`heterogeneity()`

returns an heterogeneity or dominance index.`evenness()`

returns an evenness measure.

## Usage

```
heterogeneity(object, ...)
evenness(object, ...)
index_berger(x, ...)
index_boone(x, ...)
index_brillouin(x, ...)
index_mcintosh(x, ...)
index_shannon(x, ...)
index_simpson(x, ...)
# S4 method for matrix
heterogeneity(
object,
method = c("berger", "boone", "brillouin", "mcintosh", "shannon", "simpson"),
j = NULL
)
# S4 method for data.frame
heterogeneity(
object,
method = c("berger", "boone", "brillouin", "mcintosh", "shannon", "simpson"),
j = NULL
)
# S4 method for matrix
evenness(object, method = c("shannon", "brillouin", "mcintosh", "simpson"))
# S4 method for data.frame
evenness(object, method = c("shannon", "brillouin", "mcintosh", "simpson"))
# S4 method for numeric
index_berger(x, na.rm = FALSE, ...)
# S4 method for matrix
index_boone(x, j = NULL, na.rm = FALSE, ...)
# S4 method for numeric
index_brillouin(x, evenness = FALSE, na.rm = FALSE, ...)
# S4 method for numeric
index_mcintosh(x, evenness = FALSE, na.rm = FALSE, ...)
# S4 method for numeric
index_shannon(x, evenness = FALSE, base = exp(1), na.rm = FALSE, ...)
# S4 method for numeric
index_simpson(x, evenness = FALSE, na.rm = FALSE, ...)
```

## Arguments

- object
A \(m \times p\)

`numeric`

`matrix`

or`data.frame`

of count data (absolute frequencies giving the number of individuals for each class).- ...
Currently not used.

- x
A

`numeric`

vector of count data (absolute frequencies).- method
A

`character`

string specifying the index to be computed (see details). Any unambiguous substring can be given.- j
An

`integer`

giving the index of the reference type/taxa. If`NULL`

(the default), the most frequent type/taxa in any assemblage will be used.- na.rm
A

`numeric`

scalar: should missing values (including`NaN`

) be removed?- evenness
A

`logical`

scalar: should an evenness measure be computed instead of an heterogeneity/dominance index?- base
A positive

`numeric`

value specifying the base with respect to which logarithms are computed.

## Value

`heterogeneity()`

returns an HeterogeneityIndex object.`evenness()`

returns an EvennessIndex object.`index_*()`

return a`numeric`

vector.

## Details

*Diversity* measurement assumes that all individuals in a specific
taxa are equivalent and that all types are equally different from each
other (Peet 1974). A measure of diversity can be achieved by using indices
built on the relative abundance of taxa. These indices (sometimes referred
to as non-parametric indices) benefit from not making assumptions about the
underlying distribution of taxa abundance: they only take relative
abundances of the species that are present and species richness into
account. Peet (1974) refers to them as indices of *heterogeneity*.

Diversity indices focus on one aspect of the taxa abundance and emphasize
either *richness* (weighting towards uncommon taxa) or dominance (weighting
towards abundant taxa; Magurran 1988).

*Evenness* is a measure of how evenly individuals are distributed across the
sample.

## Heterogeneity and Evenness Measures

The following heterogeneity index and corresponding evenness measures are available (see Magurran 1988 for details):

`berger`

Berger-Parker dominance index. The Berger-Parker index expresses the proportional importance of the most abundant type. This metric is highly biased by sample size and richness, moreover it does not make use of all the information available from sample.

`boone`

Boone heterogeneity measure.

`brillouin`

Brillouin diversity index. The Brillouin index describes a known collection: it does not assume random sampling in an infinite population. Pielou (1975) and Laxton (1978) argues for the use of the Brillouin index in all circumstances, especially in preference to the Shannon index.

`mcintosh`

McIntosh dominance index. The McIntosh index expresses the heterogeneity of a sample in geometric terms. It describes the sample as a point of a \(S\)-dimensional hypervolume and uses the Euclidean distance of this point from the origin.

`shannon`

Shannon-Wiener diversity index. The Shannon index assumes that individuals are randomly sampled from an infinite population and that all taxa are represented in the sample (it does not reflect the sample size). The main source of error arises from the failure to include all taxa in the sample: this error increases as the proportion of species discovered in the sample declines (Peet 1974, Magurran 1988). The maximum likelihood estimator (MLE) is used for the relative abundance, this is known to be negatively biased by sample size.

`simpson`

Simpson dominance index for finite sample. The Simpson index expresses the probability that two individuals randomly picked from a finite sample belong to two different types. It can be interpreted as the weighted mean of the proportional abundances. This metric is a true probability value, it ranges from \(0\) (perfectly uneven) to \(1\) (perfectly even).

