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Rarefaction

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mothur calculates rarefaction curve data using the rarefaction.single command for the number of observed OTUs as a function of distance between sequences and the number of sequences sampled. By theory, the rarefaction curve should match the following expression:


<math>S_n = S_t - \left (\frac{\sum_{i=1}^{S_t}{N - N_i \choose n} }Template:N \choose n \right ) </math>

where,

<math>S_n</math> = Average number of OTUs observed after drawing n individuals.

<math>S_{t}</math> = Total number of OTUs in sample of N total individuals.


In the Amazonian dataset the total number of sequences was 98, and there were 84 total OTUs with 75 singletons, 6 doubletons, 1 tripleton, and 2 quadrupletons when defining OTUs at a 0.03 distance cutoff. If we were interested in the predicted observed richness after sampling 50 individuals from this distribution we would evaluate the formula:

<math>S_{50} = 84 - 75\frac{\binom{98 - 1}{50}}{\binom{98}{50}} - 6\frac{\binom{98 - 2}{50}}{\binom{98}{50}} - 1\frac{\binom{98 - 3}{50}}{\binom{98}{50}}- 2\frac{\binom{98 - 4}{50}}{\binom{98}{50}}</math>


<math>S_{50} = 84 - 75\left(0.4898 \right) - 6 \left(0.2373 \right) - 1 \left(0.1137 \right) - 2 \left(0.0539 \right) = 45.620</math>


Although this closed form solution exists, it is actually very slow compared to averaging across a large number of randomized collector's curves. The following table compares the output based on the formula and using 1,000 permutations using mothur:

n 	Theory  mothur Diff. %Error  	n 	Theory  mothur   Diff. %Error
1 	1.000   1.000 0.000  0.000  	50 	45.620  45.634 -0.014 	0.031
2 	1.996   1.995 0.001  0.049  	51 	46.461  46.477 -0.016 	0.035
3 	2.987   2.986 0.001  0.024  	52 	47.299  47.317 -0.017 	0.037
4 	3.974   3.971 0.003  0.069  	53 	48.136  48.154 -0.019 	0.038
5 	4.956   4.954 0.002  0.045  	54 	48.970  48.995 -0.025 	0.051
6 	5.935   5.934 0.001  0.010  	55 	49.802  49.822 -0.019 	0.038
7 	6.909   6.909 0.000  0.002  	56 	50.633  50.647 -0.014 	0.028
8 	7.880   7.881 -0.001 0.014  	57 	51.461  51.478 -0.016 	0.032
9 	8.846   8.847 -0.001 0.013  	58 	52.288  52.305 -0.018 	0.034
10 	9.808   9.809 -0.001 0.011  	59 	53.112  53.126 -0.014 	0.027
11 	10.767 10.768 -0.001 0.012  	60 	53.935  53.946 -0.011 	0.020
12 	11.721 11.721  0.000 0.001  	61 	54.755  54.767 -0.012 	0.022
13 	12.672 12.675 -0.003 0.028  	62 	55.574  55.586 -0.012 	0.021
14 	13.619 13.623 -0.004 0.031  	63 	56.391  56.401 -0.009 	0.017
15 	14.562 14.566 -0.004 0.027  	64 	57.206  57.213 -0.007 	0.012
16 	15.502 15.510 -0.008 0.051  	65 	58.020  58.026 -0.006 	0.010
17 	16.438 16.446 -0.008 0.046  	66 	58.832  58.840 -0.008 	0.014
18 	17.371 17.378 -0.007 0.040  	67 	59.642  59.647 -0.005 	0.009
19 	18.300 18.305 -0.005 0.027      68      60.450  60.448  0.002   0.003
20 	19.225 19.233 -0.007 0.037  	69 	61.256  61.260 -0.004   0.006
21 	20.148 20.158 -0.010 0.049  	70 	62.061  62.057  0.004   0.007
22 	21.066 21.079 -0.013 0.061  	71 	62.865  62.861  0.004   0.006
23 	21.982 21.995 -0.012 0.057  	72 	63.666  63.667  0.000   0.000
24 	22.894 22.904 -0.010 0.042  	73 	64.467  64.469 -0.003   0.004
25 	23.804 23.814 -0.010 0.042  	74 	65.265  65.273 -0.008   0.013
26 	24.710 24.719 -0.010 0.039  	75 	66.062  66.070 -0.008   0.012
27 	25.613 25.623 -0.011 0.041  	76 	66.857  66.862 -0.005   0.007
28 	26.512 26.521 -0.008 0.032  	77 	67.651  67.654 -0.002   0.004
30 	28.303 28.313 -0.010 0.034  	79 	69.235  69.241 -0.007   0.010
31 	29.194 29.204 -0.010 0.033  	80 	70.024  70.035 -0.011   0.015
32 	30.082 30.093 -0.010 0.034  	81 	70.812  70.824 -0.011   0.016
33 	30.967 30.984 -0.016 0.052  	82 	71.599  71.612 -0.013   0.018
34 	31.850 31.863 -0.013 0.041  	83 	72.384  72.391 -0.007   0.010
35 	32.729 32.745 -0.015 0.047  	84 	73.168  73.175 -0.007   0.009
36 	33.606 33.618 -0.012 0.035  	85 	73.950  73.958 -0.008   0.010
37 	34.481 34.498 -0.017 0.050  	86 	74.731  74.746 -0.015   0.020
38 	35.352 35.371 -0.019 0.052  	87 	75.511  75.530 -0.019   0.026
39 	36.221 36.241 -0.020 0.055  	88 	76.289  76.302 -0.013   0.017
40 	37.088 37.109 -0.021 0.056  	89 	77.066  77.081 -0.015   0.020
41 	37.952 37.973 -0.022 0.057  	90 	77.842  77.856 -0.014   0.018
42 	38.813 38.835 -0.021 0.055  	91 	78.616  78.619 -0.003   0.004
43 	39.672 39.687 -0.014 0.036  	92 	79.389  79.391 -0.002   0.003
44 	40.529 40.542 -0.013 0.032  	93 	80.161  80.161  0.000   0.000
45 	41.383 41.391 -0.008 0.020  	94 	80.931  80.930  0.002   0.002
46 	42.235 42.245 -0.010 0.023  	95 	81.700  81.700  0.000   0.000
47 	43.085 43.093 -0.009 0.020  	96 	82.468  82.473 -0.005   0.006
48 	43.932 43.945 -0.013 0.029  	97 	83.235  83.237 -0.003   0.003
49 	44.777 44.792 -0.015 0.034  	98 	84.000  84.000  0.000   0.000