Ecological Applications, 15(2), 2005, pp. 507–520
q 2005 by the Ecological Society of America
QUANTIFYING SPATIAL PATTERN WITH EVENNESS INDICES
LAURA X. PAYNE,1,4 DANIEL E. SCHINDLER,2 JULIA K. PARRISH,2
AND
STANLEY A. TEMPLE3
1Department
of Wildlife Ecology, University of Wisconsin, Madison, Wisconsin 53706 USA and School of Aquatic and
Fishery Sciences, University of Washington, Seattle, Washington 98195 USA
2School of Aquatic and Fishery Sciences and Department of Biology, University of Washington, Seattle,
Washington 98195 USA
3Department of Wildlife Ecology, University of Wisconsin, Madison, Wisconsin 53706 USA
Abstract. Quantifying the spatial distributions of organisms in simple and meaningful
ways is important for understanding the ecology and habitat needs of species subject to
anthropogenic disturbances. Most multi-species conservation and management programs
do not yet account for the variation of space-use patterns exhibited by species preferring
the same habitat type. To measure species’ space-use patterns as a step toward determining
habitat needs, we suggest using evenness indices. Although commonly used in characterizing communities (i.e., as a measure of species diversity), these indices are suitable for
characterizing many other ecological patterns, including patterns of site use by individuals.
We investigate the statistical properties of five indices (Camargo’s index of evenness,
E9; Simpson’s index of evenness, E1/D̂; Lloyd’s index of mean crowding, J; Smith-Wilson
index Evar; and Dispersion index, DL, a variant of the Shannon diversity index) to evaluate
their utility for quantifying broad-scale spatial patterns of migratory shorebirds. We use a
Monte Carlo simulation approach to compare these indices for their (1) ability to characterize a wide range of spatial patterns (from even to patchy); (2) ability to discriminate
among distributions; and (3) robustness to incomplete sampling. In addition, we compare
the ability of these indices to characterize spatial dispersion for four species of shorebirds
migrating through the United States: Killdeer (Charadrius vociferus), Semipalmated Plover
(C. semipalmatus), Sanderling (Calidris alba), and Red Knot (C. canutus).
Overall, we recommend Camargo, Simpson, and Lloyd indices for quantifying spatial
dispersion. All three indices gave precise and unbiased estimates once 35 or more sites
(3.5% of 1000 simulated sites) were sampled randomly, regardless of the degree of patchiness. Nonrandom sampling resulted in biased estimates until a much greater proportion of
sites (at least 30–50%) were sampled, highlighting the importance of site selection in
sampling programs. Our analysis of shorebird spatial patterns revealed that evenness indices
discriminate well among the species we considered, ranking Killdeer as most dispersed and
Red Knot as most aggregated. Our results agree with earlier (independent) assessments of
these species’ migration strategies and illustrate how simple univariate metrics may be
useful tools for characterizing complex spatial patterns.
Key words: Camargo’s index of evenness; conservation; dispersion; evenness index; Lloyd’s
index of mean crowding; migration; Shannon diversity index; shorebird; Simpson’s index of evenness;
Smith–Wilson index; spatial pattern.
INTRODUCTION
Understanding the spatial patterns of organisms on
the landscape is of central importance to ecology and
conservation (Turner 1989, Tilman 1994). Spatial pattern, or dispersion, refers to the spatial composition of
organisms on the landscape, i.e., the relative proportion
of organisms distributed among a number of habitat
patches or sites (Krebs 1999, Read and Lam 2002).
Abiotic and biotic factors, including habitat availability, abundance of resources, and species interactions,
are important determinants of dispersion (Coomes et
al. 1999, Thomas et al. 2001, Mouillot and Wilson
Manuscript received 24 January 2003; revised 25 May 2004;
accepted 11 June 2004; final version received 6 July 2004. Corresponding Editor: C. A. Wessman.
4 E-mail: [email protected]
2002). However, species’ responses to these factors are
quite variable, so that spatial patterns found in nature
are often heterogeneous. For instance, shorebirds
(sandpipers, plovers, and their allies) show diverse migratory strategies, which result in variable spatial patterns of habitat use at the landscape scale (Fig. 1). Some
species fly moderate distances each time they move,
stopping to feed opportunistically in small numbers at
wetland sites encountered at the end of each migratory
movement (e.g., Pectoral Sandpiper, Calidris melanotos [Farmer and Wiens 1999]). Others make marathon
flights, stopping at a few, traditionally used wetlands
where they concentrate in huge numbers (e.g., Red
Knot, C. canutus [Harrington 2001]).
Such variability in shorebirds and other species has
borne many insights into ecology but also presents difficulties for conservation planning, especially when
507
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LAURA X. PAYNE ET AL.
Ecological Applications
Vol. 15, No. 2
FIG. 1. Contrasting spatial patterns of two hypothetical shorebird species at stopover sites during migration. Circle size
reflects the relative number of individuals. At left, individuals are distributed quite evenly among numerous sites. At right,
distribution is patchy, with most individuals occupying a single site.
plans must integrate the complex needs of multiple
species. For instance, despite evidence that some shorebirds exhibit a wide range of spatial patterns in certain
regions during migration (Skagen and Knopf 1993,
1994, Iverson et al. 1996), conservation efforts do not
yet reflect this breadth. Current efforts, such as the
Western Hemisphere Shorebird Reserve Network
(WHSRN), focus primarily on wetlands where large
aggregations occur, an approach that favors aggregated
species (Myers et al. 1987, Harrington and Perry 1995).
The lack of detailed information on many species’ migratory habits, and the related lack of coordinated or
explicit efforts for non-aggregated species, leaves
many at risk (U.S. Shorebird Conservation Plan
[Brown et al. 2001]). Until the diversity of land-use
patterns is better documented for each shorebird species, decisions about which networks of sites to protect
for shorebirds will continue to be modeled after the
needs of the best-studied species, an approach that may
fail because it does not account for variation in space
use by different species. While our research focuses on
shorebirds, this spatial mismatch between site use and
sites identified for protection is a common problem of
integrated, multi-species approaches.
