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# Boneh

The **boneh** calculator returns the Boneh estimate of the number of additional OTUs that would be observed for an additional level of sampling. This calculator can be used in the summary.single, collect.single, and rarefaction.single commands; however, it really only makes sense to use it with the summary.single command.

<math>S_{additional}=\sum_{i=1}^S\left(1-\frac{X_i}{N}\right)^N\left[1-\left(1-\frac{X_i}{N}\right)^M\right]+\nu\left(1-\frac{f_1}{N\nu}\right)^N\left[1-\left(1-\frac{f_1}{N\nu}\right)^M\right]</math>

where,

<math>S</math> = the number of OTUs sampled in initial sampling

<math>X_i</math> = the number individuals in the i^{th} OTU

<math>N</math> = the total number of individuals in the sample

<math>M</math> = the number of additional individuals to sample

<math>\nu</math> is the solution to the equation:

<math>\nu\left(1-\left[1-\frac{f_1}{N\nu}\right]^N\right)= \sum_{i=1}^N f_i \left(1-\frac{i}{N}\right)^N</math>

provided that:

<math>f_1 > \sum_{i=1}^N f_i \exp\left(-i\right)</math>

Open the file 98_lt_phylip_amazon.fn.sabund generated using the Amazonian dataset with the following commands:

mothur > read.dist(phylip=98_lt_phylip_amazon.dist, cutoff=0.10) mothur > cluster()

The 98_lt_phylip_amazon.fn.sabund file is also outputted to the terminal window when the cluster() command is executed:

unique 2 94 2 0.00 2 92 3 0.01 2 88 5 0.02 4 84 2 2 1 0.03 4 75 6 1 2 0.04 4 69 9 1 2 0.05 4 55 13 3 2 0.06 4 48 14 2 4 0.07 4 44 16 2 4 0.08 7 35 17 3 2 1 0 1 0.09 7 35 14 3 3 0 0 2 0.10 7 34 13 3 2 0 0 3

The first column is the label for the OTU definition and the second column is an integer indicating the number of sequences in the dominant OTU. The Boneh estimator is then calculated using the values found in the subsequent columns. For demonstration we will calculate the Boneh estimator for an OTU definition of 0.03. Using GoalSeek in Microsoft Excel, you can find that ν is equal to 31.003. Therefore the predicted number of additional OTUs that would be sampled after sampling 50 additional individuals would be calculated as:

<math>S_{additional}=
75\left(1-\frac{1}{98}\right)^{98}\left[1-\left(1-\frac{1}{98}\right)^{50}\right]
+ 6\left(1-\frac{2}{98}\right)^{98}\left[1-\left(1-\frac{2}{98}\right)^{50}\right]
+ 1\left(1-\frac{3}{98}\right)^{98}\left[1-\left(1-\frac{3}{98}\right)^{50}\right]
+ 2\left(1-\frac{4}{98}\right)^{98}\left[1-\left(1-\frac{4}{98}\right)^{50}\right]
+ 31.003\left(1-\frac{75}{98\left(31.003\right)}\right)^{98}\left[1-\left(1-\frac{75}{98\left(31.003\right)}\right)^50\right]</math>

<math>S_{additional}=13.5</math>

mothur > summary.single(calc=boneh, size=50)

...and opening 98_lt_phylip_amazon.fn.summary gives:

label boneh unique 16.154259 0.00 15.933878 0.01 15.494129 0.02 14.578435 0.03 13.501034 <--- 0.04 12.847995 0.05 11.078614 0.06 10.014746 0.07 9.599282 0.08 8.209852 0.09 7.835304 0.10 7.563195

These are the same values that we found above for a cutoff of 0.03. You can change the size parameter as long as it stays as small or smaller than the number of individuals you have already sampled. The default for the size option is the number of sequences that have already been sampled.