vol. 158, no. 3
the american naturalist
september 2001
Empirical Relationships between Species Richness, Evenness,
and Proportional Diversity
Gray Stirling1,* and Brian Wilsey 2
1. Department of Biology, McGill University, 1205 Avenue Doctor
Penfield, Montreal, Quebec H3A1B1, Canada;
2. Department of Botany, Iowa State University, Ames, Iowa 50011
Submitted June 27, 2000; Accepted April 19, 2001
abstract: Diversity (or biodiversity) is typically measured by a species count (richness) and sometimes with an evenness index; it may
also be measured by a proportional statistic that combines both measures (e.g., Shannon-Weiner index or H ). These diversity measures
are hypothesized to be positively and strongly correlated, but this null
hypothesis has not been tested empirically. We used the results of
Caswell’s neutral model to generate null relationships between richness
(S), evenness (J ), and proportional diversity (H ). We tested predictions
of the null model against empirical relationships describing data in a
literature survey and in four individual studies conducted across various
scales. Empirical relationships between log S or J and H differed from
the null model when !10 species were tested and in plants, vertebrates,
and fungi. The empirical relationships were similar to the null model
when 110 and !100 species were tested and in invertebrates. If 1100
species were used to estimate diversity, the relation between log S and
H was negative. The strongest predictive models included log S and
J . A path analysis indicated that log S and J were always negatively
related, that empirical observations could not be explained without
including indirect effects, and that differences between the partials may
indicate ecological effects, which suggests that S and J act like diversity
components or that diversity should be measured using a compound
statistic.
Keywords: diversity, biodiversity, evenness, richness, path analysis,
species density.
Diversity is a community attribute related to stability, productivity, trophic structure (McIntosh 1967; McNaughton
1977; Tilman 1996), and migration (Wisheu and Keddy
1996; Caley and Schluter 1997; Colwell and Lees 2000).
However, the way we measure diversity presumes the importance of species richness compared to species relative
* E-mail: stirling@bio1.lan.mcgill.ca.
Am. Nat. 2001. Vol. 158, pp. 286–299. 䉷 2001 by The University of Chicago.
0003-0147/2001/15803-0007$03.00. All rights reserved.
abundance, the nature of relationships between richness
and abundance, and possibly whether communities are
open or closed.
Species richness, or the number of species, is currently
the most widely used diversity measure. Relative species
abundance in a community is another factor that affects
diversity (Whittaker 1965; Hurlbert 1971). It is measured
with a standardized index of species abundance (evenness
or equitability) that is typically on a scale ranging from near
0, which indicates low evenness or high single-species dominance, to 1, which indicates equal abundance of all species
or maximum evenness (Routledge 1980; Alatalo 1981). Finally, proportional abundance indices such as the ShannonWeiner index sum species weighted by their relative abundance (Magurran 1988). These indices depend on both
species richness and evenness, although they weight rare
species differently (Hill 1973). Diversity can be measured
in other ways (Magurran 1988), but authors have long argued that species abundance and proportional diversity are
simply and directly related to species richness (Sanders 1968;
Johnson and Raven 1970; Fager 1972; Hill 1973; Alatalo
1981; Schluter and Ricklefs 1993; Huston 1994).
Analytic arguments, mathematical models, and simulations indicate that relationships between species richness,
evenness, and proportional diversity are simple, positive,
and strong. De Benedictis (1973) first argued that mathematical relationships constrain correlations between species
richness (S), an evenness measure (J ), and proportional
abundance (measured by the Shannon-Weiner index H ) to
be positive and strong. De Benedictis pointed out that these
mathematical relationships are the null model, against which
the influence of empirical, biological effects should be tested.
Hill (1973) extended this idea by arguing that diversity is
fundamentally the number of species in a community and
that other diversity measures comprise a related, higherorder series. Furthermore, May (1975) derived positive relationships between S, J , and H while assuming species
abundance was either log normal or log-series distributed.
Patterns of species abundance were regularly observed to
follow these distributions (May 1975; Magurran 1988),
which suggests that species richness is a common cause of
Diversity Components
variation in relative abundance and diversity. Consequently,
it does not seem remarkable that species richness is currently
used as the sole measure of diversity in many reviews (e.g.,
Ricklefs and Schluter 1993; Harper and Hawksworth 1994;
Barthlott and Winiger 1998; McKinney and Drake 1998)
and the independent variable in experimental diversity studies (e.g., Naeem et al. 1994; Tilman 1996; Allison 1999;
Hector et al. 1999), even though other studies indicate that
richness and evenness may be independent.
Simulations suggest that some evenness indices may be
independent of species richness across a range of values
(Sheldon 1969; Smith and Wilson 1996). Furthermore, diversity indices do vary independently of species richness in
the laboratory (Hairston et al. 1968; Rainey and Travisano
1998) and in field experiments (McNaughton 1977; Wilsey
and Potvin 2000). These studies indicate that diversity can
change with key ecological processes such as competition,
predation, and succession, each of which alter proportional
diversity through changes in evenness without any change
in species richness. Consequently, many authors suggest that
we should treat richness and evenness separately (Whittaker
1965; Tramer 1969; Hurlbert 1971; Magurran 1988; Legendre and Legendre 1998; Weiher and Keddy 1999). This separation does provide meaningful insight into community
function (McNaughton 1977) and may address critical assumptions in experimental analyses of diversity change
(Huston 1997; Wilsey and Potvin 2000), but it assumes that
factor interrelations are unimportant.
