richness-index: Richness and Rarefaction

Description

index_richness returns sample richness. index_composition returns asymptotic species richness.

rarefaction returns Hurlbert's unbiased estimate of Sander's rarefaction.

Usage

index_richness(object, ...)
index_composition(object, ...)
rarefaction(object, ...)
# S4 method for CountMatrix
rarefaction(object, sample,
  method = c("hurlbert"), simplify = FALSE, ...)
# S4 method for CountMatrix
index_richness(object, method = c("none",
  "margalef", "menhinick"), jackknife = TRUE, bootstrap = TRUE,
  simulate = FALSE, level = 0.8, n = 1000, ...)
# S4 method for CountMatrix
index_composition(object, method = c("chao1",
  "ace"), unbiased = FALSE, improved = FALSE, k = 10)
# S4 method for IncidenceMatrix
index_composition(object, method = c("chao2",
  "ice"), unbiased = FALSE, improved = FALSE, k = 10)

Arguments

object

A $m \times p$ matrix of count data.

...

Further arguments to be passed to internal methods.

sample

A length-one numeric vector giving the sub-sample size.

method

A character string or vector of strings specifying the index to be computed (see details). Any unambiguous substring can be given.

simplify

A logical scalar: should the result be simplified to a matrix? The default value, FALSE, returns a list.

jackknife

A logical scalar: should jackknifed estimates be computed?

bootstrap

A logical scalar: should boostraped estimates be computed?

simulate

A logical scalar: simulated assemblages be computed?

level

A length-one numeric vector giving the confidence level. Only used if simulate is TRUE.

A non-negative integer giving the number of bootstrap replications. Only used if simulate is TRUE.

unbiased

A logical scalar. Should the bias-corrected estimator be used? Only used with "chao1" or "chao2" (improved) estimator.

improved

A logical scalar. Should the improved estimator be used? Only used with "chao1" or "chao2".

A length-one numeric vector giving the threshold between rare/infrequent and abundant/frequent species. Only used if method is "ace" or "ice".

Value

index_richness and index_composition return a '>DiversityIndex object.

If simplify is FALSE, then rarefaction returns a list (default), else return a matrix.

Asymptotic Species Richness

The following measures are available for count data:

ace: Abundance-based Coverage Estimator.
chao1: (improved/unbiased) Chao1 estimator.

The following measures are available for replicated incidence data:

ice: Incidence-based Coverage Estimator.
chao2: (improved/unbiased) Chao2 estimator.

Details

The number of different taxa, provides an instantly comprehensible expression of diversity. While the number of taxa within a sample is easy to ascertain, as a term, it makes little sense: some taxa may not have been seen, or there may not be a fixed number of taxa (e.g. in an open system; Peet 1974). As an alternative, richness ( $S$ ) can be used for the concept of taxa number (McIntosh 1967).

It is not always possible to ensure that all sample sizes are equal and the number of different taxa increases with sample size and sampling effort (Magurran 1988). Then, rarefaction ( $E (S)$ ) is the number of taxa expected if all samples were of a standard size (i.e. taxa per fixed number of individuals). Rarefaction assumes that imbalances between taxa are due to sampling and not to differences in actual abundances.

The following richness measures are available for count data:

margalef: Margalef richness index.
menhinick: Menhinick richness index.
none: Returns the number of observed taxa/types.

References

Chao, A. (1984). Nonparametric Estimation of the Number of Classes in a Population. Scandinavian Journal of Statistics, 11(4), 265-270.

Chao, A. (1987). Estimating the Population Size for Capture-Recapture Data with Unequal Catchability. Biometrics 43(4), 783-791. DOI: 10.2307/2531532.

Chao, A. & Chiu, C.-H. (2016). Species Richness: Estimation and Comparison. In Balakrishnan, N., Colton, T., Everitt, B., Piegorsch, B., Ruggeri, F. & Teugels, J. L. (Eds.), Wiley StatsRef: Statistics Reference Online. Chichester, UK: John Wiley & Sons, Ltd., 1-26. DOI: 10.1002/9781118445112.stat03432.pub2

Chao, A. & Lee, S.-M. (1992). Estimating the Number of Classes via Sample Coverage. Journal of the American Statistical Association, 87(417), 210-217. DOI: 10.1080/01621459.1992.10475194.

Chiu, C.-H., Wang, Y.-T., Walther, B. A. & Chao, A. (2014). An improved nonparametric lower bound of species richness via a modified good-turing frequency formula. Biometrics, 70(3), 671-682. DOI: 10.1111/biom.12200.

Hurlbert, S. H. (1971). The Nonconcept of Species Diversity: A Critique and Alternative Parameters. Ecology, 52(4), 577-586. DOI: 10.2307/1934145.

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

Magurran, A E. & Brian J. McGill (2011). Biological Diversity: Frontiers in Measurement and Assessment. Oxford: Oxford University Press.

Margalef, R. (1958). Information Theory in Ecology. General Systems, 3, 36-71.

Menhinick, E. F. (1964). A Comparison of Some Species-Individuals Diversity Indices Applied to Samples of Field Insects. Ecology, 45(4), 859-861. DOI: 10.2307/1934933.

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.

Sander, H. L. (1968). Marine Benthic Diversity: A Comparative Study. The American Naturalist, 102(925), 243-282.

Examples

Run this code

# NOT RUN {
## Richness
## Margalef and Menhinick index
## Data from Magurran 1988, p. 128-129
trap <- CountMatrix(data = c(9, 3, 0, 4, 2, 1, 1, 0, 1, 0, 1, 1,
                             1, 0, 1, 0, 0, 0, 1, 2, 0, 5, 3, 0),
                    nrow = 2, byrow = TRUE, dimnames = list(c("A", "B"), NULL))
index_richness(trap, method = "margalef")
## A 2.55 1.88
index_richness(trap, method = "menhinick")
## B 1.95 1.66

## Asymptotic species richness
## Chao1-type estimators
## Data from Chao & Chiu 2016
brazil <- CountMatrix(
  data = rep(x = c(1:21, 23, 25, 27, 28, 30, 32, 34:37, 41,
                   45, 46, 49, 52, 89, 110, 123, 140),
             times = c(113, 50, 39, 29, 15, 11, 13, 5, 6, 6, 3, 4,
                       3, 5, 2, 5, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1,
                       0, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0)),
  nrow = 1, byrow = TRUE
)

index_composition(brazil, method = c("chao1"), unbiased = FALSE)
## 461.625
index_composition(brazil, method = c("ace"), k = 10)
## 445.822

## Rarefaction
rarefaction(trap, sample = 13) # 6.56

richness(trap, method = c("margalef", "menhinick", "chao1"))
# }

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