## The Index

The Simpson index has a number of forms. The original, and simplest, is

where pi is the proportion of the ith species in a sample, chosen at random (with replacement). The index A has a natural interpretation: it is the probability that any two individuals from the sample, chosen at random, are from the same species (A is always <1). It is a 'dominance' index, in the sense that its largest values correspond to assemblages whose total abundance is dominated by one, or a very few, of the species present. Dominance is the complement of evenness, so the complement of A,

is thus an equitability or 'evenness' index, taking its largest value (of 1 ā Sā , where S is the number of species in the sample) when all species have the same abundance. This also has a natural interpretation, as it is the probability that any two individuals, drawn at random, are

10 100 1000 10000 100000

10 100 1000 10000 100000

(b) Richness (Margalef's d)

10 100 1000 10000 100000

0.75

0.55

0.35

Evenness (Pielou's J1)

0.65

0.55

0.45

100 1000 10000 100000 10

Simpson (1-A1)

Evenness (Pielou's J1)

0.75

0.55

0.35

Simpson (1-A1)

0.65

0.55

100 1000 10000 100000

100 1000 10000 100000

Figure 1 Values of four diversity indices (y-axis) for simulated samples of increasing numbers of individuals (x-axis, note log scale) drawn randomly without replacement from a large data set of 140344 macrobenthic organisms from the Bay of Morlaix, France. Note that the mean value for the Simpson index 1 - A' (d) is sample-size independent, whereas it is not for numbers of species (S), Margalef's d, and Pielou's J' (a-c).

from different species. An alternative is to take the reci- The slightly revised forms, procal, which gives another index, one of the family of

Hill's diversity numbers (M). A' = { ^Ni(N, - 1)}/{N(N-1)}

i and

Year

Figure 2 Diversity of groundfish caught in standardized trawl samples from an area in the central North Sea, 1980-90, showing contrasting patterns in numbers of species (cross, on log scale) and the Simpson index 1 - A' (open circle). While numbers of species show a weak (p > 0.05) increasing trend, the Simpson index shows a significant (p < 0.05) declining trend, showing that the groundfish assemblage is becoming less even (diverse) and more dominated.

Year

Figure 2 Diversity of groundfish caught in standardized trawl samples from an area in the central North Sea, 1980-90, showing contrasting patterns in numbers of species (cross, on log scale) and the Simpson index 1 - A' (open circle). While numbers of species show a weak (p > 0.05) increasing trend, the Simpson index shows a significant (p < 0.05) declining trend, showing that the groundfish assemblage is becoming less even (diverse) and more dominated.

i where N is the number of individuals of species i, are appropriate when total sample size (N) is small. In effect they correspond to choosing the two individuals at random without replacement rather than with replacement. As with the Shannon-Wiener index (Shannon-Wiener Index), the Simpson index can be employed when the {p} come from proportions of biomass, standardized abundance, or other data which are not strictly integral counts but, in that case, the A' and 1 ā A' forms are not appropriate.

Extensions of the Simpson index, which incorporate the distance between individuals through a taxonomic or (phylo)genetic hierarchy, and share the sample-size independence of the Simpson index, are detailed in Average Taxonomic Diversity and Distinctness.

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