Concentration Curves and HaveStatistics for Ecological Analysis of
Diversity: Part III: Comparisons of
Measures of Diversity
Goodwin, D.G. and Vaupel, J.W.
IIASA Working Paper
WP-85-091
December 1985
Goodwin, D.G. and Vaupel, J.W. (1985) Concentration Curves and Have-Statistics for Ecological Analysis of Diversity: Part
III: Comparisons of Measures of Diversity. IIASA Working Paper. IIASA, Laxenburg, Austria, WP-85-091 Copyright ©
1985 by the author(s). http://pure.iiasa.ac.at/2612/
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CONCENTRATION CURYES AND HAVE-STATISI'ICS
FOR ECOLOGICAL ANALYSIS OF DIVER=
PART III: COMPARISON OF ME;ASURES OF DIVERSITY
Dianne G. Goodwin
James W. k u p e l
December 1985
WP-85-91
NOTE: This is P a r t I11 of a s e r i e s of t h r e e working papers. For a pref a c e , forward, acknowledgements, and a note about t h e authors, please
see P a r t I of t h e s e r i e s , which i s subtitled "Dominance and Evenness in
Reproductive Success", IIASA WP-85-72.
Working Aapers are interim r e p o r t s on work of t h e International
Institute f o r Applied Systems Analysis and have received only limited
review. Views o r opinions expressed herein d o not necessarily
r e p r e s e n t those of t h e Institute o r of its National Member
Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
2361 Laxenburg, Austria
Contents
Orientation
Principles f o r Judging Measures of Evenness
1.The Anonymity P r i n c i p l e
2. The Relativity Principle
3. The Replication P r i n c i p l e
4. The T r a n s f e r P r i n c i p l e
Which Measures are Consistent?
S o u r c e s of Confusion
Desirable P r o p e r t i e s of Measures of Evenness
1.Standardization
2. Intelligibility
3. Decomposibility
4 a n d 5. Sensitivity a n d Robustness
Diversity
Xeasures of Diversity
Applications
1.Lifetime Reproductive S u c c e s s of Male vs. Female Bullfrogs
2. Diversity in a Community of Herbaceous P l a n t s
3. Birds of a F e a t h e r
Correlations Between Measures of Evenness
Conclusion
References
CONCEMTRATION CURVES AND HAVE-ETATISi7CS
FOR ECOLOGICAL ANALYSIS OF DIVERSITY:
PART III: COMPARISON OF MEASURES OF DrVERSlTY
Dianne G. Goodwin a n d James W. VaupeL
Given t h e c e n t r a l importance of diversity in ecology and t h e life sciences
more generally, i t is not surprising t h a t a variety of methods and measures have
been developed t o describe and summarize diversity. In t h e two previous p a r t s of
this s e r i e s of papers, comparisons were drawn between concentration curves and
frequency distributions, t h e most widely used graphical display of variation, and
between concentration curves and dominance-diversity curves. This final p a r t of
t h e t h r e e p a p e r s e r i e s compares various statistics t h a t might be used t o summarize
diversity, with a focus on t h e usefulness of have-statistics as a supplement t o more
traditional measures. The f i r s t section of o u r discussion lays out some reasonable
c r i t e r i a and principles t h a t good measures of diversity should satisfy: some traditional measures violate at least one of t h e c r i t e r i a ; t h e have-statistics pass t h e
hurdles and have some desirable properties in addition. W e then illustrate t h e use
of different measures by way of examples drawn from Howard's studies of bullfrogs
(discussed in P a r t I), t h e study of species diversity among diatoms (discussed in
P a r t 11), an analysis of mating systems of various birds, and a survey of human f e r tility in 41 countries.
PRINCIPLES FOR JUDGING MEASURES OF EVENNESS
The l i t e r a t u r e on measures of diversity is s o vast and chaotic (see, e.g., Hurlb e r t 1971, Patil and Taillie 1982, P e e t 1974, and Rao 1982b f o r overviews) t h a t i t
is impossible t o make headway without a c l e a r goal and some principles of navigation. Our goal is t o t r y t o gain some understanding of t h e uses and limitations of
have-statistics and concentration curves by comparing them with o t h e r kinds of
measures of diversity. We will base this comparison on some principles and desir-
able p r o p e r t i e s of diversity measures.
For o u r purposes. i t is convenient t o begin with t h e a s p e c t of diversity known
as evenness. There is widespread agreement among r e s e a r c h e r s who have thought
about t h e principles t h a t a measure of evenness should satisfy (e.g., Marshall and
Olkin 1979, Foster 1985) t h a t t h e following f o u r principles a r e reasonable. For expository simplicity, w e use x t o mean individual, species, o r any o t h e r "have" and
w e use y t o mean offspring, zygotes, mates, members of a species, o r any o t h e r
"had".
1. T h e Anonymity Principle
Consider any two x ' s in a population. Suppose they c a n b e identified: f o r
convenience, call one H a r r y and t h e o t h e r Larry. Suppose one has y and t h e oth-
er has y
'.
An evenness measure should not change if H a r r y is t h e one with y rath-
er than t h e one with y
'.
2. T h e R e l a t i v i t y Principle
Evenness should depend only on t h e relative amount, i.e., t h e proportion of
t h e total, each x has and not on t h e absolute amount. Consider, f o r instance, a population with two x ' s , one having 70% of t h e y ' s and t h e o t h e r having 30%. Evenness should b e t h e same r e g a r d l e s s of whether t h e t o t a l number of y n s is ten, a
thousand, o r a million.
3. T h e R e p l i c a t i o n Principle
Suppose a population is replicated s o t h a t t h e r e are now m identical populations. Suppose t h e m populations are combined into a new population with m times
as many x ' s . The evenness of this new population should b e t h e same as t h e evenness of t h e original population. For instance, suppose t h e original population consists of two x ' s , one having 70%of t h e y ' s and t h e o t h e r having 30%. After a single
replication and combination, t h e new population will consist of f o u r x ' s , t h e top
half (i.e., top two) having: 70%of t h e y 's and t h e bottom half having 30%. Evenness
should remain t h e same.
4. The Transfer Principle
Consider any two x ' s in a population such t h a t one h a s y and t h e o t h e r h a s
Suppose a t r a n s f e r i s made such t h a t t h e f i r s t x now has
somewhat fewer, y -4.
even more, y + c , and t h e second x h a s even fewer, y 4--c.
