OIKOS 104: 377 /387, 2004
A unified mathematical framework for the measurement of richness
and evenness within and among multiple communities
Thomas D. Olszewski
Olszewski, T. D. 2004. A unified mathematical framework for the measurement of
richness and evenness within and among multiple communities. / Oikos 104: 377 /387.
Biodiversity can be divided into two aspects: richness (the number of species or other
taxa in a community or sample) and evenness (a measure of the distribution of relative
abundances of different taxa in a community or sample). Sample richness is typically
evaluated using rarefaction, which normalizes for sample size. Evenness is typically
summarized in a single value. It is shown here that Hurlbert’s probability of
interspecific encounter (D1), a commonly used sample-size independent measure of
evenness, equals the slope of the steepest part of the rising limb of a rarefaction curve.
This means that rarefaction curves provide information on both aspects of diversity. In
addition, regional diversity (gamma) can be broken down into the diversity within local
communities (alpha) and differences in taxonomic composition among local
communities (beta). Beta richness is expressed by the difference between the
composite rarefaction curve of all samples in a region with the collector’s curve for
the same samples. The differences of the initial slopes of these two curves reflect the
beta evenness thanks to the relationship between rarefaction and D1. This relationship
can be further extended to help interpret species-area curves (SAC’s). As previous
authors have described, rarefaction provides the null hypothesis of passive sampling for
SAC’s, which can be interpreted as regional collector’s curves. This allows evaluation of
richness and evenness at local and regional scales using a single family of wellestablished, mathematically related techniques.
T. D. Olszewski, Dept of Geology and Geophysics, Texas A&M Univ., TAMU 3115,
College Station, TX 77843 /3115, USA ([email protected]).
Diversity is a fundamental property of ecological communities. Although the identities and characteristics of
species in a community are the cornerstone of ecological
data, the number and variety of species in an assemblage,
independent of their identities, provides a means of
comparing assemblages from different times and places.
This, in turn, can provide information on changes in
community structure, as opposed to species membership,
over ecological as well as evolutionary timescales (May
1976, Powell and Kowalewski 2002) and on the effect of
natural and anthropogenic environmental change on
species assemblages (Kempton 1979). The emergence of
macroecology (Brown 1995, Gaston and Blackburn
2000) and recent theoretical developments in neutral
theory (Caswell and Cohen 1993, Hubbell 2001, Bordade-Água et al. 2002, Chave et al. 2002, Mouquet and
Loreau 2002) explicitly relate processes like local species
interactions, speciation and extinction, and migration
between local patches to species diversity from the scale
of local communities to entire landscapes.
Ecologists have developed numerous ways to measure
ecological diversity (reviewed by Washington 1984,
Magurran 1988, Smith and Wilson 1996, Hayek and
Buzas 1997, Krebs 1999) that provide the tools for
testing and improving predictive models of diversity.
Rather than provide another review of diversity measurement, the aim of this paper is to show how certain
established methods of evaluating richness, evenness,
Accepted 23 July 2003
Copyright # OIKOS 2004
ISSN 0030-1299
OIKOS 104:2 (2004)
377
and among-sample diversity that are generally treated as
distinct approaches comprise a unified family of mathematically related metrics. Specifically, it will be shown
that Simpson’s (1949) bias corrected measure of evenness
(Hurlbert’s [1971] probability of interspecific encounter)
is a component of the rarefaction method of evaluating
richness (Hurlbert 1971, Heck et al. 1975). Further, the
additive partitioning property of Simpson’s metric
among within-community diversity and betweencommunity diversity (Lande 1996, Veech et al. 2002)
can be related to the relationship between rarefaction
and collector’s curves as described by Gotelli and
Colwell (2001). Lastly, because rarefaction curves provide a null hypothesis of passive sampling for speciesarea curves (McGuiness 1984, Hart and Horwitz 1991),
the well-developed body of literature on SAC’s provides
an established foundation for interpreting diversity at
multiple scales.
The relationship between richness and evenness
Some measures of ecological diversity, like the Shannon
information index, attempt to subsume all aspects of
diversity into a single value (Hayek and Buzas 1997).
