Compute the following diversity indices.
Simpson's and related indices
Simpson's Index (\(d\)) (Simpson 1949; Peet 1974)
Simpson's Index of Diversity or Gini's Diversity Index or Gini-Simpson Index or Nei's Diversity Index or Nei's Variation Index (\(D\)) (Gini 1912, 1912; Greenberg 1956; Berger and Parker 1970; Nei 1973; Peet 1974)
Maximum Simpson's Index of Diversity or Maximum Nei's Diversity/Variation Index (\(D_{max}\)) (Hennink and Zeven 1990)
Simpson's Reciprocal Index or Hill's \(N_{2}\) (\(D_{R}\)) (Williams 1964; Hill 1973)
Relative Simpson's Index of Diversity or Relative Nei's Diversity/Variation Index (\(D'\)) (Hennink and Zeven 1990)
Shannon-Weaver and related indices
Shannon or Shannon-Weaver or Shannon-Weiner Diversity Index (\(H\)) (Shannon and Weaver 1949; Peet 1974)
Maximum Shannon-Weaver Diversity Index (\(H_{max}\)) (Hennink and Zeven 1990)
Relative Shannon-Weaver Diversity Index or Shannon Equitability Index (\(H'\)) (Hennink and Zeven 1990)
McIntosh Diversity Index
McIntosh Diversity Index (\(D_{Mc}\)) (McIntosh 1967; Peet 1974)
Usage
diversity.indices(
x,
index = c("simpson", "simpson.cmp", "simpson.max", "simpson.inv", "simpson.rel",
"shannon", "shannon.max", "shannon.rel", "mcintosh"),
base = 2
)Details
The diversity indices and the corresponding statistical
tests implemented in diversity.indices are as follows.
Simpson's and related indices
Simpson's index (\(d\)) which estimates the probability that two accessions randomly selected will belong to the same class of a qualitative trait, is computed as follows (Simpson 1949; Peet 1974) .
\[d = \sum_{i = 1}^{k}p_{i}^{2}\]
Where, \(p_{i}\) denotes the proportion/fraction/frequency of accessions in the \(i\)th class of a qualitative trait and \(k\) is the number of classes for the qualitative trait.
The value of \(d\) can range from 0 to 1 with 0 representing maximum diversity and 1, no diversity.
The complement of \(d\) (\(d\) is subtracted from 1) is called the Simpson's index of diversity (\(D\)) (Greenberg 1956; Berger and Parker 1970; Peet 1974; Hennink and Zeven 1990) originally suggested by Gini (1912, 1912) and described in literature as Gini's diversity index or Gini-Simpson index. It is the same as Nei's diversity index or Nei's variation index (Nei 1973; Hennink and Zeven 1990) . Greater the value of \(D\), greater the diversity with a range from 0 to 1.
\[D = 1 - d\]
The maximum value of \(D\), \(D_{max}\) occurs when accessions are uniformly distributed across the classes in the qualitative trait and is computed as follows (Hennink and Zeven 1990) .
\[D_{max} = 1 - \frac{1}{k}\]
Reciprocal of \(d\) gives the Simpson's reciprocal index (\(D_{R}\)) (Williams 1964; Hennink and Zeven 1990) and can range from 1 to \(k\). This was also described in Hill (1973) as (\(N_{2}\)).
\[D_{R} = \frac{1}{d}\]
Relative Simpson's index of diversity or Relative Nei's diversity/variation index (\(H'\)) (Hennink and Zeven 1990) is defined as follows (Peet 1974) .
\[D' = \frac{D}{D_{max}}\]
Shannon-Weaver and related indices
An index of information \(H\), was described by Shannon and Weaver (1949) as follows.
\[H = -\sum_{i=1}^{k}p_{i} \log_{2}(p_{i})\]
\(H\) is described as Shannon or Shannon-Weaver or Shannon-Weiner diversity index in literature.
Alternatively, \(H\) is also computed using natural logarithm instead of logarithm to base 2.
\[H = -\sum_{i=1}^{k}p_{i} \ln(p_{i})\]
The maximum value of \(H\) (\(H_{max}\)) is \(\ln(k)\). This value occurs when each class for a qualitative trait has the same proportion of accessions.
\[H_{max} = \log_{2}(k)\;\; \textrm{OR} \;\; H_{max} = \ln(k)\]
The relative Shannon-Weaver diversity index or Shannon equitability index (\(H'\)) is the Shannon diversity index (\(I\)) divided by the maximum diversity (\(H_{max}\)).
\[H' = \frac{H}{H_{max}}\]
McIntosh Diversity Index
A similar index of diversity was described by McIntosh (1967) as follows (\(D_{Mc}\)) (Peet 1974) .
\[D_{Mc} = \frac{N - \sqrt{\sum_{i=1}^{k}n_{i}^2}}{N - \sqrt{N}}\]
Where, \(n_{i}\) denotes the number of accessions in the \(i\)th class for a qualitative trait and \(N\) is the total number of accessions so that \(p_{i} = {n_{i}}/{N}\).
References
Berger WH, Parker FL (1970).
“Diversity of planktonic foraminifera in deep-sea sediments.”
Science, 168(3937), 1345–1347.
Gini C (1912).
Variabilita e Mutabilita. Contributo allo Studio delle Distribuzioni e delle Relazioni Statistiche. [Fasc. I.].
Tipogr. di P. Cuppini, Bologna.
Gini C (1912).
“Variabilita e mutabilita.”
In Pizetti E, Salvemini T (eds.), Memorie di Metodologica Statistica.
Liberia Eredi Virgilio Veschi, Roma, Italy.
Greenberg JH (1956).
“The measurement of linguistic diversity.”
Language, 32(1), 109.
Hennink S, Zeven AC (1990).
“The interpretation of Nei and Shannon-Weaver within population variation indices.”
Euphytica, 51(3), 235–240.
Hill MO (1973).
“Diversity and evenness: A unifying notation and its consequences.”
Ecology, 54(2), 427–432.
McIntosh RP (1967).
“An index of diversity and the relation of certain concepts to diversity.”
Ecology, 48(3), 392–404.
Nei M (1973).
“Analysis of gene diversity in subdivided populations.”
Proceedings of the National Academy of Sciences, 70(12), 3321–3323.
Peet RK (1974).
“The measurement of species diversity.”
Annual Review of Ecology and Systematics, 5(1), 285–307.
Shannon CE, Weaver W (1949).
The Mathematical Theory of Communication, number v. 2 in The Mathematical Theory of Communication.
University of Illinois Press.
Simpson EH (1949).
“Measurement of diversity.”
Nature, 163(4148), 688–688.
Williams CB (1964).
Patterns in the Balance of Nature and Related Problems in Quantitative Ecology.
Academic Press.