The `berger`

, `mcintosh`

and `simpson`

methods return a *dominance* index,
not the reciprocal or inverse form usually adopted, so that an increase in
the value of the index accompanies a decrease in diversity.

## References

Berger, W. H. & Parker, F. L. (1970). Diversity of Planktonic Foraminifera
in Deep-Sea Sediments. *Science*, 168(3937), 1345-1347.
doi:10.1126/science.168.3937.1345
.

Boone, J. L. (1987). Defining and Measuring Midden Catchment. *American
Antiquity*, 52(2), 336-45. doi:10.2307/281785
.

Brillouin, L. (1956). *Science and information theory*. New York:
Academic Press.

Kintigh, K. W. (1989). Sample Size, Significance, and Measures of
Diversity. In Leonard, R. D. and Jones, G. T., *Quantifying Diversity
in Archaeology*. New Directions in Archaeology. Cambridge:
Cambridge University Press, p. 25-36.

Laxton, R. R. (1978). The measure of diversity. *Journal of Theoretical
Biology*, 70(1), 51-67.
doi:10.1016/0022-5193(78)90302-8
.

Magurran, A. E. (1988). *Ecological Diversity and its Measurement*.
Princeton, NJ: Princeton University Press.
doi:10.1007/978-94-015-7358-0
.

McIntosh, R. P. (1967). An Index of Diversity and the Relation of Certain
Concepts to Diversity. *Ecology*, 48(3), 392-404.
doi:10.2307/1932674
.

Peet, R. K. (1974). The Measurement of Species Diversity. *Annual Review of
Ecology and Systematics*, 5(1), 285-307.
doi:10.1146/annurev.es.05.110174.001441
.

Pielou, E. C. (1975). *Ecological Diversity*. New York: Wiley.
doi:10.4319/lo.1977.22.1.0174b

Shannon, C. E. (1948). A Mathematical Theory of Communication. *The
Bell System Technical Journal*, 27, 379-423.
doi:10.1002/j.1538-7305.1948.tb01338.x
.

Simpson, E. H. (1949). Measurement of Diversity. *Nature*, 163(4148),
688-688. doi:10.1038/163688a0
.

## See also

Other diversity measures:
`occurrence()`

,
`plot_diversity`

,
`rarefaction()`

,
`richness()`

,
`similarity()`

,
`simulate()`

,
`turnover()`

## Examples

```
data("cantabria")
## Shannon diversity index
(h <- heterogeneity(cantabria, method = "shannon"))
#> [1] 3.269200 2.955298 2.491683 2.485604 2.329187
(e <- evenness(cantabria, method = "shannon"))
#> [1] 0.8987278 0.8966760 0.8462335 0.9178574 0.9373336
## Bootstrap resampling (summary statistics)
bootstrap(h, f = NULL)
#> original mean bias error
#> Altamira 3.269200 3.278572 0.009372753 0.08139834
#> Cueto de la Mina 2.955298 2.973351 0.018053466 0.14926835
#> El Juyo 2.491683 2.518580 0.026897111 0.20273474
#> El Cierro 2.485604 2.473950 -0.011653780 0.23410481
#> La Paloma 2.329187 2.309843 -0.019343136 0.26864233
bootstrap(h, f = summary)
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> Altamira 2.990875 3.220065 3.278226 3.273893 3.331302 3.509979
#> Cueto de la Mina 2.396126 2.875449 2.983203 2.974501 3.076795 3.343352
#> El Juyo 1.844609 2.400987 2.534521 2.523273 2.647435 3.059489
#> El Cierro 1.661323 2.342185 2.508510 2.482021 2.636816 3.049500
#> La Paloma 1.213008 2.145842 2.333873 2.309081 2.492706 2.940143
quant <- function(x) quantile(x, probs = c(0.25, 0.50))
bootstrap(h, f = quant)
#> 25% 50%
#> Altamira 3.225820 3.285701
#> Cueto de la Mina 2.871980 2.979696
#> El Juyo 2.401866 2.536377
#> El Cierro 2.337588 2.501459
#> La Paloma 2.138198 2.331548
## Jackknife resampling
jackknife(h)
#> original mean bias error
#> Altamira 3.269200 3.246457 -0.9779158 0.08145529
#> Cueto de la Mina 2.955298 2.932856 -0.9649994 0.15907033
#> El Juyo 2.491683 2.469588 -0.9500688 0.19670002
#> El Cierro 2.485604 2.462723 -0.9838709 0.22195573
#> La Paloma 2.329187 2.305876 -1.0023704 0.25266291
```