The sign and strength of relationships between different
diversity measures are assumptions that are largely based
on simulations, models, and analytical work. In the following study, our objective is to test whether or not empirical
estimates of richness, evenness, and proportional diversity
are strongly, positively, and highly interrelated. Alternatively,
species richness and evenness may be independent, or variation in significant relationships may be due to ecological
effects. It may not be possible to specify exact, expected
values for S, J , and H (De Benedictis 1973), but a null
model can be used as a benchmark against which empirical
parameters can be tested. We use the published values of
Caswell’s (1976) neutral model to provide expected (null)
relationships between S, J, and H (De Benedictis 1973). In
the neutral model, migration (species richness) largely governs J and H , whereas scale (e.g., alpha, beta, or gamma
diversities) and biotic interactions (e.g., competition or predation) do not affect these variables.
Methods
The Neutral Model as a Null Hypothesis
The number of species depends on time available for immigration in the neutral model. Species population size is
287
the result of a stochastic process in which the expected birth
rate equals the expected death rate (Caswell 1976). Thus,
model values are generated in the absence of environmental
effects, biotic interactions, spatial or temporal variation, or
higher-order complexity (Caswell 1976). Caswell’s published results were used to generate the expected univariate
relationships and the multiple-regression model. Since our
diversity estimates come from a range of communities with
different genetic backgrounds and because these communities are under different environmental constraints and
sampled with different strategies or effort, empirical trends
should regress to averages generated by Caswell’s neutral
model if these variables are constrained to be strongly and
positively related.
Caswell’s (1976) neutral model generates strong positive
correlations between species richness (S), evenness (J ), and
proportional diversity (H ); this is similar to the results
obtained by De Benedictis (1973). We log transformed S so
that it was on the same scale as H and J (Alatalo 1981).
These log-transformed, first-order models account for the
same proportion of variation explained by corresponding
second-order (polynomial) regressions that use the nontransformed values of S. More importantly, extrapolation
from a first-order model is not as uncertain as extrapolation
from a second-order model. The first-order predictive models relating log-transformed species richness (log 2 S) or J to H are positive, highly significant, and explain most of
the variation in H (H p 0.96 ⫹ 0.75 # log 2 S; r 2 p
0.96, F p 475.9, df p 1, 22, P ! .0001; H p ⫺2.56 ⫹
6.62 # J ; r 2 p 0.91, F p 222.6, df p 1, 22, P ! .0001).
Finally, the slope of the regression of H on log 2 S
(mean Ⳳ SE; 0.75 Ⳳ 0.04) should be the same as the av
p H / log S). The 95% conerage value of J (J p H /H max
fidence limits of the slope enclose the mean value of J
(0.73 Ⳳ 0.03), which indicates that results of the neutral
model also fit the analytic expectation that log 2 S, H , and
J are positively and directly related.
The multiple-regression model (which uses log 2 S and
J as independent variables) is highly significant and explains
almost all of the variation in H (H p ⫺1.8 ⫹ 2.83 #
J ⫹ 0.47 # log 2 S; R 2 p 0.99, F p 1,185.94, df p2, 21,
P ! .0001). Although the residual distribution of this model
is not significantly different from normal (Shapiro-Wilk Wtest p 0.92, P p .08), it is U-shaped with a minimum near
H p 2 and a maximum at the lowest and highest values
of H . This distribution suggests a bias in the model, so
both J and log 2 S were transformed (squared), and the
model was recalculated. There was no evidence of a bias in
the distribution of residuals in the model with the independent parameters squared, and both multiple-regression
models had similar multiple coefficients of determination
(parameters squared: H p ⫺0.03 ⫹ 2.05 # J 2 ⫹ 0.06 #
log 2 S 2; R 2 p 0.99, F p 8166.6, df p 2, 21, P 1 .0001). We
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The American Naturalist
used this transformed model even though bias indicated by
the residual variation did not appear to have a strong effect
on the model strength. Evenness (J ) and log 2 S are highly
correlated (r p 0.89, P ! .0001), which suggests that these
parameters may be collinear, but we could not detect related
variance inflation.
Data Sources
We compiled diversity estimates from a computer literature search (1994–1997). To be included in the search,
articles had to have the words “evenness” or “equitability”
and “species number,” “richness,” or their derivatives in
the title, abstract, or keywords. We compiled these statistics
from journals in the collections of the Smithsonian Tropical Research Institute (Balboa, Panama) and McGill University (table 1).
A second data set was analyzed based on individual articles chosen from older, published data sets. To be included,
data either had to have had a different species number range
or they had to have been collected over different scales (e.g.,
local, regional, biogeographical, or temporal).
Tests
We performed two tests on the first data set (size range
and taxonomic categories). First, if diversity components
are simply and positively related to the number of species
(S), we would expect that these relationships would be
consistent across the range of species numbers used to
estimate diversity. To estimate diversity, we divided the
data set into three groups: low (≤10), intermediate (110
and !100), and high (≥100) numbers of species. Each
group was then tested against the null model. Second,
while diversity is measured using different numbers of
species, it is generally estimated within guilds, functional
groups, or trophic levels. Consequently, we recast the survey data into four broad taxonomic categories and reran
the analysis with the expectation that relationships within
taxa, between S, J , and H , would be similar to the null
model, assuming mathematical relationships between
these variables constrained them to be positively and
strongly related.