According t o t h e
Fully-Responsive T r a n s f e r Principle, a n evenness measure should d e c r e a s e . According t o t h e Partially-Responsive Transfer Principle, a n evenness measure
should not i n c r e a s e and t h e r e should b e at least one p a i r of x's such t h a t such a
t r a n s f e r would r e s u l t in a d e c r e a s e in t h e evenness measure. Note t h a t t h e FullyResponsive T r a n s f e r Principle implies a t r a n s f e r from a "poor" z t o a v e r y r i c h x
should d e c r e a s e evenness by more than a n equal t r a n s f e r t o a not-so-rich z,and
t h e Partially-Responsive T r a n s f e r Principle implies t h a t such a t r a n s f e r from t h e
poor t o t h e v e r y r i c h should d e c r e a s e evenness by at least as much as a t r a n s f e r
t o t h e less rich.
A measure of evenness t h a t i s consistent with principles 1 , 2, and 3 and with
t h e partially-responsive version of principle 4 might b e called a consistent measu r e . A measure consistent with 1 , 2 , and 3 and t h e fully-responsive version of 4
might b e called a strictly-consistent measure. A s documented below, t h r e e of t h e
most commonly used "measures of evenness" are n e i t h e r consistent n o r strictlyconsistent.
I t t u r n s out t h a t t h e f o u r principles have a close relationship with concentration curves, as follows. If and only if t h e concentration c u r v e for one population
lies between t h e c u r v e f o r a n o t h e r population and t h e diagonal line at all points
between 0 and 1, will any consistent measure of evenness indicate t h a t t h e evenness of t h e f i r s t population i s g r e a t e r than t h e evenness of t h e second population.
Hence, if one concentration c u r v e lies under a n o t h e r i t c a n b e said t h a t t h e f i r s t
population i s definitely more even (regardless of t h e measure of evenness used)
than t h e second population. This i s one of t h e key reasons t h a t concentration
c u r v e s are s o useful and why t h e c e n t r a l role of concentration c u r v e s in t h e
analysis of evenness is, as Allison (1978) put i t , "virtually unquestioned" by
economists, o t h e r social scientists, and mathematicians who have studied inequality.
If one concentration c u r v e i s lower than a second c u r v e at some points but at
o t h e r points t h e c u r v e s e i t h e r touch o r r u n along t o g e t h e r , then all strictlyconsistent measures of evenness will indicate t h a t t h e f i r s t population is more even
than t h e second. Depending on t h e measure and on where t h e c u r v e s touch, a
measure t h a t is consistent but not strictly-consistent may indicate e i t h e r t h a t t h e
two populations are equally even o r t h a t t h e f i r s t population is more even than t h e
second. Consequently, i t is possible t o reformulate t h e c r i t e r i a f o r a measure of
evenness as follows:
--
a measure of evenness is strictly-consistent if and only if t h e measure gives a
lower value of evenness t o a concentration c u r v e t h a t lies outside a n o t h e r
concentration c u r v e at at least some points and never lies inside t h e o t h e r
concentration curve.
-
a measure of evenness is consistent if and only if t h e measure gives a lower
value of evenness t o a concentration c u r v e t h a t lies outside of a n o t h e r concentration c u r v e at all points between 0 and 1 and gives a lower o r equal
value of evenness t o a concentration c u r v e t h a t e i t h e r lies outside o r touches
another concentration c u r v e at all points.
These two c r i t e r i a might b e called t h e concentration-curve c r i t e r i a .
In o u r empirical analyses, on occasion t h e concentration c u r v e s crossed over.
In these cases two different summary measures may give t h e two populations a diff e r e n t ordering:' according t o some measures, t h e f i r s t population may b e more
concentrated and according t o o t h e r measures, t h e second population may b e more
concentrated o r , at least, equally concentrated. The differences in t h e measures
w e r e s m a l l and when a number of populations w e r e considered t h e rankings accord-
ing t o different measures tended t o b e more o r less t h e same. This highlights t h e
importance of concentration c u r v e s themselves as compared with any p a r t i c u l a r
summary measure.
WHICH MEASURES ARE CONSISTENT?
Many measures have been used t o c a p t u r e t h e evenness of a population; w e
consider only t h e most commonly used measures, as w e l l as various have-statistics.
Among t h e s e measures t h e following distinctions can b e made:
--
The havehalf, haveall, and o t h e r have-x measures are consistent measures of
evenness.
--
The halfhave and o t h e r y -have measures are consistent measures of unevenness. That is, t h e s e measures are consistent with t h e f o u r principles of evenness, e x c e p t t h a t t h e measures d e c r e a s e as evenness increases.
--
The havenone is a consistent measure of unevenness.
The Gini coefficient, which i s usually defined as t h e proportion of t h e a r e a
above t h e diagonal line t h a t lies between t h e diagonal line and a concentration
curve, is a strictly-consistent measure of unevenness. An alternative expression f o r t h e Gini coefficient is
where pi is t h e proportion of total y 's attributable t o t h e i 'th x and N is t h e
number of x 's.
The coefficient of variation and Crow's I, which equals t h e s q u a r e of t h e coefficient of variation, a r e both strictly-consistent measures of unevenness. CrownsI
is usually defined as t h e r a t i o of t h e variance in number of offspring divided by t h e
s q u a r e of t h e mean number of offspring. This r a t i o reduces to:
The c o r e of this expression, i t might b e noted, i s Simpson's well-known index of
dominance:
Another expression f o r Crow's I is:
This expression i s intriguingly analogous t o t h e formula f o r t h e Gini coefficient.
--
One of t h e entropy measures proposed by Thiel, namely
(where In denotes t h e natural logarithm) is a strictly-consistent measure of
unevenness.
--
The most commonly used "measure of evenness", Pielouss J n ,is not a consistent
measure of evenness. The measure J', given by
where H' is Shannon's measure of information ( o r entropy),
and
violates t h e replication principle. Consider, f o r instance, t h e following example. A population consists of t w o x's with 60%and 20% of t h e y 's, respectively. Another population consists of twenty x ' s , t h e f i r s t t e n having 8%of t h e
y 's each and t h e second t e n having 2%of t h e y 's each. The second population
clearly can b e c r e a t e d by replicating t h e f i r s t population t e n times. In both
populations t h e t o p half have 80%and t h e bottom half have 20%of t h e y 's, and
t h e concentration c u r v e s f o r t h e t w o populations are identical. However, J'
f o r t h e f i r s t population i s 0.72, whereas i t is 0.94 f o r t h e second population.
In general, f o r any distribution of y ' s among t h e x ' s , if t h e number of x's in-
creases but t h e concentration c u r v e remains t h e same (i.e., as a population i s
increasingly replicated), J' will asymptotically a p p r o a c h 1. I t might b e noted
t h a t although Thiel's entropy and J' are both simple transforms of Shannon's
measure of entropy, Thiel's entropy i s a strictly-consistent measure of evenness whereas J' violates t h e replication principle.
