices compatible with
the Lorenz order meant to quantify reduced portions of
196
C. Ricotta / Journal of Theoretical Biology 222 (2003) 189–197
the area beneath the Lorenz curve. For mathematical
details, see Gosselin (2001). In this view, Gosselin’s
indices can be considered as a restriction of the Gini
index that may be used for constructing additional
.
parametric evenness measures sensu Kroel-Dulay
(1998).
5. Conclusion
This paper was an attempt to illustrate the basic
properties that a parametric evenness index should
possess to adequately summarize the relative abundances distribution of a biological community. However, from a statistical viewpoint, condensing a
multivariate data set into one (parametric) evenness
figure always result in a loss of information, and there is
no ideal evenness measure capable of satisfying even the
most basic of these properties such as Lorenz compatibility and continuity. Of course, this is not to say that
these are the only criteria available for selecting evenness
indices. For instance, additional criteria are unbiasedness relative to the whole community (Engen, 1979) and
sensitivity to sample size (Magurran, 1988; Kva( lseth,
1991). Nonetheless, I think the number of parametric
evenness indices listed in this paper is wide enough so as
to have a sufficient choice.
Finally, it is again worth noting that diversity as an
unequivocal and inherent property of a biological
system does not exist. Instead, as it is the case with
any other statistic, diversity values are merely numbers
and their relevance to ecological problems must be
judged on the basis of observed correlations with other
environmental variables (Molinari, 1989). In this sense, I
agree with Magurran (1988) that ‘‘diversity measures are
valuable, but are only a means to an end. That end is
that ecologists should be able to ask the questions and
formulate the hypotheses to help them understand, and
sensibly manage, the natural world’’.
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