Results
Overall Relationships
In the 5-yr literature survey, there were 486 estimates of
S and H ; however, we could verify estimates of J in only
339 of these cases. There were estimates for all three indices
in 323 of these cases. Consequently, for the following multivariate analyses, the sample size was smaller than the
Table 1: Data sets (1993–1997) that included estimates of species richness, evenness, and proportional diversity
Data set
1
2
3
4
5
8
9
10
11
12
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
Taxa
Study
Plants
Invertebrates
Fungi
Plants
Invertebrates
Plants
Invertebrates
Vertebrates
Vertebrates
Plants
Invertebrates
Vertebrates
Invertebrates
Invertebrates
Vertebrates
Invertebrates
Invertebrates
Invertebrates
Invertebrates
Vertebrates
Vertebrates
Vertebrates
Vertebrates
Plants
Plants
Plants
Plants
Plants
Vertebrates
Vertebrates
Invertebrates
Invertebrates
Invertebrates
Vertebrates
Plants
Invertebrates
Invertebrates
Plants
Invertebrates
Invertebrates
Invertebrates
Plants
Invertebrates
Elliot and Swank (1994)
Greenberg and Thomas (1995)
Hendrix et al. (1995)
Nunnez-Olivera et al. (1995)
Petridis and Sinis (1995)
Piper (1995)
Seguin et al. (1994)
Thiel et al. (1995)
Thiel et al. (1995)
Thiollay (1995)
Coderre et al. (1995)
Edgar and Kershaw (1994)
Hodda and Wanless (1994)
Adjeroud and Salvat (1996)
Boehring-Gause and Bauer (1996)
Collins et al. (1996)
Digweed et al. (1995)
Digweed et al. (1995)
Digweed et al. (1995)
Fujita et al. (1996)
Fujita et al. (1996)
Fujita et al. (1996)
Fujita et al. (1996)
Gilliam et al. (1995)
Lapin and Barnes (1995)
Mutangah and Agnew (1996)
Mutangah and Agnew (1996)
Tsuyuzaki and Titus (1996)
Wantiez (1996)
White et al. (1996)
Commito et al. (1995)
Aronson and Precht (1995)
Carcamo (1995)
Ashbaughe et al. (1996)
Hartnett et al. (1996)
Lobo et al. (1995)
Drake and Arias (1997)
Derkson et al. (1995)
Savage and Beamont (1997)
Coleman et al. (1997)
Pinca and Dallot (1997)
Qian et al. (1997)
Cutter et al. (1994)
sample size used for the univariate tests. However, we
compared the univariate coefficients generated using the
multivariate data set as an internal check on the sensitivity
of the univariate relationship between log 2 S or J and
H to differences in sample size. For the entire data set
(n p 486; hereafter “complete”), the average H estimate
was 1.95 Ⳳ 0.05, and the average number of species was
31 Ⳳ 2.01. The mean H of the restricted data set (n p
Diversity Components
323; hereafter “restricted”) was not significantly different
(1.91 Ⳳ 0.07), and the average number of species did not
change significantly (31.6 Ⳳ 2.1).
We entered log 2 S and J into the multiple-regression
model developed above (H p log 2 S 2 ⫹ J 2 ⫹ e) to evaluate
the combined influence of both diversity components on
H in the restricted data set. As in the null model, squaring
the independent parameters reduced the bias indicated by
the U-shaped residual distribution but did not affect the
parameter or model significance (H p ⫺0.24 ⫹ 2.2J 2 ⫹
0.06 # log 2 S 2). Both S and J had highly significant univariate effects on H in the restricted data set (table 2). The
multiple coefficient of determination was highly significant
overall and was higher than either univariate model (table
2). Log2S and J were significantly intercorrelated, but this
correlation was not high (table 2), which suggests that they
were at least partially independent (Slinker and Glantz 1985;
Neter et al. 1996). As in the null model, the empirical multiple-regression model explained most variation in H . However, in contrast to the null model, there was a much larger
difference between the multiple coefficient of determination
(R2) of the empirical data and the proportion of variation
explained by the univariate relationship between log 2 S and
H (table 2).
Species-Richness Range
Low-Group Univariate Relationships. Relationships between
log 2 S or J and H were very different from what we expected. We calculated the regression of H on log 2 S (!10)
with and without the high-diversity outliers (fig. 1A). The
outliers did not have a significant effect on this relationship,
so they were removed and were analyzed separately. The
relationship between log 2 S and H is significant, although
the slope is close to 0 (fig. 1A; r 2 p 0.08, F p 20.44,
df p 1, 219, P ! .01). The outliers have a similar slope (fig.
