Ecology Letters, (2007) 10: 1037–1045

IDEA ANDPERSPECTIVE

Jessica L. Green1 andJoshua B. Plotkin2*
1

Center for Ecology and
Evolutionary Biology, Universityof Oregon, Eugene, OR, USA
2

Department of Biology,

University of Pennsylvania, 433
S. University Ave., Philadelphia,
PA 19104, USA*Correspondence: E-mail:jplotkin@sas.upenn.edu

doi: 10.1111/j.1461-0248.2007.01101.x

A statistical theory for sampling species abundances
Abstract
The pattern of species abundances is central to ecology. But direct measurements ofspecies abundances at ecologically relevant scales are typically unfeasible. This limitation
has motivated a long-standing interest in the relationship between the abundance
distribution in a large, regional community and the distribution observed in a small
sample from the community. Here, we develop a statistical sampling theory to describe
how observed patterns of species abundances are influenced by the spatial distributionsof populations. For a wide range of regional-scale abundance distributions we derive
exact expressions for the sampled abundance distributions, as a function of sample size
and the degree of conspecific spatial aggregation. We show that if populations are
randomly distributed in space then the sampled and regional-scale species-abundance
distribution typically have the same functional form: sampling can be expressed by asimple scaling relationship. In the case of aggregated spatial distributions, however, theshape of a sampled species-abundance distribution diverges from the regional-scale
distribution. Conspecific aggregation results in sampled distributions that are skewed
towards both rare and common species. We discuss our findings in light of recent resultsfrom neutral community theory, and in the context of estimating biodiversity.
Keywords
Species-abundance distribution, random sampling, negative-binomial sampling, spatial
aggregation, biodiversity, community.
Ecology Letters (2007) 10: 1037–1045

INTRODUCTION

The distribution of species abundances is a fundamental
topic in ecological research. Species-abundance distributions
have been used to examine the influence of niche
differentiation, dispersal, density dependence, speciationand extinction on the structure and dynamics of ecologicalcommunities (Tokeshi 1993; Hubbell 2001; Chave et al.
2002; Magurran 2004; McGill et al. in press). In conservation
biology, knowledge of the species-abundance distribution
helps one to predict the likelihood of population persistence
and community stability in face of global change. Despite
the theoretical and practical importance of species-abundance distributions, it is difficult to directly measure all
speciesÕ abundances at ecologically relevant scales. For
micro-organisms, this poses a challenge at the scale of asingle environmental sample (e.g. < 1 g of soil; Prosser et al.
2007). In plant and animal communities as well, the task of
exhaustively sampling a full community is typically impossible. Therefore, ecologists have a long-standing interest in
the relationship between the underlying species-abundance

distribution of a large, regional community and the observed
abundance distribution when sampling a small proportionof the community.
Efforts to develop a sampling theory of species abundances have utilized several approaches. The most widely
utilized approach, which dates back to Fisher et al. (1943)
assumes that individuals are randomly sampled from anecological community. Fisher et al. (1943) sought tounderstand patterns of species abundance in butterfly,
beetle and moth communities sampled throughout theworld. They found that species abundances in random
samples were well described by a logseries distribution, a
distribution they mathematically derived by Poisson sampling from a gamma distribution. Their analyses laid thefoundation of biodiversity sampling theory (May 1975;
Pielou 1975) and remain at the forefront of literature on
species-abundance distributions (Hubbell 2001; Chave 2004;
Magurran 2004; McGill et al. in press).
Engineer and ornithologist Frank Preston (1948)
popularized the lognormal species-abundance distributionin ecology by demonstrating that the lognormal had 2007 Blackwell Publishing Ltd/CNRS

