IDEA AND
PERSPECTIVE
A statistical theory for sampling species abundances
Jessica L. Green
1
and
Joshua B. Plotkin
2
*
1
Center for Ecology and
Evolutionary Biology, University
of 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
Abstract
The pattern of species abundances is central to ecology. But direct measurements of
species 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 distributions
of 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 a
simple scaling relationship. In the case of aggregated spatial distributions, however, the
shape 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 results
from 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: 1–9
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, speciation
and extinction on the structure and dynamics of ecological
communities (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-abun-
dance distributions, it is difficult to directly measure all
species abundances at ecologically relevant scales. For
micro-organisms, this poses a challenge at the scale of a
single 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 impos-
sible. 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 proportion
of the community.
Efforts to develop a sampling theory of species abun-
dances have utilized several approaches. The most widely
utilized approach, which dates back to Fisher et al. (1943)
assumes that individuals are randomly sampled from an
ecological community. Fisher et al. (1943) sought to
understand patterns of species abundance in butterfly,
beetle and moth communities sampled throughout the
world. They found that species abundances in random
samples were well described by a logseries distribution, a
distribution they mathematically derived by Poisson sam-
pling from a gamma distribution. Their analyses laid the
foundation 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 distribution
in ecology by demonstrating that the lognormal had
Ecology Letters, (2007) 10: xxx–xxx doi: 10.1111/j.1461-0248.2007.01101.x
 2007 Blackwell Publishing Ltd/CNRS

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 a situation in space and time where the individuals, or
pairs, are distributed at random, not clumped on one hand
or over-regularized on the other. He noted that contagion,
or conspecific aggregation, would likely to result in a
somewhat skewed 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 spatially
aggregated populations.
Efforts to understand the sampled abundance distribu-
tion 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 with
sampling area (Banavar et al. 1999; Harte et al. 1999). Recent
analyses, however, suggest that such fractal models are
biologically 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 distribution
that matches empirical vegetation data relatively well (Harte
et al. 2005). Although the HEAP framework allows for a
range of aggregation patterns, from random to highly
clustered, this model has not been utilized to explore how
the degree of aggregation influences sampled species-
abundance distributions.
Neutral community theory (Hubbell 2001) provides a
mechanistic approach for modelling heterogeneity by
accounting for the effect of dispersal limitation. Alonso &
McKane (2004) applied Poisson sampling to the metacom-
munity multinomial relative-abundance distribution to
derive 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 for
sampling heterogeneity by modelling the probability for a
dispersal-limited species to be present in a sample with a
given abundance under the assumptions stipulated by
neutral community theory. These recent developments,
while powerful, have not explicitly addressed the sam-
pling properties of communities assembled by non-neutral
forces.
In this study, we present a general statistical framework
for understanding the effect of spatial heterogeneity (or,
equivalently, heterogeneity in the sampling scheme) on the
species-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 the
simple case in which individuals are randomly sampled from
the 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 sampled
abundance distributions as a function of sample size and the
degree of conspecific clustering. We also demonstrate two
important, generic properties of how spatial aggregation
affects the sampled species-abundance distribution.
STATISTICAL FRAMEWORK
Species-abundance distributions are measured in the labo-
ratory or field by counting the number of species in a
community represented by n individuals. For the purpose of
our analysis, we will characterize species abundances in a
large 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
normalization requires
R
1
0
/ðnÞdn ¼ 1. In reality species
abundances are discrete. We use continuous distributions to
provide 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 species-
abundance distribution at the regional scale, /(n), and the
abundance distribution observed in a sample that constitutes
a proportion a of the larger ambient region, denoted /
a
(y).
Let w
a
(y|n) denote the probability that a species will be
represented by y individuals in the sample, given that it has
abundance n in the larger region. Then, the sampled species-
abundance distribution may be expressed as:
/
a
ðyÞ¼
Z
1
0
w
a
ðyjnÞ/ðnÞdn ð1Þ
Equation 1 provides a general expression for the scaling of
the species-abundance distribution with sample size, given
an arbitrary sampling scheme w
a
(y|n). Equation 1 follows
directly from the law of total probability. By using different
sampling schemes for w
a
(y|n), we can model different
spatial distributions of conspecifics across the landscape.
2 J. L. Green and J. B. Plotkin Idea and Perspective
 2007 Blackwell Publishing Ltd/CNRS