1A). The relationship does not change significantly when
S p 2 is removed from the data (H p 0.26 ⫹ 0.39 #
log 2 S), which suggests that this relationship is robust. The
slope of the empirical data was significantly different from
what we expected (sequential ANCOVA; slope: F p 7.6,
df p 1, 240, P p .006). Figure 1A indicates that the empirical relationship between J and H is positive over the
low-species-number range but that it also increases at a
much lower rate than the null model predicts. Consequently,
it crosses the line predicted by the null model near 10 species. The r 2 value of the ordinary linear regression (H on
J ) was so high (r 2 p 0.87, P ! .0001) that the slope of the
reduced major axis, or geometric mean regression (GMR),
was not very different from the linear regression (Ricker
1984). Finally, in contrast to the null model, the slope
(0.31 Ⳳ 0.08) of the empirical relationship between H and
log 2 S did not provide a good estimate of the average value
of J (0.62 Ⳳ 0.02).
Intermediate-Group Univariate Relationships. Univariate
relationships in the intermediate group (110 and !100)
were more similar to what we expected (fig. 1B, 1B ) than
they were in the low group. Proportional diversity (H )
increased with log 2 S at a similar rate as in the null model,
but average values of H were smaller than what might
otherwise be expected (sequential ANCOVA; slope: F p
0.16, df p 1, 248, P p .69; intercept: F p 342.51, df p
1, 248, P ! .0001; fig. 1B). The difference between slopes
of empirical and expected relationships (H on J ) was
marginally significant (ANCOVA; slope: F p 5.08, df p
1, 178, P p .056). The slope of the GMR (5.62 Ⳳ 0.05)
was closer than the ordinary regression (5.0 Ⳳ 0.02; fig.
1B ) to the slope of the null relationship (6.62), but it was
still significantly different. Finally, in this species number
range, the slope of the relationship between log 2 S and
H (0.81 Ⳳ 0.05) was also significantly different from the
average value of J , in contrast to the analytic expectation
(J p H / log S).
High-Group Univariate Relationships. In contrast to the
intermediate group, the relationship between log 2 S and
H in the high group (≥100) was negative (fig. 1C; r 2 p
Table 2: Relationships among empirical data grouped by number of species
Test
Sample size
Richness
Evenness
Multivariate model
Richness-evenness
289
Variable
All data
Low
(≤10)
Intermediate
(110 and !100)
High
(≥100)
n
r2
r2
R2
r
323
.70***
.56***
.94***
.35***
150
.24***
.84***
.92***
.23*
154
.69***
.82***
.94***
.64***
19
.0008
.93***
.98***
⫺.15
Note: Proportional diversity is the dependent variable, and species richness and evenness are the
independent variables.
* P ! .05.
** P ! .01.
*** P ! .001.
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The American Naturalist
Figure 1: Univariate relationships between proportional diversity (H ) on log-transformed species richness (log2 S ; A, B, C) and H on evenness
(J ; A , B , C ) over three species number ranges: A, ≥2 and ≤10; B, 110 and !100; and C, ≥100. The dashed line indicates the null-regression
line generated by the neutral model described in the text. The solid line (or the equation) indicates the best fit to the empirical data within
each panel.
0.31, F p 12.4, df p 1, 28, P p .002). Variation in H was
positively and strongly related to variation in J (fig. 1C ;
r 2 p 0.93, F p 241.2, df p 1, 17, P ! .0001). However,
figure 1C indicates that when species numbers are high,
empirical H values are larger than those predicted by the
null model (H on J ), although the slopes are not significantly different (sequential ANCOVA; slope: F p 0.003,
df p 1, 39, P p .96; intercept: F p 211.22, df p 1, 40,
P ! .0001).
Multivariate Models. Table 2 shows that the multivariate
model for each group is highly significant and accounts for
190% of the variation in H (low: H p ⫺0.27 ⫹ 1.79 #
J 2 ⫹ 0.0 8 # log 2 S 2; F p 864.0, df p 2, 147, P ! .0001;
intermediate: H p 0.22 ⫹ 2.86 # J 2 ⫹ 0.05 # log 2 S 2;
F p 1,116.9, df p 1, 151, P ! .0001; high: H p ⫺1.85 ⫹
5.41 # J 2 ⫹ 0.0 3 # log 2 S 2; F p 399.8, df p 1, 19,
P ! .0001). However, the variation in H explained by richness (r 2) is lower in each group than the proportion explained by evenness (table 2).
Taxonomic Categories
Univariate relationships (H on log 2 S or J ) varied between
groups and also differed with respect to the expected (null
model) trend lines (fig. 2). The relationships between
log 2 S and H were positive in fungi, invertebrates, and vertebrates, but in plants, the relationship was not significant
(r 2 p 0.02, P p .11). The slope of the invertebrate (H on
log 2 S) relationship was not significantly different from the
null model (ANCOVA; F p 0.97, df p 1, 270, P p .33),
whereas the other empirical relationships were significantly
different from the expected line (ANCOVA; vertebrates:
F p 8.15, df p 1, 92, P p .005; plants: F p 36.04, df p
1, 168, P ! .0001; fungi: F p 4.64, df p 1, 36, P p .04).