1038 J. L. Green and J. B. Plotkin

scale-invariant properties upon random sampling. He
showed by tabulation (Preston 1948) that Poisson sampling
individuals from a lognormal species-abundance distribution
results in a sample distribution that is approximately
lognormal with identical variance. Preston (1962, p. 186)
recognized that Poisson sampling individuals was analogous
to Ôa situation in space and time where the individuals, orpairs, are distributed at random, not clumped on one handor over-regularized on the otherÕ. He noted that contagion,or conspecific aggregation, would likely to result in aÔsomewhat skewedÕ sample distribution (Preston 1962, p.
203); however, he did not rigorously explore the influence of
this contagion on his samples. Biodiversity sampling theory
has since predominantly assumed random sampling (e.g.
Pielou 1975; Dewdney 1998; Gotelli & Colwell 2001; Chao
& Bunge 2002), while less is known about the sampling
properties of species-abundance distributions for spatiallyaggregated populations.
Efforts to understand the sampled abundance distribution of aggregated populations have focused primarily on a
specific type of aggregation. The assumption of fractal, or
self-similar spatial distributions has been leveraged to
explore how species-abundance distributions scale withsampling area (Banavar et al. 1999; Harte et al. 1999). Recent
analyses, however, suggest that such fractal models arebiologically unrealistic (Green et al. 2003; Pueyo 2006). An
alternative statistical model known as the Hypothesis of
Equal Allocation Probabilities (HEAP) allocates individuals
across a landscape according to a set of assembly-rules,
yielding a scale-dependent species-abundance distributionthat matches empirical vegetation data relatively well (Harteet al. 2005). Although the HEAP framework allows for arange of aggregation patterns, from random to highly
clustered, this model has not been utilized to explore how
the degree of aggregation influences sampled speciesabundance distributions.
Neutral community theory (Hubbell 2001) provides amechanistic approach for modelling heterogeneity byaccounting for the effect of dispersal limitation. Alonso &McKane (2004) applied Poisson sampling to the metacommunity multinomial relative-abundance distribution toderive an analytical expression for the sample distribution
under the assumption of zero dispersal limitation. Etienne &Alonso (2005) later invoked dispersal limitation by replacing
random sampling with the dispersal-limited binomial(actually, hypergeometric) sampling. This allowed forsampling heterogeneity by modelling the probability for adispersal-limited species to be present in a sample with agiven abundance under the assumptions stipulated byneutral community theory. These recent developments,
while powerful, have not explicitly addressed the sampling properties of communities assembled by non-neutral
forces.
 2007 Blackwell Publishing Ltd/CNRS

Idea and Perspective

In this study, we present a general statistical framework
for understanding the effect of spatial heterogeneity (or,equivalently, heterogeneity in the sampling scheme) on thespecies-abundance distribution observed in a sample. Our
sampling framework does not assume a particular type of
population aggregation (e.g. fractal theory) or community
dynamics (e.g. neutral theory). We begin by analysing thesimple case in which individuals are randomly sampled fromthe larger regional community, corresponding to random
spatial distributions across the landscape. We then examine
a more realistic scenario of negative-binomial sampling,which models spatial clustering of conspecific individuals.
We apply our techniques to a wide range of abundance
distributions, deriving exact expressions for the sampledabundance distributions as a function of sample size and thedegree of conspecific clustering. We also demonstrate two
important, generic properties of how spatial aggregation
affects the sampled species-abundance distribution.
STATISTICAL FRAMEWORKSpecies-abundance distributions are measured in the laboratory or field by counting the number of species in acommunity represented by n individuals. For the purpose of
our analysis, we will characterize species abundances in alarge region using a continuous probability density function/(n). The expression /(n)dn represents the fraction of
species whose abundance
falls between n and n + dn. Proper
R1
normalization requires 0 /ðnÞdn ¼ 1. In reality speciesÕ
abundances are discrete. We use continuous distributions toprovide consistency with a sampling theory of b-diversity(Plotkin & Muller-Landau 2002). Aside from offering
analytical tractability, there is a long-standing precedent
for using continuous distributions to describe species
abundances (Pielou 1975).
Our interest lies in the relationship between the speciesabundance distribution at the regional scale, /(n), and theabundance distribution observed in a sample that constitutes
a proportion a of the larger ambient region, denoted /a(y).
Let wa(y|n) denote the probability that a species will be
represented by y individuals in the sample, given that it hasabundance n in the larger region. Then, the sampled speciesabundance distribution may be expressed as:/a ðyÞ ¼

Z1

wa ðyjnÞ/ðnÞdn

ð1Þ

0Equation 1 provides a general expression for the scaling of
the species-abundance distribution with sample size, given
an arbitrary sampling scheme wa(y|n). Equation 1 follows
directly from the law of total probability. By using different
sampling schemes for wa(y|n), we can model different
spatial distributions of conspecifics across the landscape.