However, ecologists have recognized that diversity can be
split into richness (the number of species or other taxa in
a community) and evenness (a measure that reflects the
uniformity of relative abundances of the taxa). Estimated richness is strongly dependent on both sample size
(i.e. the number of specimens in a collection) and the
relative abundances of the component taxa. In order to
compare samples of different sizes, it is necessary to
calculate their expected richness at a standardized size,
which is the purpose of rarefaction (Sanders 1968,
Hurlbert 1971, Heck et al. 1975). Rarefaction curves
based on the hypergeometric distribution assume subsampling of a collection without replacement (Eq. (1a)
and (1b); all equations and variables discussed in this
paper have been collated in Table 1 for ease of reference).
Alternatively, sub-sampling with replacement is described using the multinomial distribution (Tipper
1979) but will not be discussed further in this paper.
Note that rarefaction does not allow extrapolation to
larger samples / it can only be used to predict richness at
smaller sample sizes than the collection of interest.
Dissection of Eq. (1a) provides interesting insight into
the
nature of rarefaction. The numerator
probabilistic
Nni
represents the number of possible sub-samples
m
of size m that do not
include an individual of species i.
N
The denominator
is the total number of different
m
possible sub-samples of size m from a collection of size
N, regardless of species composition. The quotient of
these two terms is therefore the probability that a sub378
sample of size m will not contain species i; the
complement of this number is the probability that
species i will occur in the sub-sample. This is equivalent
to the expected contribution of species i to the richness
of the sub-sample, which when summed over all species,
is the expected richness of the sub-sample. An equivalent
interpretation of Eq. (1b) from Brewer and Williamson
(1994) leads to the same conclusion: P(1/m/N/j) is
the probability that no individual of species i occurs in a
sub-sample of size m.
The results of rarefaction are typically depicted as
curves of expected richness as sub-sample size increases
(Fig. 1). The incremental increase in richness from a subsample of size m to a sub-sample of size m/1 (i.e.
E(sm1) /E(sm)) is the probability that the additional
individual in the larger sub-sample represents a previously unsampled species.
This probabilistic interpretation of the rarefaction
method allows it to be directly related to a commonly
used measure of evenness, Hurlbert’s (1971) probability
of interspecific encounter. This metric is based on
Simpson’s (1949) dominance index (Eq. (3)), which is
the probability that two specimens picked at random
(with replacement) from a sample are of the same
species. Converting this from a dominance index to an
evenness index by taking its complement (Eq. (4)) and
accounting for finite collection size leads to D1 (Eq. (5);
Simpson 1949, Hurlbert 1971). D1 can be readily
interpreted as the probability that the second specimen
randomly picked from a sample (without replacement of
the first specimen) will be of the same species as the first
specimen.
This metric is not biased by sample size or species
richness (Nei and Roychoudhury 1973, Gotelli and
Graves 1996, Lande 1996), in contrast to a number of
other common diversity and evenness metrics (Fig. 2).