The tools available for examining spatial pattern
vary widely, ranging from spatially non-explicit measures (e.g., heterogeneity, evenness, dominance, and
aggregation indices; similarity coefficients), to spatially explicit metrics (e.g., quadrat variance methods,
correlograms, geostatistics [variograms and Kriging],
angular correlation, wavelets, SADIE [spatial analysis
by distance indices], etc.; Krebs 1999, Dale et al. 2002).
Spatially explicit metrics account for the geographic
locations of individuals (or patches), while spatially
non-explicit metrics quantify the relative distribution
of individuals among a set of patches. Current trends
in ecology emphasize the incorporation of spatial information into ecological analyses in increasingly sophisticated ways (Wiens 1989, Liebhold and Gurevitch
2002). While we welcome such a trend, not all pattern
analyses require complex approaches to answer the
questions at hand. Furthermore, because practitioners
of conservation and management often seek easily interpreted metrics describing the ecology of focal species, increasingly sophisticated analyses are not always
the most practical. Here, we propose a simple approach
for quantifying species’ spatial patterns on a coarse
scale, the key being to obtain some measure of how
the distribution of shorebirds at wetland stopover sites
varies among species, or across years.
Many indices have been developed to quantify heterogeneity (Krebs 1999), including diversity, evenness,
and patchiness indices. Diversity indices, traditionally
used to quantify species diversity in communities, combine evenness (i.e., distribution of individuals among
species) and richness (i.e., number of species present)
into a single metric. Patchiness indices quantify the
relative patchiness of an environment from the perspective of an organism occupying that environment
(Lloyd 1967). Despite its importance, the challenge of
measuring diversity so that index values are easily interpreted and different-sized samples may be compared
directly remains difficult, and current tools are not universally accepted. This difficulty has prompted some
to divide diversity into its two separate and measurable
components, richness and evenness (Lloyd and Ghelardi 1964). While richness is sensitive to sample size
and not usually recommended as a reliable index of
diversity (without rarefaction), many evenness indices
have robust properties and are well accepted (Krebs
1999). The most common approach to measuring evenness is to scale diversity indices by their maximum
value, where all species contain the same number of
individuals and are thus equally abundant (Krebs
1999).
Various studies investigate the mathematical and biological properties of evenness indices, and have led
to new insights in their utility (Heip and Engels 1974,
April 2005
SPATIAL PATTERN AND EVENNESS INDICES
Magurran 1988, Smith and Wilson 1996, Hubalek
2000). Smith and Wilson (1996) and Hubalek (2000)
examined the statistical properties of over 30 evenness
indices by testing them for essential and desirable features, including that indices must be independent of
richness (number of sites), be sensitive to the inclusion
or exclusion of sites of minor importance, and be unaffected by the units used. Indices should also reach
maximum values when distributions are even, minimum values when indices are extremely uneven, intermediate values when evenness is intermediate, and
respond intuitively to symmetrical or asymmetrical
changes in evenness; see Smith and Wilson (1996) for
examples. These features have been recognized as important for heterogeneity measures by numerous other
researchers (Heip and Engels 1974, Routledge 1979,
Ricklefs and Lau 1980, Magurran 1988, Smith and Wilson 1996, Drobner et al. 1998, Krebs 1999, Wilson et
al. 1999).
Despite many advances in understanding index behavior, there is little consensus on a single ideal index.
Rather, several indices seem to perform well under certain circumstances, although none is without limitation
(Smith and Wilson 1996, Krebs 1999, Hubalek 2000).
Furthermore, most analyses of the statistical behavior
of indices have focused on highly simplified hypothetical systems, so that applying these scenarios to
more complex, real-world situations is unclear. Consequently, we tested candidate indices prior to application, as suggested by Hubalek (2000) and Peet
(1975). As the spatial analog of evenness indices measures the distribution of individuals among a set of
sites, our goal in selecting evenness indices was to be
able to measure how the distribution of shorebirds at
wetland stopover sites varies among species, or across
years. Hence, we examine index performance over the
broad range of spatial distributions and sampling scenarios likely to be found in our system of interest:
spatial patterns of migratory shorebirds. Our approach
could be applied to any population where the abundance of organisms is known for multiple sites.
For use in spatial-pattern analysis, evenness indices
should be: (1) able to characterize a broad range of
spatial patterns likely to be encountered in the environment (ranging from even to very uneven/‘‘patchy’’);
(2) able to discriminate well among possible spatial
patterns; (3) robust to variable sampling intensity (i.e.,
incomplete sampling); and (4) robust to biases in sampling (random vs. nonrandom). The final two points
are of great importance when interpreting index values
but have been overlooked almost entirely (but see Ricklefs and Lau [1980], who evaluated the effect of random
subsampling on overlap indices). In our present paper
we compare the behavior of five well-known indices–
Camargo’s index of evenness E9 (Camargo 1993,
1995), Simpson’s index of evenness E1/D̂ (Simpson
1949, Krebs 1999), Lloyd’s index of mean crowding
(scaled) J (Lloyd 1967), Smith-Wilson index Evar
509
(Smith and Wilson 1996), and Dispersion index DL
(Payne 1997), a variant of the Shannon diversity index
(Shannon and Weaver 1949, Pielou 1975)—and test
indices for sensitivity to distinct spatial patterns, robustness to sampling intensity, and robustness to type
of sampling. Finally, we compare the ability of these
indices to characterize spatial patterns of actual shorebird abundance data, using four species of migratory
shorebird.
Multiple-species surveys, such as the International
Shorebird Survey (ISS), may be used to estimate the
relative abundance and distribution of shorebird species among wetlands across North America (Howe et
al. 1989). The ISS surveys, conducted at over 1000
inland and coastal wetlands across the United States
since the 1970s, represent the best data set available
for quantifying shorebird spatial patterns at this nearly
continental scale. To that end, we sought objective metrics to quantify, in the broadest way possible, spatial
patterns of migratory shorebirds. We therefore investigate the use of evenness indices for their potential as
accessible metrics of spatial pattern.
MATERIALS
AND
METHODS
We used a Monte Carlo approach (Ripley 1987) to
evaluate the precision and accuracy of the five indices.