Idea and Perspective

Statistical theory for sampling species abundances 1039

Plotkin & Muller-Landau (2002) used an analogous
approach to derive a sampling theory for b-diversity (thechange in species composition between samples). They
analysed the probability that a species will be present withany abundance y ‡ 1 in a sample, given that it hasabundance n in the larger region. Equation 1 generalizes
their work to describe the full species-abundance distribution in a sample. Below we investigate eqn 1 for a suite ofregional-scale species-abundance distributions /(n), and fora suite of sampling schemes ranging from random toaggregated.
Random spatial distributions

To examine the scaling of the species-abundance distribution under the assumption of randomly distributed individuals (or, equivalently, randomly sampled individuals), we usey
the Poisson sampling distribution wa ðyjnÞ ¼ e an ðanÞy! ,which denotes the probability that a species will have yindividuals in a sample that constitutes a proportion a of thelarger ambient region, given that it has abundance n in thelarger region. Inserting the Poisson sampling distribution
into eqn 1 yields:/a ðyÞ ¼

Z1

ean

ðanÞy
/ðnÞdny!

ð2Þ0Equation 2 is equivalent to a mixed Poisson distribution,
which has been studied extensively by statisticians (see, forexample, Karlis & Xekalaki 2005). Mixed Poisson distributions were first introduced in the ecology literature by Fisher
et al. (1943) who derived the logseries species-abundance
distribution by mixing the Poisson distribution with agamma distribution (see also Boswell & Patil 1971).
The combination of the Poisson sampling distribution
with the continuous abundance distribution, /(n), in the
integrand of eqn 2 may result in a continuous sampledabundance distribution /a(y) that does not normalize toone, particularly at small sample sizes. ToRaccount for this1
we impose the normalization constraint 0 /a ðyÞdy ¼ 1,
although the normalization factors are typically close tounity. An alternative approach which alleviates the need
to normalize is to interpret the sample distribution /a(y)as discrete, but this results in an unwanted inconsistency
between the form of the regional-scale and sampled
species-abundance distribution. The most precise, yet leasttractable approach would entail a fully discrete version of
eqn 1, assuming that both the regional-scale and sample
species-abundance distributions are discrete, and wouldmodel wa(y|n) according to the hypergeometric distribution (Dewdney 1998). Nevertheless, provided regional
abundances are large enough to be modelled by acontinuous probability density function, eqn 2 provides

a reasonable approximation to discrete, hypergeometricsampling.
Aggregated spatial distributions

We now examine the scaling of species abundances under
the assumption that conspecific individuals are autocorrelated in space. This requires us to specify a sampling schemewa(y|n) that models samples from a spatially aggregatedpopulation. Our goal was to understand how the degree of
intraspecific aggregation influences the species-abundance
distribution observed in the sample, without assuming aparticular type of community dynamics such as that invokedin neutral theory (Etienne & Alonso 2005). For thispurpose, it is convenient to model the sampling probabilities
according to the widely used negative-binomial distribution:


y 
k
Cðk þ yÞankwa ðyjnÞ ¼;
ð3Þ
y!CðkÞan þ kan þ kwhere
the gamma function is defined by CðzÞ ¼
R 1 t z1et
dt. The parameter k reflects the degree of intra0
specific spatial clustering, and it can assume values in theintervals ()¥,)an) and (0, ¥). When k is negative, wa(y|n)
describes a regular spatial distribution of individuals; when k
is positive, wa(y|n) describes an aggregated spatial distribution of individuals, which is the typical situation for ecological populations (Plotkin 2000; Green et al. 2004; Genin
et al. 2005; Sandin & Pacala 2005). As k ﬁ ±¥, the negative-binomial distribution converges to the Poisson distribution, representing random sampling. Aside fromproviding analytical tractability, the negative-binomial distribution offers a rich historical context as it has been used
for decades to model the spatial structure of biologicalpopulations (Anscombe 1949; Krebs 1998; He & Gaston
2000) and communities (He & Legendre 2002; Plotkin &Muller-Landau 2002; Green & Ostling 2003).
Substituting eqn 3 into eqn 1 yields the following
relationship between the species-abundance distribution at
the regional scale, /(n), and the distribution observed in asample of proportion a from the region:Cðk þ yÞ/a ðyÞ ¼
y!CðkÞ

