tabula (version 1.0.0)

alpha-diversity: \(\alpha\)-diversity

Description

Measures within-sample diversity. diversity returns a diversity or dominance index. evenness returns an evenness measure. richness returns sample richness. rarefaction returns Hurlbert's unbiaised estimate of Sander's rarefaction.

Usage

diversity(object, ...)
evenness(object, ...)
rarefaction(object, ...)
richness(object, ...)
# S4 method for CountMatrix
richness(object, method = c("margalef",
  "menhinick"), simplify = FALSE)
# S4 method for CountMatrix
rarefaction(object, sample)
# S4 method for CountMatrix
diversity(object, method = c("berger",
  "brillouin", "mcintosh", "shannon", "simpson"), simplify = FALSE, ...)
# S4 method for CountMatrix
evenness(object, method = c("brillouin",
  "mcintosh", "shannon", "simpson"), simplify = FALSE, ...)

Arguments

object

A \(m \times p\) matrix of count data.

...

Further arguments passed to other methods.

method

A character string specifiying the index to be computed. This must be one or more of "berger", "brillouin", "margalef", "mcintosh", "menhinick", "shannon", "simpson" (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.

sample

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

Value

rarefaction returns a numeric vector.

If simplify is FALSE, then diversity, evenness and richness return a list (default), else return a matrix.

Richness and Rarefaction

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:

margalef

Margalef richness index: \(D_{Mg} = \frac{S - 1}{\ln N}\)

menhinick

Menhinick richness index: \(D_{Mn} = \frac{S}{\sqrt{N}}\)

Diversity and Evenness

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 evenness and 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.

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.

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.

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.

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

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

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.

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

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.

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.

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

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

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

turnover similarity

Examples

# 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))

richness(trap, method = c("margalef", "menhinick"), simplify = TRUE)
## A 2.55 1.88
## B 1.95 1.66

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

# Diversity
## 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))

diversity(birds, "shannon") # 2.40 2.06
evenness(birds, "shannon") # 0.80 0.78

## 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)

diversity(moths, "brillouin") # 1.88
evenness(moths, "brillouin") # 0.83

## 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
)

diversity(trees, "simpson") # 1.19
evenness(trees, "simpson") # 0.21

## 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
)

diversity(invertebrates, "mcintosh") # 0.71
evenness(invertebrates, "mcintosh") # 0.82

## 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)
)

diversity(fishes, "berger") # 0.71 0.63 0.50 0.60 0.51
# }