We sought to determine the ability of evenness indices
to detect differences among spatial patterns likely to
be encountered in nature, and the robustness of these
indices to incomplete sampling, in an approach similar
to those of Smith and Wilson (1996) and Ricklefs and
Lau (1980). First, we generated data sets representing
different spatial patterns of organisms among sites,
ranging from even to extremely patchy. Hereafter, we
refer to each of these simulated spatial distributions as
an underlying distribution, and to the entire suite of
simulated distributions ranging from even to patchy as
the distribution gradient. To test for sensitivity across
the distribution gradient, we generated index values for
20 different distributions, and tested for discrimination
ability among those distributions, using a minimal detectable difference (MDD) approach (Zar 1999). Second, we tested for robustness to sampling intensity by
randomly subsampling from five representative distributions from the gradient: one even distribution, two
intermediate distributions, and two patchy distributions. In other words, we sampled a fraction of sites
from each underlying distribution and then increased
the sampling fraction until all sites were sampled.
Third, we tested for robustness to type of sampling by
including nonrandom sampling where sites with large
numbers of birds had a higher probability of being
sampled; we repeated the subsampling procedures for
variable sampling intensity. Finally, to evaluate the
utility of these indices in differentiating among spatial
distributions of species with a range of ecological and
migratory strategies, we computed index values using
actual data on four shorebird species: Killdeer (Char-
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LAURA X. PAYNE ET AL.
510
TABLE 1. Typical characteristics of four shorebird species during fall migration through the United States, as described by
independent sources.
Gregariousness
during migration†§
Wetland habitat
preference
Extent of occurrence
at wetlands
Sanderling
low: mainly small groups
moderate: variable group
sizes
moderate
Red Knot
high
wide range; grassy to estuarine†§
coastal, also inland wetland margins†
sandy ocean beaches, also inland
wetlands†§
mainly coastal†
widespread†‡§
intermediate†§ to widespread†‡
intermediate‡ to widespread†
relatively few sites†‡§
Species
Killdeer
Semipalmated Plover
† Hayman et al. (1986). Information is for global range.
‡ Skagen et al. (1999). Information is for migration through central U.S. states (between 908 and 1208 W. latitude).
§ Birds of North America species accounts (variable authors and years); information is for North America during migration.
Authors, by species, are: Killdeer (Jackson and Jackson 2000), Semipalmated Plover (Nol and Blanken 1999), Sanderling
(MacWhirter et al. 2002), and Red Knot (Harrington 2001).
adrius vociferus), Semipalmated Plover (C. semipalmatus), Sanderling (Calidris alba), and Red Knot (Calidris canutus). These species exhibit a range of habitat
preferences, social behaviors, and overall abundances
(Table 1) shared by many of the 40 species of shorebird
that migrate commonly through North America each
spring and fall.
Index descriptions
We compared four commonly used evenness indices,
all with slightly different mathematical properties. We
based our selection on results from the Smith and Wilson (1996) analysis, and we included an additional
patchiness index, Lloyd’s index of mean crowding ( J).
While other indices exist, we limited our analysis to
the following five, as they are commonly used, well
understood, and have been proven in a variety of ecological or statistical studies (Krebs 1999).
Camargo’s index of evenness, E9 (Camargo 1993,
1995):
[O O 1z p 2s p z 2]
s
E9 5 1 2
s
i
j
(1)
i51 j5i11
where pi 5 proportion of individuals of a species at
site i; pj 5 proportion at site j; and s 5 total number
of sites in sample. Index values range from 0 (patchy)
to 1 (even). This index is relatively unaffected by sites
with very few organisms, and is unaffected by site
richness (Krebs 1999).
Simpson’s measure of evenness, E1/D̂ (variant of
Simpson 1949):
E1/Dˆ 5
ˆ
1/D
s
(2)
where D̂ 5 Sp2i and pi 5 proportion of total individuals
of a species at site i, and s 5 number of sites in the
sample. This evenness index is a variant of the reciprocal Simpson index, D̂ (Simpson 1949). Index values
range from near 0 (1/s) (patchy or skewed) to 1 (even),
and the index is relatively unaffected by sites with very
few individuals.
Lloyd’s index of mean crowding, J (Lloyd 1967, Ives
1991):
[O n (n(Ns2) 1)] 2 N
s
i
J5
i
i51
N
5
V
1
2
N2
N
(3)
where V 5 the sample variance, ni 5 number of individuals of a species occupying site i (i 5 1 to s), N
5 mean number of individuals occupying sites, and s
5 number of sites occupied by that species. For comparative purposes with the other four indices, we
bounded the index between 0 (patchily distributed individuals) and 1 (randomly distributed) by transforming J with:
1/(1 1 J)
(4)
In this scaled form, Lloyd’s index calculates the amount
of increase in the expected number of individuals occupying the same site, above what the expected number
would be if individuals were randomly and independently distributed. This index is used to assess the average degree of crowding experienced per individual
(Lloyd 1967).
Smith and Wilson index, Evar (Smith and Wilson
1996):


Evar 5 1 2
O [ ln(n ) 2 O ln(sn )] 
s
 i51
2
arctan 
p

s
j
i
2

j51


s
(5)
where ni 5 number of individuals of a species at site
i in sample (i 5 1, 2, 3, 4, . . . s), nj 5 number of
individuals at site j in sample ( j 5 1, 2, 3, 4, . . . s), s
5 total number of sites in sample. This evenness index
ranges from 0 (patchy or skewed) to 1 (even). The index
is mathematically symmetrical (equally sensitive to
sites with many organisms and those with few organisms), and estimates evenness at occupied sites only.
Dispersion index, DL (Payne 1997), a variant of J9
of Pielou (1975), and based on H9 of Shannon (Shannon
and Weaver 1949):
O p ln( p )
s
DL 5 H9/ ln Ss
where
H9 5 2
i51
i
i
(6)
and pi proportion of individuals of a species at site i,
SPATIAL PATTERN AND EVENNESS INDICES
April 2005
s 5 number of sites occupied by that species, Ss total
number of sites surveyed. Values range from 0 (patchy)
to 1 (even). This index is asymmetrical (somewhat sensitive to sites with fewest hypothetical organisms), and
mathematically incorporates sites surveyed but with
zero individuals, unlike the original J9 (Pielou 1975).