Z1 

y 
k
an
k
/ðnÞdnan þ kan þ kð4Þ

0

As before, a continuous distribution /a(y) is obtainedfrom
R 1 eqn 4 by enforcing the normalization constraint
0 /a ðyÞdy ¼ 1.
Asymmetric aggregated spatial distributions

In the preceding section we examined the properties ofsampled species abundances assuming that all species are
symmetric, i.e. that all species have the same sampling 2007 Blackwell Publishing Ltd/CNRS

1040 J. L. Green and J. B. Plotkin

Idea and Perspective

properties regardless of their abundance at the regional
scale. But the spatial distributions of species are often notsymmetric in this sense. For example, more abundantspecies typically exhibit less spatial clustering or, equivalently, larger values of the negative-binomial parameter k(Condit et al. 2000). How does such asymmetry affect theproperties of expected species abundances in a sample?
We can address the asymmetric situation by modifying
eqn 4 to account for a clustering parameter k(n) that
depends upon a speciesÕ abundance, n, at the regional
scale:/a ðyÞ ¼

Z1 anan þ kðnÞ

y 

kðnÞ
kðnÞ
CðkðnÞ þ yÞ/ðnÞdnan þ kðnÞ
y!CðkðnÞÞ

0

ð5Þ

most important finding, apparent from graphing theexpressions in Table 1, is that when individuals are sampled
randomly the shape of the sampled abundance distribution/a(y) exhibits a strong resemblance to that of the regionalscale distribution /(n). This scaling behaviour was firstrecognized by Preston (1948) in the special case of thelognormal species-abundance distribution, and later byDewdney (1998). Here, we mathematically characterize thisbehaviour by positing a simple relationship:
1 y 
/a ðyÞ  /ð6Þ
a a
We can derive this scaling rule and its associated error
expansion by generalizing the approach of Etienne &Alonso (2005). Substituting z ¼ an into Equation 2, we find:1/a ðyÞ ¼
a

Z1
0

RESULTS¼

Random spatial distributions

Table 1 catalogues closed-form solutions to eqn 2 for arange of regional species-abundance distributions. These
expressions can be used to quantify the exact relationshipbetween large-scale species-abundance distributions and thatobserved in a random sample of individuals. Some of thesample distributions in Table 1 have been previously
derived as Mixed Poisson distributions in the statistics
literature (Johnson et al. 2005; Karlis & Xekalaki 2005). The

1
ay

Z

ez

z 
z
e z
/
zdz
CðyÞ a1
E ½f ðzÞ ¼ f ðz Þ þ VarðZ Þf }ðz Þ þ . . .
2

Sampledabundance
distribution [/a(y)]
 a y
yLðxÞ
/a ðyÞ ¼  alnxDistribution nameRegional
abundance
range

Continuous logseriesxnnCð0; ln xÞ0n‡1Otherwise

Exponential*

ke)kn

0£n£¥

ak
/a ðyÞ ¼ ðaþkÞyþ1

Gamma*

kb nb1 ekn
CðbÞ

0£n£¥

a k CðyþbÞ
/a ðyÞ ¼ y!CðbÞðaþkÞ
yþb

Lognormal*

exp  2r2
pﬃﬃﬃﬃ
nr 2p

0£n£¥

–

Truncated hyperbolic1n lnðM =nÞm£n£M
OtherwiseÞ/a ðyÞ ¼ Cðy;amÞCðy;aM
y! lnðM =mÞyy b

ðln nlÞ2

Species-abundance models were chosen to parallel thoseR described in Plotkin & Muller1
Landau (2002). C denotes the gamma functionCðzÞ ¼ 0 e t t z1 dt; C(a,x) denotes theR 1 t a1
incomplete
R xgamma function Cða; xÞ ¼ x e t dt; L(x) denotes the logarithmic integralLðxÞ ¼ 0 dt= lnðtÞ. Dashes indicate that a closed-form solution is unavailable. Sample
distributions marked with an asterisk (*) have been examined previously (see Pielou 1975;
Johnson et al. 2005 for a comprehensive review). The closed-form
R 1 solution for the continuous logseries is an approximation which is valid provided 0 wa ðyjnÞ/ðnÞdn << 1 (seeAppendix A for details on analytical and numerical integration).
 2007 Blackwell Publishing Ltd/CNRS

