Equitability measures

SIMPSON"S INDEX

Assume we have a collection of N individuals w/ s spp. and N_i individuals in the i^th species

Select 2 individuals at random: if Prob [indivs. are of same sp.] is high, then community is not diverse

Random sample w/o replacement:	c = [Ni(Ni-1)]/[N(N-1)], where N = total pop'n size and Ni = indivs in ith species
w/ replacement:	= (Ni/N)2 Ni/N represents a relative abundance value --> pi (which = Ni/N, proportion of indivs. of ith spp.) is often used
	this is the most common expression of Simpson's index (note: it is a biased estimator)
	c is an unbiased estimator which assumes only N indiv. of s spp. (doesn't assume we're "sampling" entire pop'n)

e.g.,:	Community I has 5 species w/ 10 individuals each
	Community II has 1 species w/ 42 indivs, 4 species w/ 2 indivs each cI = cII = I = II =

c = Prob[2 randomly-selected individuals are of the same species]

probability loses interpretability w/ non-count data (e.g., basal cover) unless they are converted to relative abundance values (i.e., p_i)

Note that increased diversity --> decreased Prob[2 indivs. being of same spp.]

therefore, 1-c is sometimes used to express diversity

Also, 1/ = no. of equally abundant spp. necessary to produce same diversity as that observed in the sample

e.g., 1/_I = 1/0.2 = 5.0 species

1/_II = 1/0.712 = 1.4 species

c is most affected by the importance values of the first few dominant spp.

e.g., 0.9² = 0.81 (common sp.)

0.1² = 0.01 (rare sp.)

the latter sp. contributes little to the sum

thus, Simpson's has been called a dominance index (affected primarily by few dominant spp.)

c is relatively insensitive to sampling variability if sample is adequate to represent dominant spp.

1. all spp. in pop'n are represented in sample

2. there is a pop'n of size N from which we can draw an infinite number of samples w/o replacement (i.e., pop'n is infinitely large)

3. these samples represent random samples of the pop'n

Assumptions identical to Simpson's index

Prob[selecting an indiv. of i^th sp.] = N_i/N

Prob[selecting 2 indivs. of i^th sp.] = (N_i/N)²

Prob[selecting all N_i indivs. of i^th sp.] = (N_i/N)^N_i

Prob[selecting all N_i indivs. of all s spp.] = p = (N_i/N)^N_i

= (N₁/N)^N × (N₂/N)^N × ... × (N_s/N)^N

H' = -ln p = N(lnN) - [N_i(lnN_i)]

H' is Shannon's index

e.g.,	Species	Collection I	Collection II
	1	6	12
	2	3	6
	3	1	2
	Total	10	20

note that richness and proportions are identical in the two collections

H_I' = 10ln10 - (6ln6 + 3ln3 + 1ln1) = 8.979

H_II' = 20ln20 - (12ln12 + 6ln6 + 2ln2) = 17.958

therefore H' = f(n), an undesirable characteristic

use proportions of spp.:

H_I' = H_II' = 1ln1 - (.6ln.6 + .3ln.3 + .1ln.1) = 0.8979

but ln1 = 0, so H' simplifies to H' = -p_ilnp_i, where p_i = N_i/N

e^H' is interpreted as the number of equally abundant spp. necessary to produce the same diversity (H') as observed in sample

e.g., from previously examples

Community I has s = 5, N₁=N₂=N₃=N₄=N₅=10

Community II has s = 5, w/ N_i = 42 and N₂=N₃=N₄=N₅=2

H_I' = -(10/50)ln(10/50) × 5 = 1.6094

H_II' = -{(42/50)ln(42/50) + [(2/50)ln(2/50) × 4]} = 0.66147

e^H' = e^1.6094 = 5

e^H' = e^0.66147 = 1.937

Behavior of H'

1. H' is sensitive to changes in rare spp.

2. H' is logarithmically related to s

Whittaker preferred H' for ease of calculation and interpretation, esp. compared to indices developed later