Because D1 represents a probability, it can theoretically
range from 0 to 1. As Rosenzweig (1995) and Hayek and
Buzas (1997) point out, however, the range of D1 is
constrained by the size (N) and the richness (S) of the
sample being evaluated. Its maximum value occurs when
all species make up an equal proportion, pi, of a sample
(Eq. (7a)). (If S cannot be divided into N without any
remainder, Eq. (7a) will slightly overestimate the maximum possible evenness. This is because Eq. (7a)
assumes that the ni values of all species are the same,
which means that the equation will only be exact if N is
an integer multiple of S.) D1’s minimum value occurs
when all species are represented by a single individual
except for one dominant species that makes up the rest
of the sample (Eq. (7b)). Hurlbert (1971) indicated that
D1 could be adjusted for the constraints on its range by
dividing D1 by max(D1) as in Eq. (8a) or adjusting for
both min(D1) and max (D1) as in Eq. (8b). However,
unlike D1, both Eq. (8a) and (8b) are sample size
OIKOS 104:2 (2004)
Table 1. Equations and variable definitions
3
2
N ni
N ni
6
7
m
S
m 7
ai1 6
/E(sm )S
41
5
N
N
m
m
Yni 1
m
m
S Qni 1
S
Þai1 ½1 j0 ð1
Þ
/E(sm )Sa
j0 ð1
i1
Nj
Nj
3 2
39
3
82
2
N ni
N ni >
N ni nj
N ni
N nj
>
>
>
=
<6
7
6
7
6
7
m m
S
6 m 761 m 7 2aS aj1 6
7
m
/Var(sm )a
i1 4
j2 i1 4
5 4
5>
5
N
N
N
N
N
>
>
>
;
:
m
m
m
m m
S
ai1
(1a)1
(1b)2
(2)3
2
ni
N
(3)4
D2 1ai1 p2i 1l
(4)4
S
S
lai1 p2i ai1
/
S
/
ni
ni 1
N S
1ai1 p2i
N1
N N1
(5)4
4(N 1)a p3i 2 a p2i 2(2n 3)(a p2i )2
N(N 1)
(6)4
S
D1 1ai1
/
Var(D1 )
/
N S1
N1 S
(7a)1
N2 S 1 (N S 1)2 (N S)2 (N 1)2 S 1
N(N 1)
N(N 1)
(7b)1
max(D1 )
/
min(D1 )
/
D1
S S
1ai1 p2i
max(D1 ) S 1
(8a)1
D1 min(D1 )
N2 (1 S a p2i )
N2 S a n2i
max(D1 ) min(D1 ) (S N)2 S(N S)2 (S N)2 S(N S)2
(8b)1
V0 (D1 )
/
V(D1 )
/
(9)5
Db D2;g D2;a
/
k
Db /
a D
Ng 1
D1;g D1;a j1 1;a
Ng
Ng
S
D1;g 1vai1
/
v
/
ni (ni 1)
vN
1
N
1
S
(1lg )1
a n2 N 1
l vN(vN 1) vN 1
N(vN 1) i1 i
vN 1 a vN 1
la 1 1
=
lg lg la
Db D2;g D2;a (v1)(1D2;g )
/
v
/
(10)
(11)
(12)6
D2;a 1
1D2;a
v
D2;g D2;a
Db
1
1
(1 D2;g )
(1 D2;g )
(13b)
D1;g D1;a /
N
(v1)(1D1;g )
N1
(13a)
OIKOS 104:2 (2004)
(13c)
379
v
/
N 1 D1;g D1;a
1
N
(1 D1;g )
(13d)
E(sm) /expected richness of sub-sample size m, S /richness of entire sample (i.e. local richness), N /number of specimens in entire
sample (i.e. local sample size), ni /abundance of species i in local collection, m /size of sub-sample, Var(sm) /variance of expected
richness of sub-sample size m, l /Simpson’s dominance metric, pi /proportion of species i in a sample, D2 /Simpson’s evenness,
D1 /bias-corrected Simpson’s evenness (equivalent to Hurlbert’s probability of interspecific encounter), var(D1) /variance of D1,
max(D1) /maximum of D1 given N and S of sample, min(D1)/minimum of D1 given N and S of sample, V’(D1) and V(D1) /
proposed evenness metrics taking account for restricted range of sample D1, D2,g /D2 of the composite regional sample, D2;a /
weighted average of D2 of local samples (weighted by sample size), D1,g /D1 of the composite regional sample, D1,a /D1 of local
samples, Ng /total number of individuals in composite regional sample, k/number of local collections, v /number of
compositionally non-overlapping local collections of equal size, richness and relative abundance distribution, la /Simpson’s
dominance for local samples, lg /Simpson’s dominance for the composite regional sample. 1Hurlbert 1971, 2Brewer and
Williamson 1994, 3Simpson 1949, 4Heck et al. 1975, 5Lande 1996, 6MacArthur 1972.
dependent (Fig. 2a) and are therefore not recommended
for general use in evaluating evenness (Sheldon 1969,
Peet 1975).
Note that if species abundances can be fractional and
less than one, then D1 can range from 0 to 1 unrestricted.
In other words, if species abundances are measured on a
continuous scale, like biomass or area, rather than on an
integer scale, like counts of individual organisms, then
there are no constraints on the maximum or minimum
values of D1 (Krebs 1999).
