ace
The ace calculator returns the ACE richness estimate for an OTU definition. This calculator can be used in the summary.single, collect.single, and rarefaction.single commands. The calculations for the ACE richness estimator are implemented as described by Chao in the user manual for her program spade and modified by Colwell in his program estimates. Note that this implementation uses a bias-corrected version of γ when the coefficient of variation is too large.
\[S_{ACE} = \begin{cases} S_{abund} + \frac {S_{rare}}{C_{ACE}} + \frac{n_1}{C_{ACE}}{ {\hat{\gamma}}_{ACE}^2}\mbox{, for }\hat{\gamma}_{ACE}<\mbox{0.80}\\ S_{abund} + \frac {S_{rare}}{C_{ACE}} + \frac{n_1}{C_{ACE}}{ {\tilde{\gamma}}_{ACE}^2}\mbox{, for }\hat{\gamma}_{ACE}\geqslant\mbox{0.80} \end{cases}\]where,
\[N_{rare} = \sum_{i=1}^{abund}{in_i}\] \[C_{ACE} = 1 - \frac {n_1}{N_{rare}}\] \[{ {\hat{\gamma}}_{ACE}^2} = max \left[ \frac {s_{rare}}{c_{ace}} \frac{\sum_{i=1}^{abund} i \left ( i-1 \right ) n_i }{n_{rare} \left( n_{rare} - 1 \right )} - 1,0 \right ]\] \[\tilde{\gamma}_{ACE}^2 = max \left[\hat{\gamma}_{ace}^2 \left\{1+\frac{n_{rare}\left(1-c_{ace}\right)\sum_{i=1}^{abund} i \left ( i-1 \right ) n_i }{n_{rare}\left(n_{rare}-c_{ace}\right)}\right\}\mbox{, 0}\right]\] \[var \left( S_{ACE} \right ) {\approx} \sum_{j=1}^{n} \sum_{i=1}^{n} \frac{ {\partial}S_{ACE}}{ {\partial}n_i} \frac{ {\partial}S_{ACE}}{ {\partial}n_j}\] \[cov \left( f_i, f_j \right) = f_i \left(1-f_i / S_{ACE} \right ), i = j\] \[cov\left ( f_i, f_j \right) = -f_i f_j / {S_{ACE}}, i\ne j\]\(n_{i}\) = The number of OTUs with i individuals
\(S_{rare}\) = The number of OTUs with ‘abund’ or fewer individuals
\(S_{abund}\) = The number of OTUs with more than ‘abund’ individuals
\(abund\) = the threshold to be considered an ‘abundant’ OTU; this is set to 10 by default and can be changed with the ‘abund’ parameter in summary.single, collect.single, and rarefaction.single.
Open the file 98_lt_phylip_amazon.fn.sabund generated using the Amazonian dataset with the following commands:
mothur > cluster(phylip=98_lt_phylip_amazon.dist, cutoff=0.10)
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 ACE estimator is then calculated using the values found in the subsequent columns. By default ACE is calculated using an abundance threshold of 10. This value was empirically selected in the original publications, but can be altered. You may wish to increase or decrease this value. For demonstration we will calculate the ACE estimator with an abundance threshold of 5 and an OTU definition of 0.10:
\[S_{rare} = 34 + 13+3+2+0 =52\] \[S_{abund} = 3\] \[N_{rare} = \sum_{i=1}^{5}{in_i} = 1\left(34\right)+2\left(13\right)+3\left(3\right)+4\left(2\right)+5\left(0\right) = 77\] \[C_{ACE} = 1 - \frac {34}{77} = 0.558442\] \[{ {\gamma}_{ACE}^2} = max \left[ \frac {52}{0.558442} \frac{ 1 \left ( 1-1 \right ) 34 + 2 \left ( 2-1 \right ) 13 + 3 \left ( 3-1 \right ) 3+ 4 \left ( 4-1 \right ) 2+ 5 \left ( 5-1 \right ) 0 }{77 \left( 77 - 1 \right )} - 1,0 \right ] = 0.0820\] \[S_{ACE} = 3 + \frac {52}{0.558442} + \frac{34}{0.558442}0.0820 = 101.1\]The code to calculate the confidence intervals was provided by Anne Chao and is left for masochists to work out on their own.
Running...
mothur > summary.single(calc=ace, abund=5)
...and opening 98_lt_phylip_amazon.fn.summary gives:
label ace ace_lci ace_hci
unique 2352.000000 165.123709 73725.382535
0.00 1551.666667 166.520625 29763.054270
0.01 911.400000 171.653666 8608.541528
0.02 1964.419917 1196.763495 3264.045830
0.03 543.693450 296.302894 1079.361221
0.04 362.831975 217.072510 664.727469
0.05 190.203699 131.260760 308.779744
0.06 155.277262 111.141296 244.566797
0.07 132.491876 98.445244 202.265564
0.08 101.208333 78.400541 150.829575
0.09 102.821244 77.832409 157.784620
0.10 101.109191 75.477067 158.826271 <---
These are the same values that we found above for a cutoff of 0.10.