Equitability measures (continued)

Pielou pointed out that Simpson's index and Shannon's index are not appropriate measures of equitability if 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. So she suggested Brillouin's index (H) for calculating equitability 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