`index_heterogeneity`

returns an heterogeneity or dominance index.

`index_evenness`

returns an evenness measure.

index_heterogeneity(object, ...) index_evenness(object, ...) # S4 method for CountMatrix index_heterogeneity(object, method = c("berger", "brillouin", "mcintosh", "shannon", "simpson"), jackknife = TRUE, bootstrap = TRUE, simulate = FALSE, level = 0.8, n = 1000, ...) # S4 method for CountMatrix index_evenness(object, method = c("shannon", "brillouin", "mcintosh", "simpson"), jackknife = TRUE, bootstrap = TRUE, simulate = FALSE, level = 0.8, n = 1000, ...)

object | A \(m \times p\) matrix of count data (typically a CountMatrix object). |
---|---|

... | Further arguments to be passed to internal methods. |

method | A |

jackknife | A |

bootstrap | A |

simulate | A |

level | A length-one |

n | A non-negative |

`index_heterogeneity`

and `index_evenness`

return a
DiversityIndex object.

*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.

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.

- 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.

Ramanujan approximation is used for \(x!\) computation if \(x > 170\).

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.

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

Kintigh, K. W. (1984). Measuring Archaeological Diversity by Comparison
with Simulated Assemblages. *American Antiquity*, 49(1), 44-54.
DOI: 10.2307/280511.

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.

Other diversity: `richness-index`

,
`similarity`

, `turnover-index`

# Shannon diversity index # Data from Magurran 1988, p. 145-149 birds <- CountMatrix( data = c(35, 26, 25, 21, 16, 11, 6, 5, 3, 3, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 30, 30, 3, 65, 20, 11, 0, 4, 2, 14, 0, 3, 9, 0, 0, 5, 0, 0, 0, 0, 1, 1), nrow = 2, byrow = TRUE, dimnames = list(c("oakwood", "spruce"), NULL)) index_heterogeneity(birds, "shannon") # 2.40 2.06#> <HeterogeneityIndex: shannon> #> index #> oakwood 2.407983 #> spruce 2.055991index_evenness(birds, "shannon") # 0.80 0.78#> <EvennessIndex: shannon> #> index #> oakwood 0.8038044 #> spruce 0.7790626# Brillouin diversity index # Data from Magurran 1988, p. 150-151 moths <- CountMatrix(data = c(17, 15, 11, 4, 4, 3, 3, 3, 2, 2, 1, 1, 1), nrow = 1, byrow = TRUE) index_heterogeneity(moths, "brillouin") # 1.88#> <HeterogeneityIndex: brillouin> #> index #> 1 1.876307index_evenness(moths, "brillouin") # 0.83#> <EvennessIndex: brillouin> #> index #> 1 0.8275138# Simpson dominance index # Data from Magurran 1988, p. 152-153 trees <- CountMatrix( data = c(752, 276, 194, 126, 121, 97, 95, 83, 72, 44, 39, 16, 15, 13, 9, 9, 9, 8, 7, 4, 2, 2, 1, 1, 1), nrow = 1, byrow = TRUE ) index_heterogeneity(trees, "simpson") # 1.19#> <HeterogeneityIndex: simpson> #> index #> 1 0.186547index_evenness(trees, "simpson") # 0.21#> <EvennessIndex: simpson> #> index #> 1 0.2144231# McIntosh dominance index # Data from Magurran 1988, p. 154-155 invertebrates <- CountMatrix( data = c(254, 153, 90, 69, 68, 58, 51, 45, 40, 39, 25, 23, 19, 18, 16, 14, 14, 11, 11, 11, 11, 10, 6, 6, 6, 6, 5, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1), nrow = 1, byrow = TRUE ) index_heterogeneity(invertebrates, "mcintosh") # 0.71#> <HeterogeneityIndex: mcintosh> #> index #> 1 0.7066343index_evenness(invertebrates, "mcintosh") # 0.82#> <EvennessIndex: mcintosh> #> index #> 1 0.8180305# Berger-Parker dominance index # Data from Magurran 1988, p. 156-157 fishes <- CountMatrix( data = c(394, 3487, 275, 683, 22, 1, 0, 1, 6, 8, 1, 1, 2, 1642, 5681, 196, 1348, 12, 0, 1, 48, 21, 1, 5, 7, 3, 90, 320, 180, 46, 2, 0, 0, 1, 0, 0, 2, 1, 5, 126, 17, 115, 436, 27, 0, 0, 3, 1, 0, 0, 1, 0, 32, 0, 0, 5, 0, 0, 0, 0, 13, 9, 0, 0, 4), nrow = 5, byrow = TRUE, dimnames = list(c("station 1", "station 2", "station 3", "station 4", "station 5"), NULL) ) index_heterogeneity(fishes, "berger") # 0.71 0.63 0.50 0.60 0.51#> <HeterogeneityIndex: berger> #> index #> station 1 0.7144028 #> station 2 0.6336866 #> station 3 0.4945904 #> station 4 0.6005510 #> station 5 0.5079365