As with most metrics in ecology, evenness indices
are commonly used to measure only a few types of
patterns, although they are theoretically suited for measuring many others. Because we wished to measure
evenness across all sites (not just sites occupied with
$1 individual), we made a minor adjustment by adding
a small number (0.01) to each site before computing
Camargo and Simpson indices (Lloyd and Dispersion
gave identical results). Note that this constant was arbitrarily selected from a range of others (0.1 to 0.001),
all of which we tested and gave similar results. This
simple procedure resulted in appropriate results for all
indices except Smith-Wilson, which for mathematical
reasons (ln terms; see Eq. 5) responds differently;
hence, we simply computed Smith-Wilson directly
(without adding a site constant), and include it here to
characterize evenness among occupied sites only.
Simulating spatial distributions
In order to test index performance, we used a gamma
distribution to simulate the spatial distribution of organisms across multiple sites. The gamma distribution
is described by the probability density function
f (z) 5
ms 2m z s21
e z
G (s)
(7)
where parameter sigma (s) controls the shape of the
density, and the product of s and m determines the
average density at a site (Hilborn and Mangel 1997).
This attribute of the gamma distribution enabled us to
control for the number of organisms on the model landscape. The gamma distribution provides an excellent
approximation of often-skewed spatial patterns commonly found in ecological data, such as spatial heterogeneity of plant biomass (Shiyomi et al. 2000), populations of soil mites (Benton et al. 2002), foraging
rates of wild goats (Kohlmann et al. 1999), or developmental phases of aquatic invertebrates (Souissi and
Ban 2001), among many other examples. Its simplicity
makes it desirable for simulation modeling (Hilborn
and Mangel 1997), and the gamma distribution provided suitable fits to spatial pattern data for 34 of the
35 shorebird species we tested (P . 0.05, KolmogorovSmirnov goodness-of-fit test; Zar 1999).
We simulated a wide range of distributions to approximate spatial heterogeneity, ranging from even,
where individuals were spread evenly among many
sites, to very aggregated, where most individuals occur
at just a few sites. To generate each distribution, m was
set as the complement of s, where the product of m
and s always equaled 103. The number of individuals
within a sampling universe of 103 sites was set at 106
511
FIG. 2. Changes in two attributes of the gamma distribution along a gradient from evenness to patchiness. The
falling curve (solid line) represents the average proportion of
sites occupied (i.e., sites with $1 individual), and the rising
curve (dotted line) represents the proportion of total individuals at the site with the highest number of individuals. Each
point along the curve is the mean (61 SD) of 100 replicates
at each value of s, the parameter controlling patchiness. Note
that the proportion of sites occupied remains high and constant for the range s 5 101 to 103 (even to intermediate) and
then drops rapidly as s approaches 106 (extremely patchy).
In contrast, the proportion of individuals at the top site remains very low until s 5 103 and then increases steadily,
with variability also growing larger, as s increases. Roman
numerals indicate five representative distributions used in
subsequent analyses: even (I), intermediate (II and III), and
patchy (IV and V).
6 2%. In sum, for each simulation, the model randomly
selected 103 numbers from a gamma distribution with
parameters m and s, and these numbers, when summed
across 103 hypothetical sites, totaled approximately 10 6
organisms.
Index sensitivity to different distributions
To compare indices for sensitivity to different distributions, we ran 100 simulations for 20 separate distributions (i.e., levels of s) ranging from even (s 5
101) to extremely patchy (s 5 106). We compared index
sensitivity in two ways, graphically and statistically.
First, we overlaid all index values onto a single plot,
comparing average index values at each of the 20 distributions considered (Fig. 2). Second, we compared
five representative distributions (see next section, and
Table 2) using a formal statistical test, the minimum
detectible difference (MDD) approach (Zar 1999):
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LAURA X. PAYNE ET AL.
512
d$
!
2sp2
[t 1 tb(1),v ]
n a,v
(8)
where d is the size difference between two population
means (for a specified detection probability), s2p is the
pooled variance for every pair of distributions, n is the
number of sites (n 5 1000), t is the critical value of
the t distribution, a is the level of significance (a 5
0.05), b is the chance of detecting a true difference
between population means (b 5 0.90), and v is the
degrees of freedom for each comparison (v 5 198).
The graphical display enabled us to compare index behavior across the entire distribution gradient; d values
from the MDD test yield discrimination ability, where
low values indicate superior discrimination.
Sampling simulated distributions
To test indices for robustness to nonrandom sampling, we first chose five levels of s to represent distinct
distributions ranging from even to extremely patchy:
s 5 101 (I, approximately even); s 5 103 and 104 (II
and III, intermediate); and s 5 105 and 106 (IV and V,
patchy and extremely patchy). These distributions cover a very wide range, with distinct site occupancy levels
and relative distributions of individual abundances
among sites (Fig. 2).
For each distribution, we subsampled at different
sample sizes (i.e., number of sites), increasing sample
size exponentially (step 5 n1.6) from a minimum of 10
sites to 100% of the sites in the sampling universe. For
example, in a universe of 1000 sites, the number of
sites sampled increased as follows: 10, 15, 23, 36, 57,
90, 143, 230, 374, 613, and 1000. We sampled with
greater frequency at lower coverage because indices
were most sensitive to changes in sampling intensity
when fewer sites were sampled. We subsampled at each
level of coverage 1000 times.
We repeated the procedure for two different types of
sampling (random and nonrandom), subject to the constraint that each subsample must include at least 10
individuals. For the Smith-Wilson index (which quantifies evenness among occupied sites only), we imposed
the additional constraint that each subsample include
at least three sites with $1 individual. We included
these constraints to avoid biologically meaningless index values computed from samples with too little data.