Both Simpson's index and Shannon's index are widely use to describe communities

Pielou (mathematical ecologist) does not believe that Shannon's index has any theoretical basis, because plants are not distributed randomly. H' (like c) assumes:

1. well-defined "parent" pop'n that can be sampled at random

2. all spp. in pop'n are represented in sample

Pielou: perhaps the sample (i.e., quadrat) is not a sample of "something bigger" (e.g., pop'n), but rather a measurement of something that exists on a local scale

This measure is to be interpreted on the basis of pop'n, where sample = pop'n

There is no way to normalize H

e.g.,	Abundance of	Community A	Community B
	Species 1	3	6
	Species 2	4	8
	Species 3	3	6
		s=3, N=10	s=3, N=20

same proportions in each collection (i.e., same richness-abundance relations)

H_A =

H_B =

Thus, H is dependent on N

"Normalized" H_A' = H_B' =

A further disadvantage of H is that it can "behave" strangely

e.g.,		Community A:	s=10, N=50; N1=N2=...=N10=5
		Community B:	s=9, N=1000; N1=N2=...=N8=110, N9=120
		HA =
		H'A = -pilnpi =
		HB =
		H'B = -pilnpi

Thus, the collection w/ more spp. and greater evenness is A, as reflected by H' but not by H

H is rarely used (despite the mathematical advantages), because:

1. it depends on N

2. it behaves strangely, sometimes

3. it is difficult to employ

The rarity w/ which H is used reflects the idea that the decision of which index to use must be based on more than theoretical grounds--ease and interpretability of application are important

McIntosh's Index is based on a geometric model:

u = (N_i²)

e.g., N=7; N₁=3, N₂=4

u = (N_i²) = (3² + 4²) = 5 = length of vector from origin

e.g., N=10; N₁=2, N₂=3, N₃=5

u = (N_i²) = (2² + 3² + 5²) = 6.1644

Interpretation:	For a given N, u will be maximum when all individuals belong to one species (in this case, u = N); i.e., monoculture, w/ minimum diversity, has maximum u
	For a given N, u will be minimum when each individual belongs to a different species (in this case, u = N); i.e., maximum diversity has minimum u

From this, McIntosh defined an equitability index:

M = N - u

An obvious disadvantage is that M = f(N); however, M can be normalized (Peet 1974):

M' = M/M_max = (N-u)/(N-N)

But M' behaves strangely:

e.g., Collection A: s=3, N=10, N₁=2, N₂=3, N₃=5

Collection B: s=3, N=20, N₁=4, N₂=6, N₃=10

These collections have identical richness and diversity relations, but have different M':

M'_A =

M'_B =

A transformation can be applied to correct this problem (Pielou):

M'' = M/[N-(N/s)]

For our example, M_a =

M_b =

Note M'' = f(evenness) only; richness (s) does not affect M''

0 M'' 1

Other equitability measures proposed by Pielou and commonly used in the early 1970's (but not much now):

J' = H'/H'_max, where

H' = Shannon's index and

H'_max = ln(s) (i.e., J' = observed diversity/max. diversity for that no. of spp.)

J = H/H_max, where

H = Brilloun's index and

H_max =

Disadvantages:

Dependent on sample area or number of species (and number of species must be known)

Division by H'_max or H_max, which was intended to "adjust" for richness, does not:

Alatalo (1981, Oikos 37:199-204) showed that J' and J increase for purely mathematical reasons w/ increasing richness

given s species, w/ half equally abundant and the other half "indefinitely rare" (Alatalo doesn't define this):

	s	2	4	50	100	1000
	J	0	.5	.7	.82	.90

Thus, although dominance-diversity relations are constant, J ranges from 0 to 1 depending on richness of sample

Alatalo proposed an equitability index, E

E =

E does not require an estimate of richness

Dominance/diversity models

Graphics

The number of species in a community and the relative abundance of each species are integral and intrinsic properties of communities. These properties lead to theories about aspects of community organization.

Now, the question that emerges is: where do these diversity relationships come from?