Finally, above log 2 S p 6.5, the relationship between H and
log 2 S appeared to be negative in vertebrates but not in
invertebrates (fig. 2C, 2D). The empirical relationship between H and J for invertebrates was also similar to the
expected null line (ANCOVA; F p 1.36, df p 1, 190,
P p .25), although figure 2C indicates that there are two
Diversity Components
291
Figure 2: Univariate relationships between log-transformed species richness (log2 S ; A, B, C) or evenness (J ; A , B , C ) and proportional diversity
(H ) for different taxa (fungi: A, A ; plants: B, B ; invertebrates: C, C ; vertebrates: D, D ). The dashed line indicates the null-regression line; the
solid line indicates the best fit to the empirical data.
clusters of data. Based on our analysis of the number of
species used to estimate diversity (fig. 1), we confirmed that
these clusters represent either low or intermediate and high
number of species measured. The other empirical relationships between J and H were significantly different from
the null model (ANCOVA; vertebrates: F p 8.61, df p
1, 81, P p .004; plants: F p 60.18, df p 1, 96, P ! .0001;
fungi: F p 45.33, df p 1, 36, P ! .0001). There is also evidence of different data clusters in the H on J plots, re-
sembling clusters that differed between three species-richness groups (fig. 1).
The multiple-regression model for each taxonomic group
used the restricted data set and transformed (squared) independent variables (log 2 S 2 and J 2). The correlation between richness and evenness was positive in animals (vertebrates and invertebrates) and negative in plants, although
it was not very strong in any taxonomic category (table 3).
The univariate coefficients of determination for log 2 S 2 and
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Table 3: Relationships among the survey data
Test
Variable
Fungi
Plants
Vertebrates
Invertebrates
n
r2
r2
R2
r
16
.43**
.27*
.96***
⫺.28
76
.06*
.53***
.93***
⫺.43***
61
.75***
.69***
.94***
.54***
170
.82***
.58***
.94***
.51***
Sample size
Richness
Evenness
Multivariate model
Richness-evenness
Note: Relationships are broadly grouped by taxonomic differences. Proportional diversity is the
dependent variable, and species richness and evenness are the independent variables.
* P ! .05.
** P ! .01.
*** P ! .001.
J 2 for the restricted data set were all significant. But their
magnitudes differed across taxonomic categories, and neither variable consistently explained more of the variation
across the taxonomic categories (table 3). When both evenness and richness were in the multivariate model, the multiple coefficient of determination was very high in each
taxonomic group (≥0.94; table 3), which is similar to the
analyses of data grouped by number of species (invertebrates: H p ⫺0.13 ⫹ 2.17 # J ⫹ 0.06 # log 2 S; vertebrates: H p ⫺0.33 ⫹ 2.51 # J ⫹ 0.05 # log 2 S; plants:
H p ⫺0.28 ⫹ 2.01 # J ⫹ 0.06 # log 2 S; fungi: H p
⫺0.38 ⫹ 1.40 # J ⫹ 0.12 # log 2 S).
Individual Diversity Studies
Data reported in individual studies are typically more restricted. It could be argued that results based on a survey
of unrelated studies may not apply to more restricted taxonomic categories measured at different scales (e.g., across
transects in a community, within communities across
years, between local communities, or between communities in geographically distinct locations). We used Loya’s
(1972) coral diversity data, which comprised 84 transect
estimates of coral diversity at Eilat in the Red Sea (diversity
variation within a community). We dropped seven transects that had numbers of species that were !10, so the
number of species ranged from 10 to 30. Harrel et al.’s
(1967) data on fish diversity (two to 13 species) are from
intermittent Oklahoma streams of several different orders
(H in different local communities). Karr (1971) reported
bird diversities in different habitats located in Illinois and
Panama (diversity changes between geographically distinct
locations). We dropped the community with the lowest
number of species (11) and used the other 10 estimates
that ranged from middle to high species numbers
(47–165). Finally, Patrick (1968) reported the diversity of
freshwater diatom communities measured over 2 yr; all
her species counts were 1100 (diversity variation within a
community and over time).
If the species number was !50 (Harrel et al. 1967; Loya
1972), slopes of the relationships between log 2 S and H were not significantly different from the null model, which
is unlike the results obtained from the literature-compiled
data set. However, when species numbers were 1100, the
slopes were negative (table 4). The first-order regression for
Karr’s (1971) data was not significantly different from the
null model (table 4), but the second-order coefficient was
negative and highly significant (R 2 p 0.98, log 2 S 2: t p
⫺5.5, P p .0009, H p ⫺22.4 ⫹ 7.5 # log 2 S ⫺ 0.53 #
log 2 S 2). The shape of the curve suggests there is a maximum
between log 2 S p 6.5 and log 2 S p 7.0. This is the same
range in which there was a negative relationship between
S and H in Patrick’s (1968) data (H p 9.4 ⫺ 0.06 #
log 2 S) and in the larger data set compiled from the literature
review (fig. 1C).
Slopes of relationships between J and H were highly
significant (table 4) and crossed the expected line in the two
studies having lower species numbers (Harrel et al. 1967:
H p ⫺0.14 ⫹ 3.3 # J ; Loya 1972: H p 0.82 ⫹ 3.8 #
J ). Studies with higher species numbers had regressions
with slopes similar to the null model but with significantly
higher intercepts (Karr 1971: H p ⫺1.3 ⫹ 8.9 # J ; Patrick 1968: H p 0.07 ⫹ 6.5 # J ; table 4). Although the
slope of the relationship between J and H for Patrick’s
data was significantly different from the null model (table
4), there was very little variation around the empirical line,
and the two slopes were almost identical (slope for Patrick’s
data p 6.5; slope for the null model p 6.6). These results
were also similar to the large, literature-compiled data set.