ð7Þ

y1

If we let f (z) ¼ z/(z /a), then the expression above isequal to the expected value of f (Z ), where Z is a gammadistributed random variable with shape parameter y andscale 1. Expanding f (z ) around the mean of Z leads to thefamiliar expansion:

Regional
abundance
distribution [/(n)]0z y z 
/
dz
a
y!

ð8Þ

Table 1 Regional and sampled abundance

distributions under Poisson samplingIdea and Perspective

As the mean of variance of Z both equal to y, this yields
11f ð yÞ þ yf 00 ð yÞ þ . . .
/a ðyÞ ¼
ay
2
  

1
y
1
¼
/
:y þ yf 00 ðyÞ þ . . .
ay
a
2
y 
 y i
1  y 
y h/¼:y þ 2 2a/0
þ y/00
þ ...
ay
a
2a
a
a
 
1 y 
1
¼ /
þO 2
ð9Þ
a a
a
This scaling relationship, or Ôrule of thumbÕ, is intuitive
because it suggests that under random sampling theabundance distribution scales according to a change ofvariables by proportion a. Figure 1 illustrates a comparison
between the Poisson-sampled species-abundance distribution (eqn 2) and the rule of thumb (eqn 6). The sampledabundance distribution and the rule of thumb are indistinguishable except in the limit of small sample sizes. In
general, given the error term derived in eqn 9, we expect therule of thumb scaling relationship to approximate eqn 2
accurately except when the proportion of area sampled isvery small. In this regime, combining Poisson sampling witha continuous abundance distribution yields inaccurate
results, and a fully discrete approach is required.
Our proposed rule of thumb (eqn 6) is also supported byresults from applied statistics stating that the probability

Statistical theory for sampling species abundances 1041

function of a mixed Poisson distribution often resembles
that of its mixing distribution (Johnson et al. 2005; Karlis &Xekalaki 2005). In particular, the asymptotic tails of mixed
Poisson distributions agree with the tails of their mixing
distributions provided /(n) satisfies /(n) ¼ C(n)nae)bn, as
n ﬁ ¥, where C(n) is locally bounded and varies slowly at
infinity (Willmot 1990).
Aggregated spatial distributions

Figure 2 illustrates the relationship between the sampled
distribution /a(y) and the regional-scale distribution /(n),when conspecific individuals are aggregated in space. Unlike
the case for randomly sampled individuals, the shapes of thesampled species-abundance distributions significantly differ
from the regional-scale distribution. In particular, intraspecific aggregation skews the abundance distribution towards
both more rare and more abundant species in the sample,
compared to random sampling.
Table 2 catalogues closed-form analytical solutions to eqn
4 for the regional species-abundance distributions listed inTable 1. Table 2 provides a means of quantifying therelationship between the regional-scale distribution and thesample distribution for any degree of sampling heterogeneity as specified by the clustering parameter k. These sample
distributions may be readily utilized in future theoreticalwork aimed at linking small-scale sample distributions tolarge-scale biodiversity patterns.
Despite the complexity of the derived sample distributions, they each exhibit a similar behaviour under clusteredsampling. The generality of this behaviour is illustrated inFig. 3: clustered sampling results in a greater proportion of

Figure 1 Scaling of relative distributions in the case of random

sampling. This example assumes a gamma distribution at the regional
scale with parameters k ¼ 0.000075, b ¼ 0.75. Sampling proportion a decreases from right to left as a ¼ 1 (regional scale), a ¼ 2)2,
a ¼ 2)4, a ¼ 2)6, a ¼ 2)8, a ¼ 2)10, a ¼ 2)12. The scaling
relationship is represented by the solid lines, and the solution to
eqn 2 assuming Poisson sampling is represented by the dashed lines.
The scaling relationship /a(y)»/(y/a)/a and eqn 2 are indistinguishable except in the limit of small sample size and abundance, where
the Poisson approximation breaks down. All distributions areplotted as probability density f(y) vs. y, where y ¼ log10n.