Pielou's J i s also not a consistent measure of evenness. J i s defined by
where H i s Brillouin's measure of information (or entropy),
where K i s t h e t o t a l number of y 's and ki i s t h e number of y 's of t h e i 'th x .
The formula f o r Hmax, which i s t h e maximum value H can attain, c a n b e found
in Pielou (1969, p. 233). I t i s not difficult t o show t h a t J violates both t h e relativity principle and t h e replication principle. Consider, f o r instance, t h e
following t h r e e populations:
1)
a population with two z 's, one with 5 y 's and t h e o t h e r with 1y .
2)
a population with two x 's, one with 50 y 's and t h e o t h e r with 1 0 y 's.
3)
a population with twenty x 's, half with 5 y 's each and half with 1y each.
The concentration c u r v e s f o r t h e s e t h r e e populations are identical and any
consistent measure of evenness should b e t h e same f o r all t h r e e . The value of J,
however, i s 0.60 f o r t h e f i r s t population, 0.64 f o r t h e second, and 0.92 f o r t h e
third. P e e t (1974) provides a n o t h e r example demonstrating t h a t J violates t h e relativity principle.
-
McIntosh's "index of evenness" (McIntosh 1967; Pielou 1969), which w e will
denote by Mc, i s not a consistent measure of evenness because i t violates t h e
replication principle. The index can b e expressed as:
A s a n example, consider a population which consists of two z 's, with 90% and
10%of t h e y ' s respectively. Suppose this population i s replicated t e n times
to produce a population in which t h e top ten x's each have 9% of t h e y 's and
t h e bottom t e n each have 1%.Evenness should b e t h e same in both cases, but
McIntosh's index i s 0.32 f o r t h e f i r s t population and 0.92 f o r t h e second.
SOURCES OF CONFUSION
The t h r e e measures t h a t are not consistent measures of evenness, J', J, and
Mc, were all derived by Pielou by standardizing a measure of diversity-Shannon's
entropy, Brillouin's entropy, and McIntoshBsindex of diversity, respectively--so
t h a t t h e standardized measure r a n g e s from zero. when one x h a s all t h e y ' s , to
one, when all x ' s have t h e same number of y 's. This approach i s unsatisfactory on
t h r e e counts.
First, standardization of a measure of diversity does not guarantee t h a t t h e
resulting measure will b e a consistent measure of evenness. If standardization i s
desired, t h e c o r r e c t approach is t o appropriately standardize a consistent measu r e of evenness: t h e resulting measure will then also b e a consistent measure of
evenness.
Second, standardizing a measure s o t h a t i t s range is from z e r o t o one does not
imply t h a t t h e measure itself is independent of N (i.e., t h e number of 2 ' s ) . Such
standardization merely implies t h a t t h e range of t h e measure is independent of N.
Pielou a r g u e s t h a t a measure of evenness should b e independent of N, but h e r
measures a r e not. The replication principle is a way of defining and operationalizing t h e idea t h a t evenness should not vary a c r o s s populations of different sizes N
t h a t a r e identical in t h e i r distribution of t h e y ' s . The measures J', J, and Mc all
violate t h e replication principle.
Third, a measure t h a t is standardized s o t h a t its r a n g e s t r e t c h e s from z e r o t o
one (or any o t h e r interval t h a t does not depend on N) will necessarily b e inconsistent with t h e replication principle, i.e., with t h e idea t h a t evenness should not
depend on population size. A population in which one z out of two h a s all t h e y 's is
more even than a population in which one z out of 20 h a s all t h e y 's because:
1.
according t o t h e replicat.ion principle, a population in which one z out of two
has all t h e y 's is just as even as a population in which 10 z's out of 20 have
all t h e y 's, and
2.
according t o t h e t r a n s f e r principle, a population in which 10 z's out of 20
have all t h e y 's is more even t h a t - a population in which one z out of 20 has a l l
t h e y 's.
Thus, a measure t h a t gives all populations t h e same value when one z dom-
inates cannot b e a consistent measure of evenness. On t h e o t h e r hand, a consistent
measure must always give t h e s a m e value t o populations t h a t are perfectly even
(because of t h e replication principle). A consistent, reasonable and intelligible
way t o standardize a measure of evenness is t o s e t t h e measure equal t o 1/N when
one z out of N h a s all t h e y ' s and s e t t h e measure equal t o 1 when all z's have
equal numbers of y 's.
The measure J, which is a standardized version of Brillouin's entropy H,
violates both t h e replication principle and t h e relativity principle. It violates t h e
relatively principle because it depends on t h e total number of y's. Pielou a r g u e s
t h a t J should be used f o r fully-censused collections whereas J', which i s a standardized version of Shannon's entropy H', should b e used f o r samples from very
l a r g e communities. S h e bases this position on t h r e e notions:
1.
In information theory, Shannon's entropy "is s t r i c t l y defined only f o r a n infini t e population", whereas Brillouin's entropy is a p p r o p r i a t e f o r messages of
finite length (Pielou 1969, p. 231). However, as Pielou notes, "analogies with
information t h e o r y
... d o not, of course, provide a compelling reason f o r using
H' and H in t h e way just outlined" (Pielou 1975, p. 10). Indeed, why should
t h e diversity o r evenness of a population b e measured t h e same way as t h e information content of a message o r code?
"A value of H i s determined from a complete census and hence i s f r e e of sta-
tistical e r r o r whereas a value of H' i s estimated
... and thus h a s sampling er-
r o r ; estimates of H' should always b e accompanied by estimates of t h e i r stand a r d e r r o r s " (Pielou 1975, p. 11). Any measure of evenness t h a t i s estimated
h a s a sampling e r r o r and could b e accompanied by a n estimate of i t s standard
e r r o r . Thus, Pielou's argument h e r e i s simply a n argument f o r calculating
standard e r r o r s and not a n argument in f a v o r any p a r t i c u l a r kind of measure.
"If, from a n indefinitely l a r g e community, w e t a k e two samples, one small and
one l a r g e , and treat both as collections, t h e small collection would b e expected t o have a lower value of H than t h e l a r g e collection. This r e s u l t a c c o r d s
with what w e intuitively r e q u i r e of a diversity index ..." (Pielou 1975, p. 11).