As was the case in the large, literature-compiled data set
(tables 2, 3), neither the magnitude nor the sign of the
correlation between evenness and richness was consistent
in the individual studies (table 5). When the correlation
between richness and evenness was low (not significant) or
negative (table 5), species richness was not as highly related
to H as it was when the correlation was high. This pattern
is similar to variation between the number of species groups
(table 2) and different taxonomic categories (table 3). With
both diversity components in the multivariate model
(log 2 S 2 and J 2), the predictive relationships were much
Diversity Components
293
Table 4: Significance tests of univariate coefficients (intercept and slope) of relationships
in four studies and the null model described in the text
log2 S and H relationship
Study
Harrel et al. (1967)
Loya (1972)
Karr (1971)
Patrick (1968)
J and H relationship
Intercept F-test
Slope F-test
Intercept F-test
Slope F-test
187.2***
67.8***
58.04***
***
.12 NS
.03 NS
2.46 NS
*** (negative)
119.1***
31.8***
299.8***
661.8***
21.5***
26.8***
.26 NS
11.8**
Note: NS indicates a nonsignificant F-test.
* P ≤ .05.
** P ≤ .01.
*** P ≤ .0001.
stronger. Each multivariate model explains 190% of the
variation in H (table 5). This is, once again, similar to the
results of the large literature-compiled data set analyses (tables 2, 3). In summary, individual studies differ from the
null model in much the same ways as do different size range
groups or taxonomic categories in the other data set.
Other Explanations for Deviations from
the Null Model
In contrast to derived or simulated relationships (De Benedictis 1973; May 1975), empirical univariate relationships of log 2 S or J on H were consistent neither in their
magnitude nor in their sign. This was not merely a property of unrelated empirical data because we saw similar
patterns both within taxa in general and in individual
data sets in particular. These results also indicate that
scale dependence is not a plausible, alternative explanation. The magnitude of the correlation between
log 2 S and J varied across different groups of diversity
estimates, and the sign of this correlation was not always
positive, nor were these variables generally independent.
But when they were combined in a multivariate relationship, most variation in H was explained.
This suggests that indirect relationships between S, J ,
and H may contribute to direct (univariate) relationships
between richness, evenness, and proportional diversity. In
order to test this hypothesis, we calculated standardized
partial coefficients of determination and used a path analysis to decompose relationships between richness (S),
evenness (J ), and proportional diversity (H ). We used
this analysis to test whether causal models that included
indirect effects were consistent with the observed results
(Li 1975). Another advantage of this analysis is that standardized partial-regression coefficients of S and J can be
compared since they are both on the same scale.
There are several ways that three variables can be related.
The simplest causal model is a chain of uncorrelated causes
(Li 1975). The hypothesis that S governs evenness and, in
turn, proportional diversity is a (causal) chain model. This
model can be rejected since correlations between H and
log 2 S do not equal the products of the other two correlations, which they should if were is a causally dependent
path. Next, S may be considered to be the common cause
of both H and J (which are uncorrelated) or uncorrelated
joint causes of H , but these models can also be rejected
for the same reason as the chain model (Li 1975).
There are two other models with a (small) residual effect, two other direct effects, and an intercorrelated effect
that are consistent with the observed correlations: ecological relationships (ecological model) and mathematical relationships (mathematical model). The ecological model
assumes there are direct effects of S and J on H in addition
to the intercorrelated effects of log S and J on H (fig. 3).
The mathematical model assumes a different path with a
direct effect of S and H on J (H / log S p J ), in addition
to the intercorrelated effects of log S and H on J (fig. 3).
In the ecological model, table 6 indicates that the direct
effect of log S on H is lower than the effect of J on H in the low (≤10), medium (110 and !100), and high
groups (≥100). However, the difference between the direct
effect of log S and J did not have a consistent direction
when the data were put into taxonomic categories (table
6). Direct effects of both log S and J on H were always
positive in the ecological model, whereas the correlation
between them was either positive or negative. However,
log 2 S had a consistently direct negative effect on J in the
mathematical model. This effect was remarkably similar
to the size of log S on H in the ecological model in the
low, medium, and high species number groups. This was
not the case in the taxonomic categories (table 6).
Discussion
Mathematical relationships between species richness (S),
evenness (J ), and proportional diversity (H ), together
with results of simulations, verbal models, and species
abundance distributions all suggest that S, J , and H are
294
The American Naturalist
positively and highly correlated. If all these indices are
related, why collect abundance data, particularly when
there are logistical problems in just making a detailed species inventory (e.g., Lawton et al. 1998) and when most
communities appear to be open, or unsaturated, rather
than closed (Ricklefs and Schluter 1993; Caley and Schluter
1997; Colwell and Lees 2000; Gaston 2000)? Yet, a species
count as the sole index of diversity has long seemed to be
intuitively dissatisfying. That is, a community with evenly
distributed species abundance appears more diverse than
a community with the same number of species that is
dominated by a few species (Hurlbert 1971; Magurran
1988; Purvis and Hector 2000).
Our analysis of empirical data indicates that this intuitive
dissatisfaction with S as the index of diversity is justified.