For the nonrandom sampling protocol, we used a
simple logistic model to simulate sampling routines
where the probability of an individual site being sampled was proportional to the number of organisms at
that site (Fig. 3). We simulated this scenario to reflect
the possibility that sites with more birds may be sampled more frequently than sites without birds. We set
a constant minimum sampling probability level (i.e.,
minimum number of individuals below which the probability that a site will be sampled is unchanged), and
a maximum sampling probability threshold (i.e., number of individuals above which additional individuals
FIG. 3. Plot illustrating random (dashed line) vs. nonrandom (solid line) sampling probability functions. In random
sampling, all sites with $1 individual have an equal probability of being sampled; the probability value of 0.5 was
chosen only for graphical purposes here. In nonrandom sampling, sites with more individuals have a correspondingly
higher probability of being sampled. The values for the lower
and upper probability limits are arbitrary but reflect the realistic assumption that sampling (especially at the continental
scale) is always incomplete.
will not increase the probability of a site being sampled). Within those bounds, sampling probability increases logistically as the number of birds per site increases. We then compared results for the random and
nonrandom scenarios to determine the robustness of
each index to nonrandom sampling.
For all analyses, we used a simulated universe of
1000 sites, and we set 10 sites as the arbitrary minimum
number of sites sampled. This allowed us to gauge
index performance at sample sizes of 10 and greater,
approximating small samples common to field studies.
The output of each simulation procedure included all
five index values (mean and 1 SD of 1000 replicates),
percentage of total sites sampled, and the underlying
distribution of organisms on the simulated landscape.
Shorebird data set
The International Shorebird Survey provided shorebird data (see Howe et al. [1989], for ISS survey methodology and details of geographic coverage) consisting
of visual counts of individual shorebird species from
.1000 inland and coastal wetland sites in the conterminous United States during fall migration. We selected four species that stop to feed at wetlands in the
interior and coastal United States during fall migration;
these species peak in numbers during August. Despite
timing similarities, these species vary in many other
characteristics, including (but not limited to) gregariousness (typically forming small groups vs. extensive
flocks), habitat preference (coastal vs. inland/fresh),
and overall occurrence on the landscape (widespread
vs. localized; Table 1).
SPATIAL PATTERN AND EVENNESS INDICES
April 2005
FIG. 4. Comparison of index values (mean and standard
deviations) of five different heterogeneity indices across a
range of distributions, from even (I) to extremely patchy (V).
Note that lines for Simpson E1/D and Lloyd (J ) indices overlap
entirely.
To standardize ISS counts in space, we averaged
counts for each species separately across wetlands surveyed within the same 10-minute latitude/longitude
‘‘block’’ (area ; 100 km2). We quantified the spatial
pattern of four shorebird species with each index, subject to the same constraints as those used for the simulations (see Sampling simulated distributions, above).
For this study, we present data for a single year for
which sampling coverage was excellent (1977, n 5 105
blocks reporting). Because early ISS efforts surveyed
wetlands east of the Rocky Mountains preferentially
(Howe et al. 1989), these results reflect species’ spatial
patterns for the midcontinental and eastern United
States. We used bootstrapping (n 5 1000 replicates) to
generate estimates for mean index values and 95% confidence intervals. Bootstrapping (Smith and Van Belle
1984) enabled us to estimate confidence intervals associated with a single estimate, an approach that is
widely recommended for ecological analysis (Krebs
1999).
RESULTS
Index sensitivity to different distributions
Graphical comparisons.—All five indices were sensitive to differences among most of the simulated distributions (Fig. 4). All indices approached a maximum
value (close to 1) when distributions were nearly even,
513
and minimum value (close to 0) as distributions became
increasingly patchy. (Note that, although the average
value of the Dispersion index at s 5 1 000 000 was
;0.15, some runs of the model (i.e., 10%) produced
values ,0.05 at this s level, a result we consider adequately sensitive to extremely patchy distributions).
Additionally, all indices yielded intermediate values for
intermediate distributions, although index values differed owing to the different mathematical properties of
each index.
Discrimination analysis.—All indices discriminated
among all but one pair of simulated distributions at the
power (90%) and sample sizes (100 replicates of 1000
sites each) that we specified (Table 2a). However, Camargo, Simpson, and Lloyd indices discriminated best,
with discrimination ability highest across all distributions (especially II to V). The Smith-Wilson index gave
consistently low d values (Table 2a), but when these
were compared to actual differences (Table 2b), that
index’s ability to discriminate declined as distributions
became patchier, indicating weak discrimination ability
for patchy distributions. In contrast, the discrimination
ability of the Dispersion index increased with patchiness, but that index failed to detect actual differences
at the even end of the gradient (d 5 0.1420, difference
5 0.0601). For every pair of distributions compared,
the Dispersion index had weaker discrimination ability
than the other indices.
Sensitivity to incomplete sampling: random sampling
Across all sampling intensities, estimates were unbiased for the most even distributions, where ‘‘bias’’
is defined as the difference between an index value at
TABLE 2. Minimum detectable difference values (d) and observed differences among population means, generated for
five distributions (I, II, III, IV, and V).
Distributions
Dispersion Simpson Lloyd
SmithWilson Camargo
a) Minimum detectable differences, d†
I to II
0.1420 0.1148 0.1150
II to III
0.1232 0.0526 0.0528
III to IV
0.0872 0.0098 0.0100
IV to V
0.0462 0.0013 0.0013
0.1092
0.0376
0.0101
0.0072
0.1106
0.0532
0.0123
0.0015
b) Differences
I to II
II to III
III to IV
IV to V
0.6400
0.2725
0.0270
0.0129
0.4442
0.3822
0.1033
0.0118
in index
0.0601
0.2081
0.3140
0.2766
means‡
0.4907
0.4061
0.0819
0.0091
0.4903
0.4059
0.0835
0.0090
† d values indicate the minimum detectible size difference
between two population means when the probability of detecting a difference is 90%. Each population is composed of
100 replicates of each index, generated using the same parameter values of the gamma distribution. Smaller d values
indicate superior discrimination. Specific comparisons are
named in the first column; see Fig. 2 for characteristics of
each distribution.
‡ Differences in index means among populations generated
for each distribution. Specific comparisons are named in the
first column; mean values for each index and distribution (I
to V) are plotted in Fig. 4.
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LAURA X. PAYNE ET AL.
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Vol. 15, No. 2
Dispersion and Smith-Wilson indices gave less precise
estimates for both the patchy (1 SD . 0.1 when coverage ,5%) and extremely patchy (1 SD . 0.1 for all
coverages ,60%) distributions.