The underlying idea h e r e , as w e understand i t , i s as follows. Consider a community t h a t consists of a l a r g e number of different species (the x ' s ) , some of
which have l a r g e populations of individuals (the y ' s ) and some of which have
small populations. If a small sample i s taken, many of t h e rare species are
likely t o b e missed. Hence t h e diversity of t h e sample will tend t o b e l e s s than
t h e diversity of t h e e n t i r e community. This, however, does not imply t h a t a n
index of d i v e r s i t y - o r a n index of evenness-should
tend t o d e c r e a s e with t h e
size of t h e sample: indeed, such variation would violate t h e relativity principle. R a t h e r , t h e diversity ( o r evenness) of t h e sample should b e summarized
by a measure t h a t i s consistent with underlying principles. If i t i s desired, a n
estimate of t h e diversity of t h e e n t i r e community might then b e made, using
a p p r o p r i a t e statistical methods f o r drawing inferences about a universe from
a sample. I t might b e possible t o develop a short-cut a p p r o a c h t o estimating
t h e diversity of t h e e n t i r e community: in such a n a p p r o a c h , a measure of estimated diversity would have t o tend t o increase as sample size decreased, in
o r d e r t o counterbalance t h e tendency f o r small samples t o b e l e s s diverse
than t h e e n t i r e community.
DESIRABLE PROPERTIES OF MEASURES OF JCWNNESS
Although t h e "measures of evenness" most commonly used by ecologists are
not consistent with t h e f o u r principles of evenness o r t h e concentration-curve criterion, a variety of consistent measures of evenness exist. In choosing amongst
t h e alternatives, a n analyst might want t o consider how t h e measures compare according t o some desirable properties. W e consider five such p r o p e r t i e s below:
standardization, intelligibility, decomposibility, sensitivity, and robustness.
1. Standardization
A s discussed above, i t is often desirable t o use standardized measures of
evenness t h a t r a n g e in value from 1 / N when one z out of N has all t h e y ' s t o 1
when all x ' s have t h e same number of y's. Only one of t h e measures discussed
above has this p r o p e r t y , t h e haveall statistic. I t is not difficult, however, t o
standardize o t h e r measures.
-
The havehalf ranges from 1 / 2 N t o 1 / 2 . Thus, twice t h e havehalf (a measure
which w e will r e f e r t o as t h e double-havehalf) is a standardized measure of
evenness as is, more generally, z times any have-z statistic.
--
The Gini coefficient ranges from 1 - 1 / N t o 0. Hence t h e complement of t h e
Gini Coefficient (i.e., one minus t h e Gini coefficient) is a standardized measu r e of evenness. By analogy t o terms such as cosine and colog in which c o indicates complement, w e will call this measure t h e co-Gini index of evenness.
-
Crow's I ranges from N -1 ( f o r a population in which one z has a l l t h e y 's) t o
0 for- a perfectly even population. Hence,
is a standardized measure of evenness. This measure can also b e interpreted
as t h e inverse of N times Simpson's index of concentration. We will r e f e r t o i t
as t h e reciprocal-Simpson index of evenness.
--
Thiel's entropy v a r i e s from In N t o zero. One transformation t h a t might b e
used t o convert t h i s measure into a standardized measure of evenness is:
(where, by convention, z e r o times t h e log of z e r o is taken as z e r o and z e r o
raised t o t h e z e r o power is taken as one.) Buzas and Gibson (1969) proposed
this measure as a measure of evenness. It c a n also b e derived by raising e t o
t h e Shannon index and then dividing by N. We will call t h i s measure t h e
exponential-Shannon index of evenness.
It might b e noted t h a t concentration c u r v e s are standardized in t h a t t h e vertical and horizontal a x e s both r u n from 0 t o 1. Thus, i t is easy t o compare the
evenness of two populations merely by examining t h e i r concentration curves.
2. Intelligibility
A second desirable p r o p e r t y of measures of evenness is intelligibility. Ideal-
ly, a measure should b e easy t o comprehend, intuitively meaningful, simple t o explain t o o t h e r s , and naturally relevant t o t h e problems being addressed. Although
t h e r e is no disputing t a s t e , and intelligibility is clearly a matter of t a s t e and personal opinion, have-statistics, especially t h e havehalf and t h e haveall ( o r
havenone), achieve t h e s e goals f o r us b e t t e r than any o t h e r measures of evenness
w e are f a m i l i a r with.
Gini's coefficient h a s a simple geometric interpretation on a concentration
graph as t h e proportion of t h e area above t h e diagonal line t h a t lies between t h e
concentration c u r v e and t h e diagonal line. Y e t i t s biological interpretation is not
directly c l e a r . What does i t mean if t h e Gini coefficient is .3as opposed t o .4?
Simpson's index of dominance,
forms t h e c o r e of two of t h e indices discussed above, Crow's 1,
which is a measure of unevenness, and t h e "reciprocal-Simpson index",
which is a standardized measure of evenness. Simpson's index can b e i n t e r p r e t e d
as t h e probability t h a t two randomly selected y 's belong t o t h e same z,e.g., t h e
probability t h a t two individuals in a population belong t o t h e same species. This is
a helpful, ecologically-relevant interpretation, but unfortunately t h e i n t e r p r e t a tion pertains t o Simpson's index r a t h e r than t o t h e measures of evenness themselves. Suppose, f o r instance, t h a t t h e value of Crow's I w a s 9.26 and, correspondingly, t h a t t h e value of t h e reciprocal-Simpson measure w a s 0.097.
Without
knowledge of N, i t i s impossible to convert t h e s e values into t h e i r Simpson
equivalent and even if i t w a s known t h a t N w a s thirty-eight, say, t h e calculation of
t h e value of 0.27 of Simpson's index t a k e s a bit of effort.
Crow's I h a s a d i r e c t interpretation t h a t h a s some ecological meaning. Define
t h e "importance" of each z as t h e amount of y ' s t h a t z h a s and, similarly, define
t h e importance of each y as t h e total amount of y's t h e z t h a t h a s t h a t y has. Let
X b e t h e a v e r a g e of t h e f i r s t of t h e s e importance variables and let Y b e t h e a v e r a g e of t h e second. In t h e case, f o r example, of a population of females having
broods of children, X would b e t h e a v e r a g e brood size p e r female and Y would b e
t h e a v e r a g e brood size p e r child. Then i t can b e shown t h a t
If, as above, I i s 9.26, t h e n t h i s implies t h a t t h e a v e r a g e child h a s 10.26 times as
many siblings (including itself) as t h e a v e r a g e mother h a s children. Such a situation could a r i s e if most females have no children and if almost all children come
from l a r g e families. The relationship between X and Y implies t h a t in a stationary
population females on a v e r a g e only have 1/ (I+1) as many offspring as t h e i r own
a v e r a g e brood size. Hence, in t h e example given, t h e a v e r a g e child would have
less t h a n a tenth t h e offspring h e r mother had-perhaps
because more than nine-
t e n t h s of each b i r t h c o h o r t leaves no offspring. P r e s t o n (1976) provides a n interesting discussion of t h e relationship, f o r humans, between family sizes of child r e n and family sizes of women.