As an evenness metric, D1 can be directly related to
rarefaction: D1 /E(s2) /E(s1)/E(s2) /1 (see appendix;
Smith and Grassle 1977), which leads to the following
interpretation. A rarefaction curve grows by adding the
probability that each consecutively larger subsample will
include a new species. A sub-sample of m /1 will
necessarily have E(s1) /1. The expected richness of a
subsample of m /2 is the richness of E(s1) /1 plus the
probability that the second specimen will be a different
species than the first, i.e. D1. The difference is simply the
slope of the steepest segment of the rarefaction curve
and can never exceed a value of 1. Graphically, this
means that when comparing the rarefaction curves of
two collections, the curve that initially rises more steeply
is the more even of the two (as measured by D1) no
matter what the total richness of the samples is (Fig. 1).
In summary, using rarefaction curves to compare the
diversity of two samples provides information on both
richness and evenness, because D1 is calculated in the
process of rarefaction.
Additive partitioning of diversity
Fig. 1. Rarefaction of samples of equal size but different
richness and evenness. In all three plots, the middle line is a
real sample of S /30, N/295, and a typical concave up
abundance distribution, the top line is a sample of the same size
and richness but with maximum evenness, and the bottom line is
a sample of the same size and richness but with minimum
evenness. Rarefaction curves for all three are shown using (a)
arithmetic axes, (b) semi-log axes, (c) log-log axes for comparison.
380
In addition to the diversity of single samples, ecologists
are interested in diversity due to changes in species
composition across ecological landscapes (Loreau 2000,
Koleff and Gaston 2002). Following the terminology of
Whittaker (1972), alpha diversity describes diversity
within local habitats, beta diversity describes diversity
due to differences between habitats, and gamma diversity describes the total diversity of an entire region.
OIKOS 104:2 (2004)
Fig. 3. The relationship among alpha, beta, and gamma
diversities expressed by rarefaction and collector’s curves.
Curves a, b, and c are local samples. Curve a/b/c is the
sum of the three local samples. The heavy solid line is the
rarefaction curve of (a/b/c). The heavy dashed line is the
expected collector’s curve of (a/b/c). The rarefaction curve is
greater than a, b, or c or their composite collector’s curve due to
compositional differences among the local samples, although
their evennesses are approximately equal. D1,g is E(s2) /1 of the
composite curve (a/b/c). Mean D1,a is approximately the
average E(s2) /1 of local curves a, b, and c. The difference
between the two is approximately Db.
Fig. 2. Selected evenness metrics as a function of sample size.
D1 /Hurlbert’s probability of interspecific encounter. (a) D2 /
Simpson’s
evenness,
V(D1) /(D1 /min(D1))/(max(D1)/
min(D1)), V?(D1)/D1/max(D1), J?/H/log(S). (b) H/
/Spilog(pi)/log(S), a /Fisher’s log-series a. All metrics have
been applied to the same data as used to produce the middle
rarefaction curve in Fig. 1.
Although alpha and gamma diversity are typically
thought of in terms of richness, they can be assessed
using any aspect of diversity (Whittaker 1972). However,
because beta diversity represents the amount of difference in species composition among multiple samples, it
has often been measured using similarity metrics designed to evaluate turnover along an environmental
gradient (Wilson and Mohler 1983, Wilson and Shmida
1984, Vellend 2001, Koleff et al. 2003). This difference in
units of measurement can make it difficult to interpret
beta relative to alpha and gamma diversity (Veech et al.
2002).