Figure 2 Scaling of relative-abundance distributions in the case ofclustered sampling. This example assumes a gamma distribution ofabundances at the region scale with parameters k ¼ 0.000075,
b ¼ 0.75, and a clustering parameter k ¼ 0.5. Sampling proportion a decreases from right to left as a ¼ 1 (regional scale), a ¼2)2, a ¼ 2)4, a ¼ 2)6, a ¼ 2)8, a ¼ 2)10.
 2007 Blackwell Publishing Ltd/CNRS

1042 J. L. Green and J. B. Plotkin

Idea and Perspective

Table 2 Regional and sampled abundance distributions under negative-binomial sampling

Name

Sampled abundance distribution

Continuous logseries/a ðyÞ ¼

Cð1kÞCðkþyÞ1 F1 ðkþy;1þk;k lnðxÞ=aÞðk lnðxÞÞk
lnðxÞ=aÞ
 1 F1 ðy;1k;k
yLðxÞa k Cð1þyÞLðxÞ

GammaðkkÞk pcscðkpÞCðkþyÞ1 F1 ðkþy;k;kk=aÞ/a ðyÞ ¼ kk1 F1 ð1þy;2k;kk=aÞþaðk1Þ
a k CðkÞ2 Cðyþ1Þ
h
i

kk b

pðkþy1ÞcscðpðbkÞÞ kk k/a ðyÞ ¼ k1CðbÞCðkþyÞCðk þ yÞ1 F1 k þ y; 1  b þ k; kk
Cðb þ yÞ1 F1 b þ y; 1 þ b  k; kk
a
a  a
a

Lognormal–

Truncated hyperbolic/a ðyÞ ¼ y lnðM1 =mÞ

Exponentialkk CðkþxÞ
Cðxþ1Þðk1Þ



m2 F1 ðk1;kþx;k;k=amÞÞ M2 F1 ðk1;kþx;k;k=aMðamÞkðaM Þk

Regional-scale distributions are defined in Table 1. F1(a,b,z) denotes the Kummer confluent hypergeometric function, and F2(a,b,z) denotes
the Gauss hypergeometric function. Dashes indicate thatRa closed-form solution is unavailable. The closed-form solution for the continuous
1
logseries is an approximation which is valid provided 0 wa ðyjnÞ/ðnÞdn << 1 (see Appendix A for details on analytical and numericalintegration).

both rare and abundant species, compared to random
sampling or compared to the regional abundance distribution. This phenomenon is more exaggerated as the degree ofconspecific aggregation increases. Although a surplus ofboth rare and common species may seem surprising, there isan intuitive explanation for this fundamental result: whenconspecific individuals are aggregated, there is a greater
chance of either ÔhittingÕ or ÔmissingÕ a cluster of individuals
in a sample, resulting in a sampled abundance distributionthat is skewed towards both rare and common species.
Asymmetric aggregated spatial distributions

Figure 4 illustrates how asymmetry in clustering properties(eqn 5) can affect the species-abundance distribution in asample. In the figure we compare the regional speciesabundance distribution to the abundance distribution under
Poisson sampling, negative-binomial sampling with fixed k,
and negative-binomial sampling with varying k(n). The
sampled abundance distribution is clearly influenced byasymmetries in speciesÕ clustering properties, but the resulting
sampled abundance distribution is very similar to the situation inR which all species have the mean clustering parameter ¼ 1 kðnÞ/ðnÞdn. In other words, the symmetric andk0
asymmetric models yield qualitatively similar results, for kequal to the average aggregation tendency of species.
DISCUSSION