Sensitivity to incomplete sampling:
nonrandom sampling
FIG. 5. Bias, or change in mean index value (left column),
and precision, or 1 SD (right), as a function of sampling intensity. Bias is the difference between an index value at 100%
coverage and its value at a lower sampling intensity. Each
row represents one spatial distribution, from even (I) to extremely patchy (V).
100% sampling coverage and its value at a lower sampling intensity. However, as the patchiness of the underlying distribution increased, the effect of reduced
sampling intensity on bias also increased (Fig. 5). Bias
was smallest for Camargo, Simpson, and Lloyd indices,
and greatest for Dispersion and Smith-Wilson. However, even for the distributions at which bias was greatest (i.e., intermediate and patchy), bias was negligible
(i.e., #0.1) for Camargo, Simpson, and Lloyd once 2–
3% of sites were sampled; hence, these three indices
performed best under this criterion (Fig. 5).
Results for precision were similar, with all indices
yielding precise estimates for even distributions, but
with precision decreasing slightly as distributions became patchier (Fig. 5). However, even for the distributions at which precision was poorest (distributions
II through IV), Camargo, Simpson, and Lloyd indices
gave acceptably precise estimates (i.e., 1 SD # 0.1)
once 2–3% of sites were sampled (Fig. 5). In contrast,
We detected a potentially strong effect of nonrandom
sampling on index performance at intermediate to
patchy distributions (Fig. 6). In general, nonrandom
sampling yielded biased estimates for all indices until
20–30% of sites were sampled. The magnitude of bias
for Camargo, Simpson, Lloyd, and Smith-Wilson indices was greater for the nonrandom than for the random case, until 20–30% of sites were sampled. For the
Dispersion index, the direction of bias was opposite to
the random case and magnitude was similar, but index
values did not converge until 100% sampling, indicating that all subsampling resulted in biased estimates
(for distributions II to IV). In sum, sampling schemes
that overrepresent large sites appear to have the greatest
effect on indices for intermediate and patchy distributions, with Camargo, Simpson, Lloyd, and SmithWilson indices performing similarly. Note that the lower biases generated for extremely patchy distributions
are likely due to the sampling constraints imposed by
our pseudo-random sampling protocol (i.e., that subsamples must have $3 sites occupied and $10 birds
to be included for computation). Although it is simple
to run unconstrained subroutines at extremely patchy
distributions (for which we would expect the highest
bias), such an exercise would produce unrealistic results, because it is biologically meaningless to compute
index values on entirely empty samples or on samples
with very few birds. Hence, we imposed the modest
constraint that we thought best reflected the way these
indices might be used, and the bias responded as expected: it became lower, because the scant sites that
actually have birds were selected disproportionately
often in the same sampling routine.
Overall index performance
All indices responded to differences in underlying
spatial distribution and sampling intensity, but indices
differed in: (a) sensitivity to underlying distribution;
(b) bias; (c) precision; and (d) robustness to several
sources of error (b and c), as a function of sample size
and underlying distribution (Table 3). In general, patchiness tends to exacerbate any weaknesses (i.e., lack of
robustness) in index behavior, as does low sampling
intensity. These weaknesses are further exacerbated by
nonrandom sampling, with bias considerable until 20–
30% of sites are sampled. As sampling intensity increases and distributions become more uniform, bias
decreases. Camargo, Simpson, and Lloyd indices performed the best and similarly under most circumstances.
April 2005
SPATIAL PATTERN AND EVENNESS INDICES
515
FIG. 6. Effect of nonrandom (dotted lines) and random (dashed lines) sampling on index behavior. Each row represents
one spatial distribution, from even (I) to extremely patchy (V). Nonrandom sampling increases the magnitude of bias, so
that the minimum number of sites that must be sampled to reduce bias to a negligible level exceeds 100 sites (10%).
Shorebird data
DISCUSSION
All five indices ranked the spatial patterns of the
species we tested in the same sequence, with Killdeer
appearing most dispersed, followed by Semipalmated
Plover, Sanderling, and Red Knot (Fig. 7). Camargo
and Dispersion indices yielded significant differences
among all species, with relatively narrow confidence
intervals. Simpson and Lloyd indices yielded significant differences among all species except Killdeer and
Semipalmated Plover, for which confidence intervals
overlapped; confidence intervals were generally wider
for all species. The Smith-Wilson index, which characterized dispersion at occupied sites only (and had the
narrowest confidence intervals, as expected), showed
significant differences among all species.
Simulations
Results of our model simulations corroborate earlier
findings (e.g., Smith and Wilson 1996) that the indices
we selected respond unequally to different underlying
distributions. Our results also reveal new findings: that
some evenness indices are more robust to incomplete
sampling and type of sampling. Overall, no index outperformed all others; however, Camargo, Simpson, and
Lloyd performed better for most criteria tested (Tables
2 and 3). Our results differ from earlier findings (Smith
and Wilson 1996) in that the Smith-Wilson index did
not perform as well under the conditions we tested
(except for discrimination ability, Table 2a). However,
LAURA X. PAYNE ET AL.
516
Ecological Applications
Vol. 15, No. 2
TABLE 3. Comparison chart of index behavior across the
criteria we tested.
Sampling intensity (percentage of
sites sampled)
Index
1%
3% 5–6% 8–11% 30–40% 50–60%
Camargo, E9
Precision
Biasr
Biasnr
o
o
2
1
o
2
1
1
2
1
1
2
1
1
o
1
1
1
Lloyd, J
Precision
Biasr
Biasnr
o
o
2
o
o
2
o
1
2
1
1
2
1
1
o
1
1
1
Simpson, E1/D̂
Precision
Biasr
Biasnr
o
o
2
o
o
2
1
1
2
1
1
2
1
1
2
1
1
o
Dispersion, DL
Precision
Biasr
Biasnr
2
2
2
o
2
2
1
2
2
1
o
2
1
1
o
1
1
1
Smith-Wilson, Evar
Precision
2
2
Biasr
Biasnr
2
2
2
2
2
2
2
2
2
2
2
o
o
o
1
1
Notes: We tested for precision, bias with random sampling
(Biasr), and bias with nonrandom sampling (Biasnr). Symbols
reflect poor (2), adequate (o), and good (1) performance at
each level of sampling intensity, based on the following cutoff
values (given here as absolute values): precision, 2 [1 SD .