However, rarefaction and D1 can be applied to the
assessment of beta diversity, thereby providing a means
of evaluating among-sample diversity using the same
techniques as within-sample diversity. Beta diversity can
be graphically assessed among a set of collections by
comparing the expected collector’s curve with the
rarefaction curve of all the collections combined (Gotelli
and Colwell 2001). Collector’s curves and rarefaction
curves differ in the way that expected richness is
accumulated as cumulative sample size increases. Rarefaction curves depict how richness is expected to
accumulate as individual specimens are added to a
OIKOS 104:2 (2004)
growing random sub-sample. Collector’s curves grow
by adding entire real samples rather than individual
specimens and keeping track of the cumulative S as the
number of incorporated samples increases (Fig. 3). If
individuals of different species are randomly distributed
among local samples, then the collector’s curve will
match the predicted rarefaction curve (i.e. each local
sample is simply a random, smaller sub-sample of the
composite regional sample and therefore must fall on the
same rarefaction curve). However, if individuals of
different species are not randomly allocated among
collections (i.e. different habitats have different species
compositions), the expected collector’s curve will fall
below the individuals-based rarefaction curve (Gotelli
and Colwell 2001).
The reasoning behind this pattern is that when species
are not randomly distributed among different patches in
a region, a local collection will include a higher
proportion of the species that occur in that particular
patch and fewer species common in other patches.
Richness of such a collection is expected to be lower
than richness of a random sub-sample of the same size
that draws from the entire regional species pool. Smith et
al. (1985) review several approaches to rarefaction
including work by Shinozaki (1963) and Kobayashi
(1982, 1983) that predict expected richness based on
non-random distribution of species among quadrats /
effectively, predictions of collector’s curves for random
(i.e. rarefaction) and non-random distributions of species among collections of equal size.
The difference between a collector’s curve and a
rarefaction curve is a graphical depiction of the degree
381
to which different species are not randomly distributed
among local collections (Fig. 3). This difference, in turn,
is a measure of beta diversity / the diversity contributed
to the region by differences in species composition of
local samples.
Beta diversity can also be measured by additively
partitioning gamma diversity into alpha and beta
components (Eq. (9); Nei 1973, Patil and Taillie 1982,
Lande 1996, Veech et al. 2002). To ensure that Db /0, it
must be calculated using D2 (Eq. (4)) rather than the
bias-corrected D1, because D2 is a strictly concave
measurement of evenness, whereas D1 is not. (A metric
is concave when its value for the composite regional
sample must be greater than or equal to the weighted
average of local samples; Patil and Taillie 1982, Lande
1996.) Equation (10) presents the correction to obtain Db
from D1 / as one would expect, the difference is
negligible when Ng is large. Patil and Taillie (1982)
term Db the ‘‘rarity gain’’ averaged over all communities
because it represents the average increase in probability
that two specimens randomly chosen from the composite
regional sample are more likely to be different than two
specimens randomly chosen from a local sample. If this
difference is zero, then the composite sample is no more
even than the local sample, even though it may be
significantly larger and richer.
Due to the relationship between D1 and rarefaction,
Db can be interpreted graphically (Fig. 3). The initial rise
of the collector’s curve approximates the average rise of
the rarefaction curves of the individual local collections.
As depicted in Fig. 3, Db is approximately the difference
between the composite, regional rarefaction curve representing gamma diversity and the average of the local
rarefaction curves measured at m /2. In other words, Db
measures beta diversity as the difference between the
regional rarefaction curve and an approximation of the
collector’s curve. This is an approximation because the
slopes of the rarefaction curves equal the bias-corrected
D2 rather than D1, and also because the estimate of D2,a
in Eq. (9) is a weighted average whereas the rarefaction
curves are not. However, if Ng is much greater than the
number of collections, then the error is small.
So, just as rarefaction curves display information on
both richness and evenness of individual samples, they
also provide information on the partitioning of diversity,
both richness and evenness, among samples.
An alternative formulation for beta diversity
Veech et al. (2002) have strongly advocated using the
additive formulation of beta diversity reviewed by Lande
(1996) in addition to a more traditional multiplicative
approach. However, one particular multiplicative approach is directly related to Db (Eq. (9)). It addresses the
question of how many compositionally distinct local
382
samples (v) does a given value of D1,g represent? The
definition of D1,g (Eq. (11)) can be reformulated to show
that v/la/lg (Eq. (12)), and can be interpreted in the
following manner (l is defined in Eq. (3)). MacArthur
(1972) and Patil and Taillie (1982) point out that 1/l is
the number of species required for a perfectly even
community to have the same l value as the measured
sample / what MacArthur (1972) called the number of
‘‘equally common species’’. In the case of Eq. (12), v/
(1/lg)/(1/la) / i.e. the composite sample has v times as
many ‘‘equally common species’’ as any of the local
samples. MacArthur’s M (1972; page 189) is a special
case of Eq. (12) assuming only two local communities of
equal size and was used by MacArthur (1965) to assess
the relationship between bird diversity and habitat
diversity in a case study from Puerto Rico. Equation
(12) can be applied to more than two communities by
using the average la as the average of the local
communities (weighting each sample by its size).
