Page 321 - Becoming Metric Wise

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The Informetric Laws

Clearly, it is impossible for one function, one candidate diversity measure,
to satisfy all “reasonable” requirements. It seems though that a very good
choice is given by Leinster and Cobbold (2012) and Zhang et al. (2016):

1
(9.24)
1 2 D
where D is Rao’s quadratic entropy measure (Rao, 1982) defined as:
N
X
D 5 d ij p i p j (9.25)
i; j51
i 6¼ j

Rao describes this index as the expected dissimilarity between two
individuals selected randomly with replacement, where d ij is the dissimi-
larity between species i and j and p i (p j ) is the proportion of species i (j).
If there is only one cell, D is set equal to zero. Rao’s quadratic entropy
measure is a generalization of the Simpson index, formula (9.23).