simpson {abdiv} | R Documentation |

## Simpson's index and related 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

```
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 = \sum_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 \leq 1`

), Simpson index, or
inverse Simpson index. The dominance is undefined if the vector sums to
zero, in which case we return `NaN`

.

### Examples

```
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))
```

*abdiv*version 0.2.0 Index]