Because v measures the minimum number of compositionally distinct local communities of average D1 it
would take to make up the composite sample, it can be
interpreted as a measure of beta diversity related to Db.
Equation (13a), which is derived from Eq. (11), shows
that Db is a reciprocal function of v. In other words, each
local sample incorporated into the regional composite
sample increases Db by a progressively smaller degree
that depends solely on the average diversity of the local
samples.
Many other factors can also influence the measurement of beta diversity when using real data. This is
particularly relevant when using diversity measurements
to compare local and regional communities at different
times or in different places. In addition to the importance of local diversity discussed above, other influences
include differences in the sizes of local samples, differences in evenness among local communities, and different degrees of compositional overlap.
A potential source of bias in measuring beta diversity
arises from the influence of different local sample sizes.
Fig. 4 illustrates the idea: A, B, and C are three local
communities with identical relative abundance distributions (i.e. identical evenness) but no species in common.
When two are combined in equal proportions, v/2;
when all three are combined in equal proportions, v/3.
If they are combined in unequal proportions, however,
then v is less than three, implying that fewer than three
distinct local communities have been included even when
the average of the local communities is weighted by
sample size. In addition, note that v need not increase
linearly with the relative proportions of the samples
mixed.
Fig. 5 illustrates the influence of differences in local
evenness on v. Sample A has S/30 and N /295 (its D1
and rarefaction curve are shown in Fig. 1 and 2) and has
a typical hollow curve distribution of abundances.
OIKOS 104:2 (2004)
Fig. 4. Ternary diagram showing the effect on v (contoured) of
combining varying proportions of local samples with equal D1,a,
S, and N. Samples A, B, and C have no species in common.
Note that v correctly identifies when samples are combined in
equal proportions, but not when they are combined unequally.
To depict more communities would require more dimensions.
v without an increase in richness as local samples are
combined.
The effects of combining both different species
composition and different abundance distributions on
measurements of v are shown in Fig. 6. In this case, A,
B, and C have the same S and N as before, but they have
no species in common. For the given S and N, B has
maximum evenness and C has minimum evenness. In
this case, when compositionally distinct samples are
mixed in equal proportions, v faithfully indicates the
number of collections. However, differences in abundance structure also increase v beyond a value of three
when local samples do not all contribute equally.
In most real data-sets, there are likely to be more than
three local samples, so Fig. 4, 5, and 6 would require
additional dimensions. However, local samples from a
given region are also likely to have at least some
overlapping species and a much smaller range of
evenness values than the minimum and maximum
possible values used in these diagrams. This suggests
that distortions due to differences in evenness and
composition among local samples will be much less
extreme than shown in Fig. 5 and 6.
Whenever complex community data are boiled down
to a single index, there will necessarily be loss of
information and distortion of underlying community
patterns. The discussion here shows how the measurement of diversity among local communities is influenced
not only by the number of species that they have in
common, but also differences in their respective abun-
Fig. 5. Ternary diagram showing the effect on v (contoured) of
combining varying proportions of local samples with the same
species, but different D1,a. Samples A, B, and C have equal N
and S. Sample A is the same collection used as an example in
Fig. 1 and 2, while B and C are collections with maximum and
minimum evenness for the given N and S. v represents the
number of compositionally exclusive local communities with
equal D1,a (equal to the weighted average of all three local
samples) that would have to be combined to achieve the
measured level of D1,g.
Samples B and C have the same S and N, and include the
same species as A in the same rank abundance order,
although abundances are redistributed for maximum
(sample B) and minimum (sample C) possible evenness.
