• index_richness() returns sample richness. index_composition() returns asymptotic species richness.

• rarefaction() returns Hurlbert's unbiased estimate of Sander's rarefaction.

• bootstrap_*() and jackknife_*() perform bootstrap/jackknife resampling.

index_richness(object, ...)

simulate_richness(object, ...)

bootstrap_richness(object, ...)

jackknife_richness(object, ...)

index_composition(object, ...)

rarefaction(object, ...)

# S4 method for CountMatrix
rarefaction(object, sample, method = c("hurlbert"), simplify = TRUE, ...)

# S4 method for CountMatrix
index_richness(object, method = c("none", "margalef", "menhinick"), ...)

# S4 method for CountMatrix
simulate_richness(
object,
method = c("none", "margalef", "menhinick"),
quantiles = TRUE,
level = 0.8,
step = 1,
n = 1000,
progress = getOption("tabula.progress"),
...
)

# S4 method for CountMatrix
bootstrap_richness(
object,
method = c("none", "margalef", "menhinick"),
probs = c(0.05, 0.95),
n = 1000,
...
)

# S4 method for CountMatrix
jackknife_richness(object, method = c("none", "margalef", "menhinick"), ...)

# 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. A length-one numeric vector giving the sub-sample size. A character string or vector of strings specifying the index to be computed (see details). Any unambiguous substring can be given. A logical scalar: should the result be simplified to a matrix? The default value, FALSE, returns a list. A logical scalar: should sample quantiles be used as confidence interval? If TRUE (the default), sample quantiles are used as described in Kintigh (1989), else quantiles of the normal distribution are used. A length-one numeric vector giving the confidence level. A non-negative integer giving the increment of the sample size. Only used if simulate is TRUE. A non-negative integer giving the number of bootstrap replications. A logical scalar: should a progress bar be displayed? A numeric vector of probabilities with values in $$[0,1]$$ (see stats::quantile()). A logical scalar. Should the bias-corrected estimator be used? Only used with "chao1" or "chao2" (improved) estimator. 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(), simulate_richness() and index_composition() return a DiversityIndex object.

• bootstrap_*() and jackknife_*() return a data.frame.

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

## 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.

## 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.

## 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 .

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.

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.

plot_diversity()

Other diversity: heterogeneity-index, similarity(), turnover-index

N. Frerebeau

## Examples

## 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") # 2.55 1.88
#> <RichnessIndex: margalef>
#>   size    index
#> A   23 2.551432
#> B   13 1.949356index_richness(trap, method = "menhinick") # 1.95 1.66
#> <RichnessIndex: menhinick>
#>   size    index
#> A   23 1.876630
#> B   13 1.664101
## 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
#> <CompositionIndex: chao1>
#>      size    index
#> row1 1978 461.6254index_composition(brazil, method = c("ace"), k = 10) # 445.822
#> <CompositionIndex: ace>
#>      size    index
#> row1 1978 445.8224
## Rarefaction
rarefaction(trap, sample = 13) # 6.56
#>        A        B
#> 6.556735       NA