L a t t e r (1980) a r g u e s t h a t entropy "has many convenient p r o p e r t i e s from a
mathematical point of view, but i s extremely difficult t o i n t e r p r e t genetically". W e
have not been a b l e t o find any helpful biological interpretations of any of t h e various entropy measures and have not been a b l e t o develop much of a feeling f o r what
a Theil entropy of. say. 0.52 means.
3. D e c o m p o s i b i l i t y
Theil's entropy, like various o t h e r measures of entropy, does, however, have
t h e desirable p r o p e r t y of decomposibility. Foster (1985) calls decomposition "the
most useful p r o p e r t y of t h e Theil entropy measure".
Suppose a population is
comprised of s e v e r a l groups. Then, as explained by Foster and by Theil (1972), i t
i s possible t o calculate a "within-group" entropy and a "between-group" entropy.
The within-group entropy measures t h e a v e r a g e unevenness within t h e various
groups; t h e between-group entropy measures t h e unevenness of t h e distribution
where t h e group mean r e p l a c e s each group member's y value.
The desirable
f e a t u r e of TheilBsentropy is t h a t t h e value of Theil's measure f o r t h e e n t i r e population is simply t h e sum of t h e within-group and between-group measures.
A s discussed by Foster (1985), i t is also possible t o decompose Crow's I (and
some o t h e r measures described by Foster) into within-group and between-group
components, although t h e decomposition is somewhat complicated. Patil and Taille
(1982) also discuss a number of measures t h a t can b e decomposed. W e have not y e t
investigated whether i t i s possible t o find some useful decompositions based on
various have-statistics nor have w e explored t h e uses of decomposition in o u r studies.
4 and 5. S e n s i t i v i t y and R o b u s t n e s s
A measure is sensitive if i t responds t o changes in t h e underlying data. If t h e
d a t a are known t o b e a c c u r a t e , this is a desirable property. If, however, some of
t h e d a t a may b e in e r r o r , a robust measure t h a t is insensitive t o e r r o r s i s desirable. Hence, f o r some applications sensitive measures are p r e f e r a b l e and f o r oth-
er applications robust measures are indicated. Some measures are sensitive t o
d a t a in certain ranges-say
in t h e middle of t h e overall range-and
robust t o d a t a
in o t h e r ranges--say at t h e extremes. In investigating some biological questions, i t
may b e desirable t o use a measure t h a t i s sensitive t o prolific o r dominant x's but
robust t o changes in x's t h a t have little o r no y 's, but in o t h e r analyses t h e oppos i t e may b e t h e case--e.g., in studies where t h e r a r e species with small populations
are of g r e a t interest. A good introduction t o t h e concepts of sensitivity and
robustness, illustrated by a comparison of t h e mean (which is a sensitive measure),
t h e median (which i s robust) and t h e mid-mean, t h e mean of t h e middle half of t h e
d a t a values, (which is sensitive t o t h e middle r a n g e and robust t o t h e extreme ends
of t h e range), can b e found in Tukey (1978).
Strictly-consistent measures of evenness o r unevenness, like Gini's coefficient, Crow's I, o r Thiel's entropy, are more sensitive t o t r a n s f e r s of y's among
t h e z's than a r e consistent measures like t h e havehalf, haveall, o r halfhave. The
haveall is a n extreme c a s e because i t only depends on t h e proportion of z t h a t
have all t h e y 's: t h e distribution of t h e y 's among t h e s e x's is irrelevant. Similarly, o t h e r have-x and y-have statistics are insensitive t o c e r t a i n kinds of
t r a n s f e r s among t h e 2's. When field d a t a in ecological studies may b e subject t o
substantial e r r o r , this robustness of have-statistics may b e a valuable p r o p e r t y .
Although robust t o c e r t a i n kinds of t r a n s f e r s , have-statistics are sensitive t o
changes in t h e amount of y ' s any particular x has. A statistician might call t h e
havehalf t h e ".5 f r a c t i l e of t h e inverse right-hand concentration curve".
The
median is also a .5 f r a c t i l e (of a frequency distribution), but t h e havehalf differs
from the median in a key r e s p e c t . The median is a robust statistic t h a t will not
change in value if any of t h e values of t h e frequency distribution a r e changed, exc e p t f o r t h e one o r two middle values of t h e distribution.
The value of t h e
havehalf, on t h e o t h e r hand, changes if any single value of t h e underlying frequency distribution is altered. This sensitivity t o changes in any of t h e values of t h e
underlying frequency distribution holds f o r all t h e have-statistics e x c e p t t h e
haveall and i t s complement t h e havenone.
The sensitivity and robustness of different measures of evenness d e s e r v e s
f u r t h e r attention. A useful e x e r c i s e would b e t o do some empirical calculations
based on plausible changes and e r r o r s in a d a t a s e t , t o determine how different
measures respond.
Ecologists use t h e word diversity in two different senses, one broad and t h e
o t h e r narrower. In t h e b r o a d e r sense, diversity r e f e r s t o t h e differences among
individuals in a population. In i t s narrower, more technical meaning, which is typically used in studies of species diversity, diversity is defined as a measure t h a t
c a p t u r e s both "evenness" and "richness", where richness is a measure of how
many z B s(e.g., species) t h e r e are in t h e population o r community, t h a t is, t h e variable w e have been calling N. For example, Pielou (1975, p. 14) explains t h a t "the
diversity of a community depends on two things: t h e number of species and t h e
evenness with which t h e individuals are apportioned among them." It seems reasonable t h a t such a measure of diversity should satisfy t h r e e of t h e principles laid out
at t h e beginning of this p a p e r , namely t h e anonymity, relativity, and t r a n s f e r principles. Instead of t h e replication principle used f o r a measure of evenness, t h e
following replication principle might be used f o r a measure of diversity.
The Replication Principle for a Measure of D i v e r s i t y :
Suppose a population is replicated s o t h a t t h e r e are now m identical populations. Suppose t h e m populations are combined into a new population with m times
as many x ' s . The diversity of this new population should b e g r e a t e r than t h e
diversity of t h e original population. Furthermore, t h e bigger m is, t h e l a r g e r t h e
diversity should be.
This principle, plus t h e o t h e r t h r e e principles, implies t h a t a diversity measu r e D can b e considered a function of N (the number of 2 ' s ) and E , where E is
some consistent o r strictly-consistent measure of evenness.