The empirical relationship between species number (log S)
and J or H is not consistent in statistical strength, coefficient magnitude, or sign over the three orders of species
numbers we used to estimate diversity. Furthermore, simple
linear relationships vary in strength and magnitude across
four broad taxonomic categories and among four individual
studies. These results indicate that the assumption that
mathematical relationships constrain empirical richness,
evenness, and proportional diversity to be strongly and positively related may not be correct. Indeed, the mathematical
path model (table 6) indicates that S and J are always
negatively related, which provides clear evidence of a systematic empirical effect that differs from analytic expectations, simulation results, and the null model (De Benedictis
1973; Hill 1973; May 1975; Caswell 1976).
At the same time, we observed correspondence between
the results generated by Caswell’s (1976) neutral model
and some of the empirical results. Our results show the
utility of this altogether too infrequently used null model.
It provided a test for differences in simple relationships
between taxonomic groups and, within these groups,
marks where these trends may change across the number
of species used to measure diversity (fig. 2). Furthermore,
the multivariate coefficient for log S in the null model
Figure 3: Two path-analysis models (ecological path and mathematical
path). Straight arrows (in bold) indicate direct effects and their direction.
Curved arrows indicate a correlation. The standardized partial-regression
coefficients corresponding to direct effects and correlations for different
data sets are listed in table 6.
(0.06) is remarkably similar to most empirical coefficients.
However, the null model generally predicts neither the sign
nor the magnitude of univariate, empirical coefficients relating log S, J , and H , nor does empirical species richness
explain most variation in H . In other words, S is not
generally the common cause of variation in either H or
J , nor are these three variables generally independent from
each other. Thus, they seem to be acting as diversity components that may reflect different ecological processes.
Diversity Models
We used a path-analytic approach to test whether or not
indirect effects between these variables were a necessary
Table 5: Regression statistics for four diversity studies that differed in the scale at
which diversity was estimated and the number of species used to estimate proportional diversity
Test
Sample size
Richness
Evenness
Multivariate model
Richness-evenness
* P ! .05.
** P ! .01.
*** P ! .001.
Variable
Harrel et al.
(1967)
Loya
(1972)
Karr
(1971)
Patrick
(1968)
n
r2
r2
R2
r
34
.52***
.58***
.96***
⫹.13
77
.75***
.61***
.94***
⫹.42***
10
.87***
.89***
.99***
⫹.77***
12
.46*
.99***
.99***
⫺.68*
Diversity Components
295
Table 6: Standardized partial-regression coefficients and correlations for two
path models
Ecological model
Source
Overall
≤10 species
110 and !100 species
≥100 species
Plants (2–168)
Vertebrates (2–488)
Invertebrates (3–187)
Fungi (3–10)
Mathematical
model
S
pH S
pH J rSJ pJ S
pJ H rSH 31.9
7.4
39.1
165.5
17.0
47.6
37.8
8.0
.67
.29
.43
.18
.69
.59
.70
.87
.51
.85
.64
.99
1.03
.51
.41
.76
.35
.24
.64
⫺.15
⫺.43
.54
.51
⫺.28
⫺.97
⫺.28
⫺.40
⫺.18
⫺.66
⫺.72
⫺.99
⫺1.09
1.56
1.06
1.24
.97
.90
1.45
1.67
1.23
.84
.49
.83
.03
.25
.87
.91
.66
Note: The subscripts on the standardized partial-regression coefficients (p) indicate the
direction of the direct effect (from the right). Correlations (r) are between the subscripted
variables; S is the average number of species in each source listed in the left hand column.
For a diagram of the models, see figure 3.
part of an empirical diversity statistic. Neither the common
cause (i.e., S governs H and J directly), nor the independent joint cause (both directly affect H ), nor the chain
of uncorrelated causes (S governs H and J in sequence)
are sufficient models because they do not take correlations
between diversity components into account. However,
these models underlie assumptions that diversity factors
are strongly, positively, and fundamentally related to S, on
the one hand, or that S and J are independent, on the
other. These models also underlie some explanatory models of empirical trends. For example, it has been argued
that an increasing number of species (i.e., through migration) must have a positive but diminishing effect on
(increasing) proportional diversity because of an increasing positive effect of evenness (Monk 1967; Sager and
Hasler 1969; Loya 1972). However, this diminishing curve
is caused by a difference in scale of H and S at lower
species numbers (see Alatalo 1981), and it does not predict
or explain the negative relationship of log S with H at high
species numbers (fig. 1). This negative relationship must
manifest itself through the intercorrelated effect of log S
and J since the direct effect of log S on H is positive,
while the direct effect of S on J is negative and has the
same magnitude (table 6).
In the early diversity literature, there was also a disagreement over the scale at which species richness (Monk
1967; Tramer 1969; Loya 1972) or evenness (Sager and
Hasler 1969) governs proportional diversity in communities. The partial coefficients indicate that direct effects
of S are much smaller than J in the low group (!10 species;
table 6). However, there is also a difference of similar
magnitude in the high group (1100 species), which indicates that the predominant effect of evenness is not due
to differences in species richness or a result of H being
systematically insensitive to rare species (Sager and Hasler
1969). What is observed at high species numbers is a very
close positive relationship between J and H , and none of
the evenness values for the literature-compiled data or
individual data sets are remarkably low. This result provides no indication that guilds with higher species richness
generally have low evenness (see Weiher and Keddy 1999),
even though the standardized partial-regression coefficients indicate that the direct relationship between these
two parameters is always negative. We suggest that these
results cannot be rationalized unless indirect or intercorrelated effects between richness and evenness are considered to be meaningful and direct effects of both factors
are compared on the same scale.