Niche pre-emption model

Suppose the percent of total available resources used by a species is determined by the species' success in pre-empting for its own use some portion of available resources

Less successful spp. utilize resources that are left. And so on, for all spp.

Graphically, w/ k=0.5:

In model: let	I1 = importance value of most successful sp.,
	I2 = I.V. of 2nd-most important sp., and so on ...
	Is = I.V. of least successful sp.
	Ii = N

e.g., given k = 0.4:

Sp. i	Resources available	Resources pre-empted by sp. i	Ii/Ii-1 = c	k = 1-c = Ii/res. avail.
1	100	40	--	--
2	60	24	24/40 = 0.6	24/60 = 0.4
3	36	14.4	14.4/24 = 0.6	14.4/36 = 0.4
4	21.6	8.64	8.64/14.4 = 0.6	8.64/21.6 = 0.4
5	12.96	5.184	0.6	0.4
.
.
.
s

= 100

Ii = Nk(1-k)i-1; e.g.,	I1 = 100(.4)(1-.4)1-1 = 40
	I2 = 100(.4)(1-.4)2-1 = 24

Note Ii = Nk(1-k)i-1 = I1ci-1; e.g.,	I1 = 40(.6)1-1 = 40
	I2 = 40(.6)2-1 = 24

This model predicts that spp. importance values look like this:

Question: do they really?

Conclusion: this model appears to represent reality in:

1. coniferous spp. of low species richness; e.g.,:
   • Frasier fir forest (GSMNP)
   • pine-heath forest
   • pygmy conifer-oak scrub type (AZ)

2. early-successional stages of old-field succession

3. may be appropriate for some strata not very rich in spp. even though community as a whole is not geometric (i.e., strata w/in a community)

4. in "severe" environments

In these situations, dominance tends to be strongly developed, and spp. may be related in resource use thru strong competitive interactions

More graphics

If the relative abundances of spp. are governed by relatively independent factors, then we might expect the relative importances of spp. to be approx. normally-distributed

Note that old-field succession tends to: [Whittaker 1972, Taxon 21:213-251]

geometric -------> log-normal -------> increased dominance by

trees -------> steepening of log-normal distribution

According to May (1975 in Cody and Diamond, eds., Ecology and Evolution of Communities):

... the lognormal distribution is associated with [results] of random variables, and factors that influence large and heterogeneous assemblies of species indeed tend to do so in this fashion.

... if the environment is randomly fluctuating, or alternatively as soon as several factors become significant ..., we expect the statistical Law of Large Numbers to take over and produce the ubiquitous lognormal distribution.

Colinvaux (1986 Ecology text, p. 676):

Log-normal populations result when the abundance of each species pop'n is determined at random relatively independently of other species populations. The prevalence of log-normal distributions shows that the relative distributions of animals and plants very often is determined by random processes.

Thus, interactions appears to be less important in the lognormal model than in the niche pre-emption (geometric series) model

The log-normal model may be expected in relatively spp.-rich communities where many factors (which are largely independent) influence spp. performance

Broken-stick model

P_r = (N/s) [1/(s-i+1)], where

s = no. spp.,

N = no. individuals,

i = spp. sequence from least to most important, and

P_r = I.V. of sp. n in sequence from least important (i=1) thru sp. in question

e.g., given N=100 individuals of s=3 spp.,

Spp.	Pr
1	(100/3)[1/(3-1+1)] = 11.1
2	(100/3)[1/(3-1+1) + 1/(3-2+1) = 27.8
3	(100/3)[1/(3-1+1) + 1/(3-2+1) + 1/(3-3+1)] = 61.1
Note Pr =	100

Acc. to May (1975), this model is applicable to communities comprising a limited number of taxonomically similar spp. ... (Colinvaux 1986 refers to spp. of a guild)

... in competitive contact w/ each other in a relatively homogeneous habitat; in this situation, a more structured pattern of relative abundances of spp. (other than that resulting from conditions associated w/ log-normal distribution) may be expected. The basic picture is one of intrinsically even resource division of some major environmental resource. In this model, species are limited by competition at randomly located boundaries.