Hill (1973a) and Pielou (1975) noted that the three diversity indices mostly used by ecologists are specific cases of the generalized entropy formula of Renyi (1961)
where: a = 0, 1, ...; q is the number of species; pi is the relative frequency or proportion of species i. Hill (1973a) prefers the corresponding diversity numbers:
It can be shown that the first three entropies (order 0 to 2) and corresponding diversity numbers are:
(a) H2 = -log ^ p2 = -log (concentration) (b) N2 = concentration 1 (6.35)
Hill (1973a) noted that increasing the order a diminishes the relative weights of rare species in the resulting index. In a review of the topic, Peet (1974) proposed other ways of creating families of diversity indices. Let us examine the first three orders of eq. 6.31 in more detail.
Number l) Entropy of order a = 0 — The number of species q (eq. 6.33b) is the index of of species diversity most often used in ecology. It goes back to Patrick (l949):
It is more sensitive to the presence of rare species than higher-order indices. The number of species can also be seen as a component of other diversity indices (e.g. H, Subsection 2).
As the size of the sampling units increases, additional rare species appear. This is a problem with all diversity indices and it is at its worst in eq. 6.36. It is incorrect to compare the diversities of sampling units having different sizes because diversity measures are not additive (Subsection l.4.2). This point has been empirically shown by He et al. (l996). This problem can be resolved by calculating the numbers of species that the sampling units would contain if they all had the same size, for example l000 organisms. This may be done using Sanders' (l968) rarefaction method, whose formula was corrected by Hurlbert (l97l). The formula computes the expected number of species q' in a standardized sampling unit of n' organisms, from a nonstandard sampling unit containing q species, a total of n organisms, and ni organisms belonging to each species i:
i = l where n' < (n _ nl), nl being the number of individuals in the most abundant species (yl), and the terms in parentheses are combinations. For example:
Shannon's 2) Entropy of order a = 1 — Margalef (l958) proposed to use Shannon's entropy H
entropy (eqs. 6.l and 6.34a) as an index of species diversity.
The properties of H as a measure of diversity are the following:
• H = 0 (minimum value), when the sampling unit contains a single species; H increases with the number of species.
• For a given number of species, H is maximum when the organisms are equally distributed among the q species: H = log q. For a given number of species, H is lower when there is stronger dominance in the sampling unit by one or a few species (e.g. Figs. 6.1a and b). The actual value of H depends on the base of logarithms (2, e, 10, or other). This base must always be reported since it sets the scale of measurement.
• Like the variance, diversity can be partitioned into different components. It follows that the calculation of diversity can take into account not only the proportions of the different species but also those of genera, families, etc. Partitioning diversity into a component for genera and a component for species within genera allows one to examine two adaptive levels among the environmental descriptors. Such partitioning can be done using eqs. 6.10-6.12. Total diversity, H = A + B + C, which is calculated using the proportions of species without taking into account those of genera, is equal to the diversity due to genera, H (G) = A + B, plus that due to species within genera, H(S | G) = C, which is calculated as the sum of the species diversities in each genus, weighted by the proportions of genera. The formula is:
This same calculation may be extended to other systematic categories. Considering, for example, the categories family (F), genus (G), and species (S), diversity can be partitioned into the following hierarchical components:
Using this approach, Lloyd et al. (1968) measured hierarchical components of diversity for communities of reptiles and amphibians in Borneo.
Most diversity indices share the above first two properties, but only the indices derived from eq. 6.31 have the third one (Daget, 1980). The probabilistic interpretation of H refers to the uncertainty about the identity of an organism chosen at random in a sampling unit. The uncertainty is small when the sampling unit is dominated by a few species or when the number of species is small. These two situations correspond to low H.
In principle, H should only be used when a sample is drawn from a theoretically infinite population, or at least a population large enough that sampling does not modify it in any noticeable way. In cases of samples drawn from small populations, or samples whose representativeness is unknown, it is theoretically better, according to Pielou (1966), to use Brillouin's formula (1956), proposed by Margalef (1958) for computing diversity H. This formula was introduced in Section 6.1 to calculate the information per symbol in a message (eq. 6.3):
where the « is the number of individuals in species i and n is the total number of individuals in the collection. Brillouin's H corresponds to sampling without replacement (and is thus more exact) whereas Shannon's H applies to sampling with replacement. In practice, H computed with either formula is the same to several
Concentration decimal places, unless samples are so small that they should not be used to estimate species diversity in any case. Species diversity cannot, however, be computed on measures of biomass or energy transfer using Brillouin's formula.
3) Entropy of order a = 2 — Simpson (1949) proposed an index of species diversity based on the probability that two interacting individuals of a population belong to the same species. This index is frequently used in ecology. When randomly drawing, without replacement, two organisms from a sampling unit containing q species and n individuals, the probability that the first organism belong to species i is n,/n and that the second also belong to species i is (n - l)/(n - 1). The combined probability of the two events is the product of their separate probabilities. Simpson's concentration index is the probability that two randomly chosen organisms belong to the same species, i.e. the sum of combined probabilities for the different species:
Concentration = > — —i—----- = L=-----——
When n is large, ni is almost equal to (ni - 1), so that the above equation becomes:
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