At the outset of this analysis, we suggested that significant deviations from the null model may suggest the importance of ecological effects on diversity. Currently, an
ecological issue is whether communities are closed or
open, that is, whether communities are structured by biotic
interactions or are open to migration (Ricklefs and Schluter 1993; Caley and Schluter 1997; Colwell and Lees 2000;
Gaston 2000). Immigration is the only biological effect in
Caswell’s (1976) neutral model; otherwise, abundance is
governed by a stochastic process. In this respect, the null
model may be regarded as a test of whether or not migration regulates local diversity (measured by H ). Alternatively, biotic interactions affecting abundance (e.g.,
competition, predation) may govern diversity. Our results
indicate that univariate relationships between S or J and
H are very similar to the null model in invertebrates but
differ significantly in other taxonomic categories (fig. 2).
It has been widely reported that terrestrial and aquatic
invertebrate communities often consist of a large proportion of rare species (Williams 1964; Hughes 1984), and
296
The American Naturalist
long-term monitoring indicates that the numbers of insect
species at collecting stations continue to increase with time
(Rosenzweig 1998). Assuming that correspondence with
the null model indicates that migration governs invertebrate community diversity and that the partials reflect direct ecological effects, we would expect that the richness
partial coefficient on H in invertebrate communities
would be higher than evenness in the ecological model
(table 6). Following this logic, we note that plant communities do not have a significant relationship (fig. 2B),
which suggests that the evenness partial should have a
larger direct effect than richness. Finally, the difference
between vertebrate richness and evenness should be between invertebrates and plants. Differences between the
richness and evenness partials listed in table 6 are consistent with the hypothesis that the richness partial indicates
the direct effect of migration and that the evenness partial
indicates the direct effects of biotic interactions on diversity. Thus, both components affect diversity, but taxonomic differences in the strength of these effects may
weight empirical trends across the range of species number
or standardized abundance (fig. 1).
We Measure Diversity in Many Ways,
but What Is It?
Hurlburt (1971) argued that diversity had been defined in
so many ways that it risked becoming a nonconcept, but
he also advised taking both richness and evenness into account when measuring diversity. We treated H as the dependent variable because it takes both factors into account
and is widely used. Proportional diversity (H ) is sensitive
to both species richness and evenness (Pielou 1966; Peet
1974) and is the best measure of their joint influence (Fager
1972; De Benedeictis 1973). However, the information-theoretic relevance of H to ecological diversity is dubious
(Hairston et al. 1968; Hurlbert 1971), so it may be most
usefully thought of as a measure of the uncertainty in predicting the species of an individual (Hairston et al. 1968;
Fager 1972). Sample estimates of H are normally distributed
(Magurran 1988). Proportional diversity as measured by
H is not strongly affected by rare species, and it has a
broader region of sensitivity than the Simpson’s index (Fager 1972). Proportional diversity (H ) is also relatively sample-size independent (Sanders 1968). For our purposes,
H is a useful measure of proportional diversity because it
is widely reported and has been developed as a test statistic
for a model that is neutral with respect to physical, functional, and biotic interactions (Caswell 1976). However, the
use of the H statistic has been dismissed because it is not
intuitively related to ecological diversity (Hurlbert 1971), it
is insensitive to different species abundance distributions
(May 1975), it is descriptive (Ghilarov 1996), and it has a
small-sample bias (Routledge 1980). Other indices may have
a better ability to discriminate (Taylor et al. 1976) or may
have better statistical properties (Lande 1996; but see Magnussen and Boyle 1995). Nonetheless, H remains the most
useful empirical statistic because it is more widely used,
even though its performance and meaning are controversial.
In addition to questioning the sampling, statistical, and
measurement issues involved in choosing a diversity estimator, Kempton and Taylor (1974) also questioned
whether, in practice, the property of diversity should be
separated into its components (richness and evenness). A
division can provide insight into community function
(McNaughton 1977; Wilsey and Potvin 2000) and, possibly, into systematic biological effects (Caley and Schluter
1997; Gaston 2000). However, neither richness nor evenness are reliable independent measures of differences in
diversity compared with the performance of the combined
statistic (Kempton and Taylor 1974). Our analysis of disparate empirical data indicates that diversity reflects effects
of evenness and richness components along with their
intercorrelations.
Species abundance plots represent this integration
graphically, but they are inconvenient statistically (Weiher
and Keddy 1999). Consequently, we believe that to measure diversity empirically or to test the outcome of multispecies interactions using diversity as a community attribute, species abundance also has to be measured in a
species inventory and both components have to be integrated within a compound statistical measure. However,
the compound diversity measure that best integrates evenness and richness, that has the desired statistical properties,
and that makes the most ecological sense is a another issue.
Acknowledgments
Many thanks to A. Loan-Wilsey for help with data collection, the referees and the associate editor for their attention to the manuscript, and the authors who presented
the data we used in this work.
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