# Diversity indices

These statistics apply to association data, where number of individuals are tabulated in rows (taxa) and possibly several columns (samples). The available statistics are as follows, for each sample (see the Past manual for mathematical details):

• Number of taxa (S)
• Total number of individuals (n)
• Dominance D = 1-Simpson index. Ranges from 0 (all taxa are equally present) to 1 (one taxon dominates the community completely).
• Simpson index 1-D. Measures 'evenness' of the community from 0 to 1. Note the confusion in the literature: Dominance and Simpson indices are often interchanged!
• Shannon index H (entropy). A diversity index taking into account the number of individuals as well as number of taxa. Varies from 0 for communities with only a single taxon to high values for communities with many taxa, each with few individuals.
• Buzas and Gibson's evenness: eH/S
• Brillouin’s index HB
• Menhinick's richness index: S/sqrt(n)
• Margalef's richness index: (S-1) / ln(n)
• Equitability (also known as Pielou’s evenness). Shannon diversity divided by the logarithm of number of taxa. This measures the evenness with which individuals are divided among the taxa present.
• Fisher's alpha - a diversity index, defined implicitly by the formula S=a*ln(1+n/a) where S is number of taxa, n is number of individuals and a is the Fisher's alpha.
• Berger-Parker dominance: simply the number of individuals in the most dominant taxon relative to n.
• Chao1, bias corrected: An estimate of total species richness based on the numbers of singleton and doubleton species (Chao 1984).
• iChao1: “Improved” Chao1 estimator (Chiu et al. 2014), taking into account also the numbers F3 and F4 of species observed 3 and 4 times.
• ACE: Abundance-based Coverage Estimator (Chao & Lee 1992).

Some of these indices are explained in Harper (1999).

Approximate confidence intervals for all these indices can be computed with a bootstrap procedure. The given number of random samples (default 9999) are produced, each with the same total number of individuals as in the original sample. For each individual in the random sample, the taxon is chosen with probabilities proportional to the original abundances. A 95 percent confidence interval is then calculated. Note that the diversity in the replicates will often be less than, and never larger than, the pooled diversity in the total data set – this bias can optionally be “fixed” by centering the confidence interval on the original value.

Alternatively, analytical estimates of the 95% confidence interval are available for some of the indices.

Bootstrapped comparison of diversity indices in two samples is provided in the Compare diversities module.

#### References

Chao, A. 1984. Nonparametric estimation of the number of classes in a population. Scandinavian Journal of Statistics 11:265-270.

Chao, A., Lee, S.-M. 1992. Estimating the number of classes via sample coverage. Journal of the American Statistical Association 87:210–217.

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 70:671-682.

Harper, D.A.T. (ed.). 1999. Numerical Palaeobiology. John Wiley & Sons.

Published Aug. 31, 2020 8:05 PM - Last modified May 9, 2021 11:44 PM