0.15], o [1 SD , 0.1], 1 [1 SD , 0.05]; bias, 2 [bias . 0.15],
o [bias # 0.1], 1 [bias , 0.05].
note that our consideration of this index was for calculating evenness at occupied sites only, in contrast to
the application of the other four indices. Below, we
compare indices across all criteria.
Under complete sampling, the five indices successfully characterized a broad range of distributions, but
indices responded differently to the underlying degree
of patchiness (Fig. 4). Our analysis of discrimination
ability verified this discrepancy (Table 2), with Camargo, Simpson, and Lloyd indices performing best.
These results suggest that, prior to application, one
should first consider the inherent sensitivity of each
index to underlying distributions as well as its discrimination power—two points often overlooked by researchers, although their importance has been noted
previously (Magurran 1988, Krebs 1999).
Under incomplete random sampling, the underlying
distribution influenced robustness of some indices
(Dispersion, Smith-Wilson). However, three of the indices we tested (Camargo, Simpson, and Lloyd) consistently characterized the underlying distribution of
organisms with samples as small as 35 randomly selected sites (3.5% of 1000 sites), regardless of underlying distribution. This result, that heterogeneous spatial distributions did not affect bias in three of the indices we tested, is encouraging, and highlights the potential utility of these indices. Similar results were
FIG. 7. Dot plots for five indices using actual shorebird
data. Dots represent mean dispersion (and 95% confidence
intervals) for four shorebird species during fall migration
(August 1977) in the United States: KILL, Killdeer; REKN,
Red Knot; SAND, Sanderling; SEPL, Semipalmated Plover.
Mean and 95% confidence interval estimates were generated
by bootstrapping actual data (105 wetlands surveyed; n 5
1000 bootstrapped replicates).
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SPATIAL PATTERN AND EVENNESS INDICES
attained in a parallel analysis of four overlap indices
(Ricklefs and Lau 1980).
If sampling is nonrandom, such that there is a tendency to sample sites with the most individuals, bias
increases considerably at low to intermediate sampling
intensity. The Camargo index was most robust to nonrandom sampling, providing unbiased estimates (bias
# 0.1) once 40% of the sites were sampled; Simpson
gave unbiased estimates once 50% of sites were sampled, and Lloyd gave unbiased estimates once 60% of
sites were sampled. These results demonstrate that to
produce unbiased estimates of spatial patterns using
evenness indices, considerable care should be taken to
ensure that sampling effort is distributed randomly
across potential habitat to avoid preferentially selecting
sites with large numbers of organisms.
Overall, we recommend three indices—Camargo,
Simpson and, Lloyd—for quantifying spatial distributions of organisms among sites. All three gave precise and unbiased estimates when at least 35 (3.5% of
1000) sites were sampled from a distribution, regardless of the underlying patchiness of the distribution. If
sampling is biased towards sites with greater numbers
of organisms (i.e., nonrandom sampling), none of the
indices performed as well. However, Camargo generated values with the smallest bias, and estimates became unbiased at relatively smaller sample sizes. Camargo, Simpson, and Lloyd indices (particularly the
latter two), performed remarkably similarly despite
their fundamentally different mathematical composition.
In addition to the specific comparisons made above,
our simulation results provide several general contributions worth noting. First, discussion of the utility of
diversity and evenness indices has focused on whether
indices can be used to compare community structure
(as has been widely used), or if interpretation should
be restricted to describing patterns in ‘‘fully censused’’
pieces of biotic space at a particular scale (Smith and
Wilson 1996, Pielou 1975). Our study addresses a different, equally important component of this question.
Namely, how do indices respond to variable sampling
effort, when not all sites are sampled but the underlying
true distribution is known? Second, most previous analyses have used highly stylized distributions to compare
evenness among a handful of categories (sites; Smith
and Wilson 1996, Hubalek 2000). While this approach
has done much for our ability to grasp the mechanics
of these indices, it remains difficult to interpret these
results in a larger context. Here, we present index comparisons using 1000 sites, in which the underlying distribution ranges from individual(s) occupying few to
all sites. Third, we test index behavior over a much
broader set of conditions than has previously been examined, using a series of ecologically plausible distributions ranging from even to extremely patchy. Most
previous analyses (e.g., Smith and Wilson 1996, Hubalek 2000) have measured evenness at occupied sites
517
only (the standard way of quantifying evenness for species diversity), and then, for more even distributions
than those we consider. However, we have shown that
some indices may be used more flexibly to quantify
dispersion for several species among a set of sampled
sites. Additionally, we formally tested the discrimination ability of these indices, so that relative differences can be interpreted in a more objective context
than previously examined (i.e., as compared to Smith
and Wilson’s (1996) or Hubalek’s (2000) treatment of
‘‘desirable features’’). And finally, we use field data to
give an example of the utility of these indices.
Shorebird data
The relative differences in dispersion among the species we considered corroborate independent assessments of their biology (Table 1). All indices ranked
Killdeer as the most dispersed species, followed by
Semipalmated Plover, Sanderling, and Red Knot
(patchiest), similar to independent findings by other
researchers (Skagen et al. 1999, Harrington 2001). Furthermore, although both Sanderling and Red Knot share
a preference for coastal habitat in the geographic area
covered by the International Shorebird Survey (ISS;
Howe et al. 1989) the fact that these species have such
different spatial patterns, despite similar habitat associations, suggests important differences in their migration ecology—differences that may require a unique
conservation approach at the continental scale.
In this analysis, we sought to characterize and compare wetland use by four species in the broadest way
possible—across hundreds of wetlands in the continental United States. We asked this question to answer
one component of what conservation planners need to
know—i.e., To what extent are shorebird species different in their overall use of U.S. wetlands during migration? The relatively low index values suggest that
these shorebirds are relatively patchy in their use of
wetlands at the continental scale in August during fall
migration. Note, however, that it would be equally
meaningful to characterize species’ spatial patterns
among a subset of U.S. wetlands, for instance among
habitats of a specific biological or management type.