In this case, maximum v does not occur when the three
are combined in equal proportions, but rather along the
B-C gradient. Differences in relative abundance distributions among local communities can lead to an increase in
OIKOS 104:2 (2004)
Fig. 6. Ternary diagram showing the effect on v (contoured) of
combining varying proportions of local samples with different
species and different D1,a. Samples A, B, and C have equal N
and S. Sample A is the same collection used as an example in
Fig. 1 and 2, while B and C are collections with maximum and
minimum evenness for the given N and S. v represents the
number of compositionally exclusive local communities with
equal D1,a (equal to the weighted average of all three local
samples) that would have to be combined to achieve the
measured level of D1,g.
383
dance structures and the proportions to which they are
mixed and the diversity of habitats they come from.
Beta diversity and the species-area relationship
One of the best-known patterns in ecology concerns the
relationship between species diversity and area (reviewed
by Connor and McCoy 1979, McGuiness 1984, Williamson 1988, Hart and Horwitz 1991, Rosenzweig 1995, He
and Legendre 2002). If one assumes that the density of
individuals in a region is constant (Hubbell 2001), then
species-area curves (SAC’s) have the same axes as
rarefaction curves (Fig. 7a). The conversion from species
individuals to species density is not necessarily straightforward (Nee and Cotgreave 2002) and can influence
interpretation of species accumulation curves (Gotelli
and Colwell 2001). This allows the use of rarefaction and
Fig. 7. The relationship between rarefaction and D1 and
species-area curves (SAC). (a) Solid lines (RC) are Coleman
curves (i.e. rarefaction curves scaled to area) of individual
samples. Dashed line is the species-area curve / it need not be
linear on log-log plots but typically is (Rosenzweig 1995). A
region with uniformly distributed species would have a SAC
matching the topmost rarefaction curve. The difference between
the two reflects community patchiness. (b) The relationship
between alpha, beta, and gamma diversities on a heterogeneous
landscape: the value of beta changes with area/size of sample.
Evenness values of samples at different scales need not become
asymptotic (i.e. landscape need never become saturated with
local community patches).
384
D1 to help interpret and test several proposed explanations of the shape of SAC’s.
One explanation of increasing richness with area is
passive sampling / a larger area produces larger
samples, which will include more species (McGuiness
1984). An explicit formulation of this null expectation
was derived by Coleman (1981) and Coleman et al.
(1982), and its equivalency to rarefaction (Eq. (1b)) has
been demonstrated by Brewer and Williamson (1994).
Another common explanation for the shape of SAC’s is
that as the area sampled increases, so does the number of
different habitats and therefore the number of distinct
communities (McGuiness 1984). Essentially, this hypothesis calls upon beta diversity as the source of species
increase in SAC’s.
Rarefaction curves are typically shown using arithmetic axes, in contrast to the semi-log or log-log plots
typical of SAC’s. For reference, Fig. 1 shows the same set
of rarefaction curves all three ways: on semi-log plots,
they are concave-up or sinusoidal, whereas on log-log
plots they can be concave-up if they represent very
uneven relative abundance distributions, but they are
typically convex-up.
By comparing SAC’s with rarefaction curves, it is
possible to evaluate the compositional variability of an
ecological landscape and how it changes with the scale
evaluated (Hill et al. 1994, Plotkin et al. 2000, Scheiner
et al. 2000). If a region is compositionally uniform, then
the only reason for increase of richness with area is
sampling effect and the species-area curve should be
statistically indistinguishable from a rarefaction curve
(Coleman et al. 1982, McGuiness 1984, Williamson
1988, Hart and Horwitz 1991). If, however, species are
not randomly scattered across an ecological landscape,
then the rarefaction curves of each individual sample
should rise above the species-area curve with the
terminal ends of all the rarefaction curves defining the
species-area curve (Fig. 7a). If the SAC follows a
rarefaction trajectory, then all the rarefaction curves of
the individual samples should lie on the same trajectory.
Since the SAC is essentially a collector’s curve, the
difference between the rarefaction and the SAC is a
depiction of the landscape’s beta diversity / the component of the region’s total diversity that is due to
differences in composition between local patches. It
should be noted that the species-area relationship need
not follow a power law, as depicted in Fig. 7a, for this to
be true.
