# simpson: Simpson's index and related measures In kylebittinger/abdiv: Alpha and Beta Diversity Measures

## Description

These measures are based on the sum of squared species proportions. The function `dominance()` gives this quantity, `simpson()` gives one minus this quantity, `invsimpson()` gives the reciprocal of the quantity, and `simpson_e` gives the reciprocal divided by the number of species.

## Usage

 ```1 2 3 4 5 6 7``` ```simpson(x) dominance(x) invsimpson(x) simpson_e(x) ```

## Arguments

 `x` A numeric vector of species counts or proportions.

## Details

For a vector of species counts `x`, the dominance index is defined as

D = ∑_i p_i^2,

where p_i is the species proportion, p_i = x_i / N, and N is the total number of counts. This is equal to the probability of selecting two individuals from the same species, with replacement. Relation to other definitions:

• Equivalent to `dominance()` in `skbio.diversity.alpha`.

• Similar to the `simpson` calculator in Mothur. They use the unbiased estimate p_i = x_i (x_i - 1) / (N (N -1)).

Simpson's index is defined here as 1 - D, or the probability of selecting two individuals from different species, with replacement. Relation to other definitions:

• Equivalent to `diversity()` in `vegan` with `index = "simpson"`.

• Equivalent to `simpson()` in `skbio.diversity.alpha`.

The inverse Simpson index is 1/D. Relation to other definitions:

• Equivalent to `diversity()` in `vegan` with `index = "invsimpson"`.

• Equivalent to `enspie()` in `skbio.diversity.alpha`.

• Similar to the `invsimpson` calculator in Mothur. They use the unbiased estimate p_i = x_i (x_i - 1) / (N (N -1)).

Simpson's evenness index is the inverse Simpson index divided by the number of species observed, 1 / (D S). Relation to other definitions:

• Equivalent to `simpson_e()` in `skbio.diversity.alpha`.

Please be warned that the naming conventions vary between sources. For example Wikipedia calls D the Simpson index and 1 - D the Gini-Simpson index. We have followed the convention from `vegan`, to avoid confusion within the `R` ecosystem.

## Value

The value of the dominance (0 < D ≤q 1), Simpson index, or inverse Simpson index. The dominance is undefined if the vector sums to zero, in which case we return `NaN`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```x <- c(15, 6, 4, 0, 3, 0) dominance(x) # Simpson is 1 - D simpson(x) 1 - dominance(x) # Inverse Simpson is 1/D invsimpson(x) 1 / dominance(x) # Simpson's evenness is 1 / (D * S) simpson_e(x) 1 / (dominance(x) * richness(x)) ```

kylebittinger/abdiv documentation built on Jan. 31, 2020, 3:13 p.m.