The distribution of species abundances is a long-standing
topic of research in community ecology, informing basic
ecological theory and applied conservation management.
Most species-abundance data are based on limited samples,and it is important to understand how sampled abundance
patterns relate to the underlying species-abundance distribution in the regional community. In this study, we haveaddressed this need by developing a sampling theory of 2007 Blackwell Publishing Ltd/CNRS

species abundances for random and autocorrelated populations, based on widely used statistical sampling schemes.
Under Poisson sampling, the sampled and regional-scale
distributions have nearly the same shape, and the relationship between the two can be expressed as a simple scaling
function with an error term. In the presence of conspecificaggregation, however, the sampled abundance distribution ismarkedly different from the underlying, regional-scale
distribution: spatial clustering increases the frequency ofboth rare and common species in the sample distribution.
The scaling rule we have derived for Poisson sampling(eqn 6) is in close agreement with the empirical observations
made by Preston (1948, 1962) in the special case of thelognormal distribution. Preston found that when plotting
species-abundance distributions as the fraction of species
within the abundance interval [log(y),log(y)+dlog(y)], thesample distribution had nearly identical shape as theregional-scale lognormal species-abundance distribution,
but shifted to the left by a factor of log(y/a) – in accordancewith our Ôrule of thumbÕ (eqn 6). Our results also resolve
Dewdney’s (1998) conjecture that there exists a large familyof species-abundance distributions whose random sampling
properties are approximated by a simple scaling behaviour(Figs 1 and 3). Although the Poisson-sampled abundance
distribution can change shape at very small abundances (cf.
McGill 2003), this phenomenon only occurs in parameter
regimes and at abundances for which the use of acontinuous species-abundance distribution is not welljustified to begin with.
The generality of our rule of thumb is evidenced by recent
findings in the neutral theory of community assembly.
Alonso & McKane (2004) have shown that under the neutralmodel, when there is no dispersal limitation in the regional
species pool, the local community species-abundance distribution may be approximated by Poisson sampling from themetacommunity’s multinomial relative-abundance distribution, fM(y). For a large enough local community, the sampledIdea and Perspective

Statistical theory for sampling species abundances 1043

Figure 3 The

scaling of five different
regional abundance distributions (solid)
under Poisson sampling (solid), aggregatedsampling (dashed, negative-binomial parameter k ¼ 0.5) and highly aggregated sampling (dotted, negative-binomial parameterk ¼ 0.05). In all cases, 5% of the regional
community is sampled. The shape of theabundance distribution under random sampling is similar to that of the regional
abundance distribution. The abundance distribution under negative-binomial samplingshows an excess of both rare and common
species, especially for highly aggregateddistributions (dotted). Distribution parameters are as follows: continuous logseries
x ¼ 0.99999093735;
exponential
k¼
0.0001; gamma b ¼ 0.55, k ¼ 0.000055;
lognormal l ¼ 7.2103, r ¼ 2; truncatedhyperbolic m ¼ 100, M ¼ 5000.

Figure 4 The scaling of the regional abundance distribution (solid)

under Poisson sampling (solid), symmetric aggregated sampling(dashed, negative-binomial parameter k ¼ 0.05), and asymmetricaggregated sampling (dotted). The sampled area represents 5% of
the regional area, and the regional abundances are exponentiallydistributed with k ¼ 10)4. In the asymmetric case, the clusteringparameter depends upon abundance according to the formulak(n) ¼ n/200 000. The sampled abundance distribution in theasymmetric case is very similar to the symmetric sampled
distribution, forR k equal to the average clustering parameter across
 ¼ 1 kðnÞ/ðnÞdn ¼ 0:05 Alternative choices for k(n)species: k0and /(n) yield similar results.

distribution SJ(n), defined as the expected number of species
with abundance n represented in a sample of J individuals, isclosely approximated by the scaling relationship SJ(n) ¼
fM(n/J)/J (Ecol. Lett., 8; eqn 13). This scaling relationshipparallels our rule of thumb (eqn 6), but is expressed in thecontext of sampling from the distribution of relative (rather
than absolute) abundances. Thus, eqn 6 is applicable beyond
the family of species-abundance distributions examined inthis paper. Indeed, according to our derivation, we can
expect eqn 6 to provide an accurate approximation for the
randomly sampled abundance distribution, to first order in
1/a, provided the derivates of the regional-scale abundance
distribution are all bounded.
Our analysis of spatially autocorrelated populations also
parallels recent results for the neutral theory in the case of adispersal-limited local community (Hubbell 2001; Vallade &Houchmandzadeh 2003; Volkov et al. 2003; Chave 2004).
Neutral theory predicts that the shape of the local speciesabundance distribution sampled from the metacommunity
multinomial distribution will be a function of the immigration rate. Increased dispersal limitation leads to an increasingly left-skewed sample distribution. Dispersal limitation is
one obvious mechanistic force that induces aggregation,although habitat preferences will have similar effects. Our
results suggest that any mechanism leading to autocorrelated
 2007 Blackwell Publishing Ltd/CNRS

