untb (version 1.7-7-1)

simpson: Simpson's diversity index

Description

Simpson's diversity index

Usage

simpson(x, with.replacement=FALSE)

Arguments

x

Ecosystem vector; coerced to class count

with.replacement

Boolean, with default FALSE meaning to sample without replacement; see details section

Author

Robin K. S. Hankin

Details

Returns the Simpson index \(D\): the probability that two randomly sampled individuals belong to different species.

There is some confusion as to the precise definition: some authors specify that the two individuals are necessarily distinct (ie sampling without replacement), and some do not.

Simpson (1949) assumed sampling without replacement and gave

$$ 1-\frac{\sum_{i=1}^Sn_i\left(n_i-1\right)}{J(J-1)} $$ in our notation.

He and Hu (2005) assumed sampling with replacement: $$ 1-\frac{\sum_{i=1}^Sn_i^2}{J^2}. $$

The difference is largely academic but is most pronounced when many species occur with low counts (ie close to 1).

References

• S. P. Hubbell 2001. “The Unified Neutral Theory of Biodiversity”. Princeton University Press.

• F. He and X.-S. Hu 2005. “Hubbell's Fundamental Biodiversity Parameter and the Simpson Diversity Index”. Ecology Letters, volume 8, pp386-390. doi: 10.1111/j.1461-0248.2005.00729.x

• E. H. Simpson 1949. “Measurement of diversity”, Nature, volume 163, p688

See Also

preston

Examples

data(butterflies)

D <- simpson(butterflies)
theta <- optimal.prob(butterflies)*2*no.of.ind(butterflies)

# compare theta with D/(1-D) (should be roughly equal; see He & Hu 2005):
theta
D/(1-D)


# Second argument pedantic in practice.

# Mostly, the difference is small:
simpson(butterflies,FALSE) - simpson(butterflies,TRUE)

# Most extreme example:
x <- count(c(1,1))
simpson(x,TRUE)
simpson(x,FALSE)