(D, in these two
cases, might be called a consistent o r a strictly-consistent measure of diversity.)
The functional relationship is defined by:
where
and
That is, diversity should increase if e i t h e r richness N o r evenness E increases.
Numerous functions f satisfy these c r i t e r i a . but one seems particularly app r o p r i a t e , at least f o r ecological applications:
Diversity in this c a s e is simply richness times evenness, where richness is measu r e d by N and evenness is measured by any consistent o r strictly-consistent measu r e . If evenness is standardized t o Vary from 1 / N when one x dominates t o 1 when
a l l x ' s have equal y 's, then this measure of diversity v a r i e s from 1 t o N . Such a
measure of diversity c a n b e interpreted as t h e "equivalent number" (MacArthur
1965), "effective number", o r "even number" of z 's, i.e., t h e number of z 's t h a t in
a situation of complete evenness would produce t h e same diversity as t h e actual
diversity. For instance, consider a community of different species with differing
populations. If N, t h e number of species, i s 120 and E, t h e evenness. i s 0.5. then
t h e diversity would b e 60 and i t might b e said t h a t t h e community h a s a diversity
equivalent t o t h e diversity of a community with 60 equally numerous species.
The concept of diversity as t h e product of N and E i s s o natural t h a t t h e following principle may seem a p p r o p r i a t e :
The Proportional RepLication R i n c i p L e :
Suppose a population i s replicated s o t h a t t h e r e are now m identical populations. Suppose t h e m populations are combined into a new population with m times
as many z 's. The diversity of this new population should b e m times t h e diversity
of t h e original population.
This principle, t o g e t h e r with t h e o t h e r t h r e e principles, implies t h a t diversity
should be measured as t h e product of N and some measure of evenness. Such a
diversity measure might b e called a "proportional" measure.
MEASURES OF DIYERSITY
Numerous measures of diversity have been proposed and i t i s beyond o u r
scope to review more t h a n a f e w of t h e m o s t widely known measures as well as some
have-statistic measures. Among these measures t h e following distinctions can be
made:
-
N times a have-y measure i s a consistent, proportional measure of diversity.
Such a measure can b e i n t e r p r e t e d as t h e number of z's t h a t account f o r t h e
specified proportion of t h e y 's. Thus t h e N
t h a t have half t h e y 's. Twice t h e N
. havehalf
. havehalf
i s t h e number of z's
i s a standardized measure of
diversity t h a t v a r i e s from one t o N , and may b e i n t e r p r e t e d as t h e number of
x ' s in a n even population with t h e same diversity as t h e actual population. W e
will r e f e r to this measure as t h e double-halfhave measure of diversity.
--
N times t h e haveall i s a standardized, consistent, proportional measure of
diversity t h a t gives t h e number of x's t h a t have any y 's. F o r instance, in t h e
case of females having offspring, if t h e haveall i s .80 and N i s 50, t h e n 40 females had offspring. In studies of t h e diversity of species in a community,
where e v e r y species included h a s to have, by definition, at least one member,
the N
-
haveall i s simply equal to N:
in t h i s special case, this measure of
diversity coincides with t h e measure of richness.
--
Simpson's index of concentration i s a strictly-consistent, proportional measu r e of concentration and i t s inverse i s a standardized, strictly-consistent,
proportional measure of diversity, which might b e called t h e reciprocalSimpson index of diversity.
--
N times t h e complement of Gini's coefficient i s a standardized, strictlyconsistent, proportional measure of diversity, which might be called t h e coGini index of diversity.
--
Shannon's entropy,
i s a strictly-consistent measure of concentration, but i t i s not standardized o r
proportional.
-
By multiplying t h e exponential-Shannon index of evenness by N, t h e following
standardized, strictly-consistent, proportional measure of diversity, which
might b e called t h e exponential-Shannon index of diversity, can b e derived:
-
Brillouin's entropy i s not a consistent measure of diversity because i t violates
t h e relativity principle. P e e t (1974) h a s noted t h a t
"... t h e Brillouin
formula
does not provide a n acceptable index of heterogeneity"
--
The variance i s not a consistent measure of diversity because i t violates t h e
relativity principle.
APPLICATIONS
To illustrate some of t h e points made above about alternative measures of
evenness and diversity, w e provide t h r e e examples.
1. Lifetime Reproductive Success of Pale vs. Female Bullfrogs
Table 1 p r e s e n t s various summary statistics pertaining to lifetime reproductive success of male vs. female bullfrogs.
The original d a t a are from Howard
(1983); in P a r t I of this s e r i e s of working , p a p e r s , in Figure 9 , concentration
c u r v e s were drawn based on t h e s e data. Scrutiny of t h e two c u r v e s and t h e various summary statistics might help t h e interested r e a d e r form his or h e r own judgments of t h e merits of concentration c u r v e s and of different summary statistics.
2. Diversity ina Community of Herbaceous Plants
Table 2 i s similar to Table 1 and has a similar purpose. I t p r e s e n t s various
summary statistics pertaining to t h e diversity of herbaceous plants in a deciduous
woodlot, as described in P a r t 2 of this s e r i e s of working p a p e r s , in conjunction
with Figure 1. Note t h a t whereas Table 1 includes various measures of evenness,
Table 2 p r e s e n t s a l t e r n a t i v e measures of diversity. Only those measures of diversity t h a t were discussed above and t h a t s e e m particularly useful are included. All
of t h e standardized measures of diversity given in t h e table correspond to N times
a standardized measure of evenness.
3. Birds of a Feather
An example of t h e use of summary measures of evenness f o r ecological
analysis i s Payne and Payne's (1977) comparison of t h e distribution in mating suc-
cess of m a l e b i r d s in different mating systems. Payne and Payne a r g u e t h a t "mating systems and t h e statistics of mating success among males are closely related"
and t h a t measures t h a t summarize t h e evenness in t h e distribution of mating suc-
cess among individual b i r d s are useful tools in describing and comparing t h e mating systems of populations (Payne and Payne 1977. p. 165).
Payne and Payne use t h r e e p a r t i c u l a r measures of "evenness" in t h e i r study,
namely t h e coefficient of variation, Pielou's evenness index and t h e coefficient of
skewness. A s w e have noted, t h e coefficient of variation (often abbreviated CV),
which i s t h e s q u a r e r o o t of Crow's I. i s a strictly-consistent measure of uneven-
Table 1. Summary statistics f o r t h e predicted lifetime reproductive success of
male and female bullfrogs.
