# Individual rarefaction

Given one or more samples (in columns) of abundance data for a number of taxa (in rows), this module estimates how many taxa you would expect to find in a sample with a smaller total number of individuals. With this method, you can compare the number of taxa in samples of different size. Using rarefaction analysis on a large sample, you can read out the number of expected taxa for any smaller sample size (including that of the smallest sample). The algorithm is from Krebs (1989), using a log Gamma function for computing combinatorial terms. An example application in paleontology can be found in Adrain et al. (2000).

#### Unconditional variance

The classical rarefaction variance (defining the confidence interval) is called conditional variance. It is conditional on the reference sample, and will reduce to zero for the full sample size. In contrast, Colwell et al. (2012) described an unconditional rarefaction variance estimate that will not reduce to zero at the end of the rarefaction curve. This method is also available in Past.

There are two models for individual rarefaction described by Colwell et al. (2012), the multinomial model (classical rarefaction) and the Poisson model (Coleman rarefaction). The two methods give quite similar results. The “industry standard” rarefaction software, EstimateS, somewhat incongruously computes E(Sn) according to the multinomial equation (eq. (4) in Colwell et al., equivalent to the equation given above), while V(Sn) uses the Poisson formulation (eq. 7 in Colwell et al.), according to the EstimateS manual. This approach is followed in Past for compatibility with EstimateS. The computation also requires an estimate for total (sampled and unsampled) species richness. The Chao1 estimator is used for this (cf. Colwell et al. 2012).

#### Rarefaction of Simpson and Shannon indices

In addition to rarefaction of species richness, Past also includes rarefaction of the Simpson D and Shannon H diversity indices, following Chao et al. (2014). In the forms 1/D and e^{H}, these are special cases of so-called Hill numbers, which is a family of diversity indices. Past reports the rarefaction curves in these forms, for consistency with Chao et al. (2014), but for convenience the Shannon index can also be reported as the conventional H.

Past does not yet compute confidence intervals for these rarefaction curves (which would require bootstrapping).

#### References

Adrain, J.M., Westrop, S.R. & Chatterton, D.E. 2000. Silurian trilobite alpha diversity and the end-Ordovician mass extinction. *Paleobiology* 26:625-646.

Chao, A., Gotelli, N.J., Hsieh, T.C., Sander, E.L., Ma, K.H., Colwell, R.K. & Ellison, A.M. 2014. Rarefaction and extrapolation with Hill numbers: a framework for sampling and estimation in species diversity studies. *Ecological Monographs* 84:45-67.

Colwell, R.K., Chao, A., Gotelli, N.J., Lin, S.-L., Mao, C.X., Chazdon, R.L. & Longino, J.T. 2012. Models and estimators linking individual-based and sample-based rarefaction, extrapolation and comparison of assemblages. *Journal of Plant Ecology* 5:3-21.

Krebs, C.J. 1989. *Ecological Methodology*. Harper & Row, New York.