# simpson

From untb v1.7-2
by Robin K S Hankin

##### Simpson's diversity index

Simpson's diversity index

- Keywords
- math

##### 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

##### 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

##### 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)
```

*Documentation reproduced from package untb, version 1.7-2, License: GPL*

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