Standardized measures of evenness:
Males
Females
Haveall
.31
.46
Double-Havehalf (2 . Havehalf)
.13
.20
Co-Gini (l-Gini)
.18
.26
.18
.28
.23
.34
Reciprocal (1/N
Simpson)
Exponential-Shannon (e Shannon Index
1N )
Havequarter
Havehalf
Quarterhave
Halfhave
Havenone
Measuresof unevenness:
Crow's Index
Gini's Coefficient
Thiel's Entropy
ness. Payne and Payne (1977, p. 167) note "monogamous and polygynous species
overlap p e r h a p s less in Ws. and this may b e t h e single statistic best describing
t h e distribution of mating success in different mating systems." Pielou's evenness
index h a s already been discussed; i t i s not a consistent measure of evenness. The
coefficient of skewness is also a defective measure of evenness because i t does not
satisfy t h e t r a n s f e r principle. For example, t a k e t h r e e populations of t h r e e x's
each. In t h e f i r s t population t h e distribution of y among t h e x's is 1.7.7. In t h e
second population i t is 4,5,6 and in t h e t h i r d i t is 1.1,13. The f i r s t population h a s a
skewness of -.707, t h e second a skewness of 0 and t h e t h i r d a skewness of .707. But
t h e t r a n s f e r principle implies t h a t i t i s t h e middle population t h a t is most even.
Table 2. Summary statistics of diversity of herbaceous plant species in a deciduous woodlot.
Standardized measures of diversity
(i.e. Equivalent numbers of species)
Value
Haveall (richness; number of species)
-
Double-Havehalf (2 N . Havehalf)
Co-Gini (N . (1-Gini))
Reciprocal-Simpson (l/Simpson)
Exponential-Shannon (e Shannon Index )
Other measures of diversity:
Shannon Index (Entropy)
Simpson's Dominance Index (of lack of diversity)
Table 3 is styled a f t e r Payne and Payne's presentation and uses a n accessable
subset of t h e i r d a t a sources. However, w e relied on o u r own statistical calculations. For details of t h e specific d a t a sets t h e r e a d e r should r e f e r t o Payne and
PayneOsa r t i c l e and t h e s o u r c e material.' W e present a variety of measures of
evenness f o r t h e birds of different species, but omit Pielou's index and t h e coefficient of skewness because they a r e not consistent measures of evenness. W e have
not included t h e havehalf as i t is simply one half of t h e double-havehalf.
The species of male birds are placed in t h r e e categories:
A.
Those which generally form lek o r dispersed lek mating systems in which males
display, but form no p a i r bond and provide no parental c a r e ;
B.
Those which form mating systems in which males are sometimes polygynous,
form p a i r bonds and may provide some parental c a r e , and
'A c a u t i o n a r y n o t e is i n o r d e r . As w a s d e m o n s t r a t e d i n P a r t I, t h e c o n c e n t r a t i o n of r e p r o d u c t i o n
c h a n g e s depending o n t h e s t a g e of r e p r o d u c t i o n c o n s i d e r e d and w h e t h e r a l l , o n l y t h e r e p r o d u c t i v e l y v i a b l e o r o n l y t h e r e p r o d u c t i v e l y s u c c e s s f u l a n i m a l s a r e included. Also, c o n c e n t r a t i o n is reduced w i t h t h e u s e o f a v e r a g e d d a t a . T h e n a t u r e o f t h e d a t a m u s t o b v i o u s l y b e c o n s i d e r e d b e f o r e
a n y s u b s t a n t i v e c o n c l u s i o n s c a n b e drawn. T h e p u r p o s e of t h i s e x a m p l e is i l l u s t r a t i v e .
Table 3. Distribution of mating s u c c e s s among male b i r d s f r o m s p e c i e s w i t h d i f f e r e n t mating s y s t e m s .
STANDARDIZED MEASURES OF EVENNESS
Number of
mdulL
melem
Mean no. mmLlngm,
mmLes o r
fledgllnga/mdulL
DoubleHevmhmU
HmvmeII
Co-Glnl
Reclprocel
Slmpmon
ExponenLlel
Shannon
0.100
0.288
0.143
0.132
0.175
Crow'.
Index
Glnl'a
Coeff.
Thlel's
Entropy
Have
quarter
Have
none
HeU
have
QumrLer
hmve
Reference
Mmnecus menecus
(WhlLe-bemrded mmnekln)
0.032
0.750
1.000
1.000
L l l l 1974
(LekB.Groupl.1967)
Ymnacus menecue
(WhlLe-beerded mmnekln)
0.034
0.200
0.989
0.908
L l l l 1974
(LekB,1968)
TelmmLodyLes peluaLrle
(Mmreh wren)
0.087
0.240
0.830
0.545
Verner 1965
par aeeson
Agelmlus phoenlceus
(Red-wlnged blmckblrd)
0.084
0.039
0.779
0.529
Holm 1973
*(Herem evermgea)
Agelelue phoenlceus
(Red-winged blmckblrd)
0.114
0.019
0.789
0.472
Holm 1973
*(Herem evermgea)
Vldum chmlybemLe
(Indlgo)
14
1.0 meLlngs
observed
Vldue chmlybemLe
(Indlgo)
Tympenuchus cupldo
(Prmlrle chlcken)
Lyrurus LeLrlx
(Black grouse)
Plprm eryLhrocsphela
(Colden-headed manekln)
Melosplem melodle
(Song sparrow)
Agelmlua phoenlceus
(Red-winged blmckblrd)
B
53
3.0 female
metes
0.813
0.981
0.714
0.794
0.670
0.259
0.288
0.140
O.lNO
0.019
0.709
0.425
Holm 1973
Legopus legopus
(Red grouse)
C
72
5.2 slae o f
fled~ed
broods
0.839
0.917
0.703
0.784
0.834
0.278
0.297
0.161
0.144
0.013
0.712
0.406
Janklne eL a1 1963
(1960,hlghlenda)
Agelmlus phoenlceus
(Red-wlnged b l e c k b l r d )
B
51
2.7 female
0.641
0.981
0.725
0.801
0.869
0.248
0.275
0.140
0.135
9.039
0.899
0.420
Holm 1973
Legopus lagopus
(Red grouss)
C
0.667
0.988
0.742
0.628
0.889
0.207
0.258
0.117
0.149
0.014
0.884
0.394
Jenklns eL e l 1963
(1960,lowlenda)
MATING SYSTEMS.
a k a
74
5.0 slee of
f ledled
broods
-
A. L e k s mnd dlapersed l e k s males dlaplmy. but f o r m no pmlr bonds end p r o v l d e no perenLel cmre.
8. Polygynous male f o r m p 8 l r bonds mnd mmy provlde some perenLml cmre.
C. Monogemous mmles hmve w e l l developed pmir bonds, mmles and femmlem p r o v l d e pmrenLml care.
-
C.
