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Rarefaction

Diversity can perform both individual-based (classical) rarefaction and sample-based rarefaction. Use the first method when you want to estimate diversity for a fewer number of individuals and use the second when you want to estimate diversity for a fewer number of collections.

Individual-based (classical) rarefaction

Diversity uses the analytical formulation of expected richness given by Hurlbert (1971) and for variance in expected richness given by Heck et al. (1975). These are the same equations used by Raup (1975) and Tipper (1979), and the rarefaction engine in Diversity is the same as in the program Analytical Rarefaction. These equations assume that you have N total individuals in your sample and wish to estimate diversity at some lower number of individuals (n). Each species has an abundance of Ni. Given this, the expected number of species at a lower level of sampling is given as:

individual-based rarefaction richness The variance in this estimate is as follows: individual-based rarefaction variance

Individual-based rarefaction can be performed to a constant number of individuals by clicking the calculate metrics button. The desired number of individuals is set in the preferences, which can be selected under the Diversity menu.

Individual-based rarefaction can be used to generate a rarefaction curve showing the expected number of species for all levels of sampling smaller than the sample size. This rarefaction curve is generated by clicking on the “do rarefaction” button. The shaded region reflects the confidence interval and either 95% or 99% confidence can be selected under preferences. From this window, a comma-delimited table of the rarefaction can be exported and a pdf of the rarefaction curve can be saved. The pdf can be edited in Adobe Illustrator.

Individual-based rarefaction can be performed on all collections combined by going to the Collections menu and selecting Individual-based Rarefaction for All Collections.

Sample-based rarefaction

Rarefaction can also be performed to answer the question of what richness would have been had fewer samples been collected. Because individuals have patchy distribution, such sample-based rarefaction may not give results identical to individual-based rarefaction. This type of rarefaction is also called a collector's curve

Diversity uses the analytical solution of Colwell, Mao, and Chang (2004) to calculate the sample-based rarefaction. These equations assume that you have H total sampling areas or quadrats and that you've tallied the number of species that occur in only one quadrat (s1), only two quadrats (s2,), and so on, up to the number of species that occur in all H quadrats (sH). From that, the estimated richness (τ) had only h quadrats been sampled can be calculated.

estimated richness (1)

This can be rewritten as the total number of species observed (Sobs) minus some loss of diversity based on the occurrences of species among the quadrats.

estimated richness (2)

The alpha term is a combinatorial coefficient.

alpha

Variance can be calculated similarly, but is based partly on the extrapolated diversity estimate, given by a form of the Chao2 estimator.

variance

The Chao2 estimator is given as:

Chao-2 estimator

A sample-based rarefaction curve can be generated by clicking on the Sample Rarefaction button on the toolbar. As for the classical rarefaction, the plot can be saved as a pdf and the rarefaction results can be exported as a comma-delimited file.