1044 J. L. Green and J. B. Plotkinsamples of conspecific individuals will result in a sampledabundance distribution that is skewed towards both rare andabundant species, compared to the metacommunity. We
have shown that this phenomenon is not limited to themultinomial distribution discussed in neutral theory, butrather it applies across a wide range of regional-scaleabundance distributions.
Based on empirical data and simulations, McGill (2003)
recognized the tendency of sample autocorrelation to fatten
the rare tail of the lognormal species-abundance distribution. Here, we have used explicit sampling formulae anddemonstrated that autocorrelation produces sampled distributions that are skewed towards both rare and common
species. In addition, we have shown that this phenomenon
holds over a wide range of abundance distributions. In thecase of the lognormal distribution, the sampled distributions
are skewed towards both rarity and commonness, but therare skew is more pronounced than the common skew
(Fig. 3, red curve). These results suggest that the apparent
rare-skewed lognormal abundance distribution observed in
censused tropical forest plots (Hubbell 1997, 2001) is
consistent with the common assumption of a lognormal
regional abundance distribution (Preston 1948, 1962; May
1975) combined with observed, local aggregation ofconspecific trees (He & Legendre 1996; Condit et al. 2000;
Plotkin & Muller-Landau 2002).
Our analysis of the sampled species-abundance distribution sheds light on a topic of practical interest in applied
ecology – namely, inferring the properties of a communityfrom one or more samples. One common approach, which
has recently gained popularity in microbial ecology (Curtis
et al. 2002; Finlay 2002; Bohannan & Hughes 2003; Venter
et al. 2004), uses parametric distributions to estimate thenumber of species in a community. Sample data are fit tomodels of relative abundance (or assumed on theoreticalgrounds), and the sample frequency distribution is projected
to estimate the number of unobserved species in thecommunity (Pielou 1975; Bunge & Fitzpatrick 1993). Thisapproach is grounded in the assumption that the sample
frequency distribution is a truncated version of thecommunity-level distribution when individuals that are
randomly sampled from the community (eqn 6). In many
ecological contexts, however, this assumption may beseriously violated. Microbial communities, for example, arecommonly investigated by identifying individuals from soil
or sediment cores across a landscape. Even if the cores are
randomly distributed in space, spatial aggregation in
microbial populations would result in a non-random sample
of individuals from the community. In this case, thesampled distribution would be characterized by a relativeabundance distribution that is markedly different in shapethan the community-level distribution, complicating the
problem of estimating community-level species richness.
 2007 Blackwell Publishing Ltd/CNRS

Idea and Perspective

Estimating species richness in a single environmental sample
from clone libraries using the random sampling assumption
may also be inaccurate, because PCR-based methods may
yield a biased, non-random subsample of individuals(Wintzingerode et al. 1997; Lueders & Friedrich 2003).
Thus, parametric-based distribution approaches that ignore
conspecific aggregation for estimating species richness arelikely to yield unreliable predictions.
Our results on sampling may inform future research
aimed at leveraging abundance and aggregation patterns
measured at local scales to predict biodiversity patterns atlarger, regional scales.
ACKNOWLEDGEMENTS

We thank Steve Hubbell for useful discussions. We thankJ. Chave and two anonymous referees for substantial input.
J. L. G. acknowledges support from the National Science
Foundation (MCB 0500124). J. B. P. acknowledges support
from the Burroughs Wellcome Fund.
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SUPPLEMENTARY MATERIAL

The following supplementary material is available for thisarticle:Appendix S1 Deriving the Sampled Abundance Distribu-

tions.
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Editor, Jerome Chave
Manuscript received 22 June 2007
First decision made 5 July 2007
Manuscript accepted 30 July 2007 2007 Blackwell Publishing Ltd/CNRS