Those in which males f o r m p a i r bonds, are generally monogamous and both
males and females c a r e for t h e young.
These classifications are generally consistent with a t r e n d f r o m extreme po-
lygamy to monogamy.
W e have o r d e r e d t h e birds, according to t h e i r double-
havehalf measures, f r o m those populations in which individual mating success i s
m o s t concentrated to those in which it is least concentrated. A s suggested by
Payne and Payne, t h e r e i s a tendency f o r mating success to b e progressively more
evenly distributed in systems moving f r o m those which are extremely polygamous
to those which are monogamous.
N o t e t h a t all of t h e various measures of evenness rank t h e species in more or
less t h e s a m e o r d e r . Consequently, any one of t h e measures could b e used to draw
Payne and Payne's conclusion. If a single measure were to b e used, it s e e m s to us
t h a t t h e havehalf o f f e r s t h e advantage of being simple t o explain and easy to
comprehend. If t w o measures were to be used, t h e havehalf and t h e haveall provide, at least f o r us, m o r e readily intelligible information t h a t any o t h e r p a i r of
measures.
CORRELATIONS BETWEEN MEASURES OF EVENNESS
The comparison of mating success of b i r d s p r e s e n t e d above suggests t h a t t h e
various measures of evenness and unevenness are highly intercorrelated.
To
check t h i s conjecture, w e calculated t h e correlation (as measured by Pearson's
r2) between e a c h possible p a i r of t h e measures. Table 4 shows t h e results. N o t e
t h a t w e grouped t h e double-havehalf and t h e havehalf together and w e grouped t h e
co-Gini and Gini measures t o g e t h e r because each of t h e s e twin measures h a s identical correlations with t h e o t h e r measures.
All t h e measures a r e highly c o r r e l a t e d with all t h e o t h e r measures. This sug-
gests t h a t they all are providing m o r e or less t h e s a m e information. Indeed, t h e set
of alternative measures c a n be reduced even f u r t h e r than t h e high correlation
coefficients imply. Although t h e reciprocal-Simpson measure and Crow's I are not
perfectly c o r r e l a t e d with e a c h o t h e r , these t w o measures are simply transformations of e a c h o t h e r and e a c h one i s completely determined by t h e o t h e r . I t i s only
because t h e function linking t h e t w o measures i s not a linear function t h a t t h e
correlation between t h e measures i s not one. Similarly, t h e exponential Shannon
measure and Thiel's entropy are deterministic transformations of each o t h e r .
Table 4. Correlation coefficients (for Pearson's r2)f o r evenness measures i n Table 3.
Ihveall
Double-Havehalf A-hvehalf
Haveall
Co-Cini /Gini
Reciprocal Sinpson
Exponential Shannon
Crow's I
Thiel's Entropy
Havequar ter
Ihlfhave
,
.784
Co-Gini Gini
Reciprocal
Exponential
Si q s o n
Shannon
Crow's I
Thiel's
tfav e
tlalf
Quarter
Entropy
quarter
have
have
When t w o measures are highly c o r r e l a t e d or are perfectly determined by each
o t h e r , a choice between t h e measures can b e based on considerations of convenience, intuitiveness, comprehensibility, explainability, and t h e like. Just as saying
a glass is half-full may convey a different v e c t o r of meaning than saying t h e glass
i s half-empty, use of t h e haveall measure r a t h e r than t h e havenone measure may
highlight a different a s p e c t of evenness in a population. Thus, even in t h i s simple
case of t w o complementary measures, w e think t h a t a c a r e f u l analyst should devote
some attention to considering t h e m o s t a p p r o p r i a t e measure to use t o p r e s e n t
information--and p e r h a p s decide to p r e s e n t both measures. Tversky and Kahneman
(1981) and Vaupel (1982) provide f u r t h e r discussion of statistical insinuation and
implicational honesty in t h e use of alternative measures.
W e also used d a t a from Lutz (1985) t o investigate a n o t h e r d a t a s e t , pertaining
to t h e concentration of reproduction among human females in 4 1 different count r i e s . The correlations between t h e various p a i r s of evenness and unevenness
measures are displayed in Table 5. I t i s interesting to note t h a t all t h e measures,
with t h e exception of t h e haveall statistic, are highly c o r r e l a t e d . The haveall
measure a p p e a r s to provide a n o t h e r dimension of information. In P a r t 1 of this
s e r i e s of p a p e r s , we frequently found t h e haveall statistic ( o r i t s complement, t h e
havenone) b e useful in addition t o t h e havehalf measure; t h e havehalf plus t h e
haveall generally seemed to b e t h e m o s t informative p a i r of statistics.
CONCLUSION
In t h e t h r e e p a r t s of t h i s s e r i e s of p a p e r s w e have strived to persuasively
make a single, simply-stated point: concentration c u r v e s and various associated
have-statistics are useful in ecological analyses of diversity. In P a r t I , w e provided s e v e r a l examples of how concentration c u r v e s and have-statistics could b e used
to analyze evenness in reproductive success. In P a r t 11, w e extended t h e approach
t o species diversity and some r e l a t e d topics. Finally, h e r e in P a r t 111, w e comp a r e d have-statistics with o t h e r measures of evenness and diversity, both according t o some general principles and as applied to some specific examples.
Diversity, heterogeneity, variety, and inequality in populations is a v a s t subject, at t h e h e a r t of ecology, demography, and t h e life sciences more generally.
One approach to this subject i s to study population concentrations. What proportion of t h e population h a s a l l t h e offspring, what proportion of t h e species accounts f o r half t h e total biomass, what proportion of t h e matings are attributable
t o t h e m o s t successful q u a r t e r of t h e males? These are important questions t h a t
are directly addressed by concentration c u r v e s and have-statistics.
Table 5. Correlation coefficients (for Pearson's r 2 ) for evenness measures, World Fertility Survey.
Fhveall
Double-Fhvehalf A-Iavehalf
.816
Co-Gini G n i
.959
Reciprocal
Exponential
Simpson
Shannon
.967
.884
Crow's I
.926
Thiel's
Have
Half
Quarter
Entropy
quarter
have
have
.849
.971
.9 18
.993
I
Eiaveall
.485
.462
.669
.483
.689
.251
.498
.222
.976
.959
.942
.948
.888
.985
.947
.936
.918
.886
.953
.775
.932
.851
-819
.982
g
I
Co-Gi ni G ni
Reciprocal Sirrpson
Exponential Shannon
Crow's I
Thiel's Entropy
Havequar ter
Half have
.897
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