Based on the explicit relationship between D1 and
rarefaction curves, a uniform landscape is expected to
have a constant D1 value regardless of area because D1 is
independent of sample size (Fig. 7b). Rosenzweig (1995)
took a similar approach, but used Fischer’s log-series a,
which is not strictly sample size independent (Fig. 1b).
An increase in D1 with sample size reflects non-random
differences in species composition in a region / i.e. it is a
OIKOS 104:2 (2004)
measure of beta diversity (Fig. 7b). If D1 approaches a
constant at some sample size/area, this implies that the
area has sampled the full variety of compositionally
different local habitat patches. However, there is no a
priori reason to believe that D1 will approach an
asymptotic value. Such a pattern also makes clear that
Da and Db (Fig. 7b) can change with local sample size.
The pattern of change as a function of sample size
provides information on the ecological structure of a
landscape. This also means that reported values of alpha
diversity should be accompanied by information on the
size of collections and that beta diversity should not be
treated as a constant characterizing a region.
In summary, species-area curves, one of the most
familiar patterns in ecology, are related to rarefaction as
a null hypothesis, a point made by several authors
previously (McGuiness 1984, Hart and Horwitz 1991).
Interestingly, the interpretations of SAC’s regarding their
governing processes (passive sampling versus habitat
differentiation) converges on emerging ideas of additive
partitioning of both richness (Gotelli and Colwell 2001)
and evenness (Lande 1996, Veech et al. 2002). These
connections may prove useful in interpreting, expressing,
and testing the results of models of ecological communities and landscapes, which can predict changes in beta
diversity with sample size and area (Caswell and Cohen
1993, Bell 2000, Hubbell 2001, Chave et al. 2002).
Summary
Analyzing diversity requires assessing both richness and
evenness. Richness is effectively assessed using rarefaction, which accounts for differences in sample size.
However, rarefaction curves also depict information on
the evenness of samples in the steepness of their initial
slope, which is equal to D1, the bias-corrected estimate of
the probability that two specimens picked at random
from a sample will be different species (Simpson 1949,
Hurlbert 1971).
Lande (1996) has shown that D1 of a regional
composite sample (gamma diversity) can be partitioned
additively into within-sample (alpha) diversity and
between-sample (beta) diversity. The same mathematical
partitioning among alpha, beta, and gamma diversities
can be seen in the difference (beta) between the rarefaction curve of a regional composite sample (gamma) and
the collector’s curve constructed by cumulatively adding
all the collections from a region (alpha).
However, it should be pointed out that assessment of
beta diversity can be influenced by many factors
including local sample size differences, variation in local
evenness values, degree of compositional overlap, and
the number of local samples incorporated into the
regional composite estimate of gamma diversity. In
addition, estimates of beta diversity also need to take
OIKOS 104:2 (2004)
into account habitat diversity when comparing values
from different regions. Even when measured using the
same metric as alpha and gamma diversities (Lande
1996, Veech et al. 2002), beta diversity can be influenced
by factors that may not influence other forms of
diversity.
Lastly, rarefaction curves are equivalent to the curves
Coleman et al. (1982) used as a null hypothesis when
evaluating species-area relationships (i.e. the passive
sampling explanation; McGuiness 1984, Brewer and
Williamson 1994). Therefore, the diversity partitioning
properties of both rarefaction and D1 can be taken
advantage of when interpreting SAC’s. In addition, these
measures of richness and evenness provide a means of
interpreting the results of ecological landscape models
and comparing them with data addressing the relationship between species diversity, area, and sample size.
Acknowledgements / Thanks to Doug Erwin for providing
comments on the manuscript. This paper was prepared with
support from the Smithsonian Institution’s Walcott Fund and
the NASA Astrobiology Program. This is Paleobiology
Database Publication Number 18.
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Appendix A. The equivalence of the initial slope of a rarefaction line to the bias-corrected Simpson’s evenness
(Hurlbert’s D1) is demonstrated if it can be shown that E(sm; m /2)/E(sm; m /1)/D1.
3
2
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E(s2 )E(s1 )
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Q.E.D.
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