Under this different scenario more restricted conditions
apply, and some species—particularly habitat specialists—will be considerably more dispersed than the values we generated. To test this possibility, we consider
briefly a regional study from northern California (all
sites within 90 km of Humboldt Bay), in which researchers surveyed shorebirds at 40 randomly selected
sandy ocean beaches between January and April 1996
(Colwell and Sundeen 2000). Using their published results (the average number of individuals per beach),
we computed the five indices for Sanderling, a species
that prefers sandy ocean beaches (MacWhirter et al.
2002). In contrast to the relatively low index values
(mean 6 95% confidence intervals) that we computed
for peak fall migration at the continental scale ( DL:
518
LAURA X. PAYNE ET AL.
0.57 6 0.01; E1/D̂: 0.10 6 0.004; J: 0.09 6 0.004; Evar:
0.11 6 0.004; E9: 0.11 6 0.003, Fig. 7), Sanderlings
were significantly more dispersed within a single habitat (sandy ocean beaches) during January–April in
northern California (DL: 0.79 6 0.01; E1/D̂: 0.38 6 0.02;
J: 0.39 6 0.02; Evar: 0.32 6 0.03; E9: 0.36 6 0.02).
Although it is not surprising that a species spatial patterns may vary with the spatial and temporal extent of
each question (near-continent vs. regional, multiplehabitat types vs. a single predominant habitat type, and
August 1977 vs. January–April 1996), understanding
such variability provides important insights for conservation planning and management. More broadly, we
include results for both studies to illustrate that the
indices tested in this paper may be used to quantify a
range of patterns within the same system (or species),
be they extremely patchy patterns such as our focal
question of interest, or intermediate patterns such as
those common to other ecological processes/scales.
From a purely practical standpoint, the drawback of
these indices is that they do not tell us everything we
want to know. Whereas we can compare relative aggregation among a group of species, or dispersion
trends in time, these indices do not tell us precisely
how much habitat or which specific habitats are important. However, some of these factors can be determined from examination of the input data, e.g., which
habitats are important, and others may be estimated
through further analyses, e.g., how much habitat is required. To aid in interpretation of evenness indices, we
therefore suggest computing several simple metrics as
complementary summaries of spatial pattern—such as
richness (the proportion of sites occupied), dominance
(the proportion of individuals at the top site), and the
identities of the top sites. We see these as complementary measures, not as replacement indices, since
the simultaneous interpretation of multiple metrics is
cumbersome—requiring at least two comparisons for
every unit of information—and simpler metrics also
have problems/biases of their own (e.g., site richness;
see Magurran 1988). Hence, our analyses demonstrate
that univariate indices, such as Camargo, Simpson, and
Lloyd, can be powerful tools for characterizing broad
spatial patterns of multiple species.
Application of evenness indices to management
and conservation
Like most metrics these indices are most useful in a
comparative sense, when values are interpreted relatively. With this in mind, we believe that the primary
uses of these indices will be in quantifying gross differences in space use by co-occurring species, and in
detecting temporal changes in space use within species,
as they are subject to changing environmental conditions—such as climate conditions and availability of
suitable habitat. In the former case, managers could
benefit from knowing whether species that co-occur in
the same habitat type, locally, use similar amounts of
Ecological Applications
Vol. 15, No. 2
habitat at the continental scale, and therefore how to
prioritize local habitat availability to accommodate
those species. In the latter case, detection of temporal
changes in dispersion could inform managers in two
ways. First, if some species have inconsistent habitatuse patterns (i.e., being highly aggregated in some
years and dispersed in others), then conservation plans
will need to be expanded to accommodate both strategies in any given year, providing many adequate sites
as well as a few very high-quality wetlands. Second,
trends in dispersion patterns may serve as indicators
of potential population-level changes in habitat use or
availability through time—such as a species that was
formerly dispersed but is now becoming increasingly
aggregated. This application could be useful to conservationists because population declines are extremely
difficult to detect in these remote-nesting and migratory
species. Finally, although it is not possible to determine
unambiguously the particular processes behind the spatial patterns we observe, pattern analysis may help to
reveal which processes are important (Coomes et al.
1999). For instance, relative differences in spatial pattern among species or time periods may be a result of
changes in predator pressure on certain species (e.g.,
Lank et al. 2003) or habitat availability differences in
wet/dry years (Skagen and Knopf 1994), two factors
that could merit further investigation.
In summary, our analyses compared five indices to
identify the best ones for characterizing spatial dispersion patterns of shorebirds. While evenness indices
can only provide one dimension of information for conservation planning, the ability of these indices to simplify highly complex information into simple metrics—
encompassing multiple observations across space and
time—should not be underestimated. While more sophisticated information may be gained through the use
of spatial statistics or spatially explicit indices (e.g.,
Coomes et al. 1999), such approaches necessitate greater expertise (or, in the absence of expertise, time and
resources) than is often available to many managers or
conservation practitioners. By examining spatial patterns of multiple species, managers can begin to identify space needs of all species and to develop explicit
efforts for non-aggregated species, which are currently
assumed (though without evidence) to benefit from the
same approaches used for aggregated species.
ACKNOWLEDGMENTS
We thank B. A. Harrington, S. K. Skagen, International
Shorebird Survey volunteers, and Manomet Center for Conservation Sciences for collection and use of shorebird data.
A. R. Ives, L. L. Conquest, E. A. Logerwell, L. F. Keller, A.
M. Kilpatrick, J. R. Cary, and colleagues at the Universities
of Wisconsin-Madison (Department of Wildlife Ecology) and
Washington (Department of Biology and School of Aquatic
and Fishery Sciences) provided helpful suggestions and insights, as did three anonymous reviewers. L. X. Payne and
J. K. Parrish thank the Pacific Northwest Research Station,
USDA Forest Service for funding, and the Department of
Wildlife Ecology, University of Wisconsin-Madison, and the
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SPATIAL PATTERN AND EVENNESS INDICES
School of Aquatic and Fishery Sciences, University of Washington, for logistical support. This paper is one component
of L. X. Payne’s dissertation.
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