# Chao

The chao calculator returns the Chao1 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 bias-corrected Chao1 richness estimator are implemented as described by Chao in the user manual for her program SPADE and modified by Colwell in his program EstimateS.

$S_{chao1} = S_{obs} + \frac{{n_1}\left ({n_1}-1 \right )}{2\left ({n_2}+1 \right )}$

where,

$S_{chao1}$ = the estimated richness

$S_{obs}$ = the observed number of species

$n_{1}$ = the number of OTUs with only one sequence (i.e. "singletons")

$n_{2}$ = the number of OTUs with only two sequences (i.e. "doubletons")

To calculate the 95% confidence intervals we assume a lognormal distribution of the variance:

$var\left ( S_{chao1} \right ) = \frac{{n_1}\left ({n_1}-1 \right )}{2\left ({n_2}+1 \right )} + \frac{{n_1}\left (2{n_1}-1 \right )^2}{4\left ({n_2}+1 \right )^2} + \frac{{n_1}^2{n_2}\left ({n_1}-1 \right )^2}{4\left ({n_2}+1 \right )^4}$, when n1>0 and n2>0

$var\left ( S_{chao1} \right ) = \frac{{n_1}\left ({n_1}-1 \right )}{2} + \frac{{n_1}\left (2{n_1}-1 \right )^2}{4} - \frac{{n_1}^4}{4S_{chao1}}$, when n1>0 and n2=0

$var\left ( S_{chao1} \right ) = S_{obs} e^{\left (-N / S_{obs} \right )}\left (1- e^{\left (-N / S_{obs} \right )}\right )$, when n1=0 and n2>=0

If n1>0, then:

$C = exp \left (1.96 \sqrt{\ln \left ( 1 + \frac{var\left ( S_{chao1} \right )}{\left ( S_{chao1} - S_{obs}\right )^2 }\right )} \right )$

$LCI_{95%} = S_{obs} + \frac {S_{chao1} - S_{obs}}{C}$

$UCI_{95%} = S_{obs} + C \left ( {S_{chao1} - S_{obs}} \right )$

Otherwise:

$P = e^{\left (-N/S_{obs}\right)}$

$LCI_{95%} = max \left (S_{obs}, \frac{S_{obs}}{1-P}-1.96 \left ( \frac{S_{obs}P}{1-P} \right)^\frac{1}{2}\right )$

$UCI_{95%} = \frac{S_{obs}}{1-P} + 1.96 \left ( \frac{S_{obs}P}{1-P} \right)^\frac{1}{2}$

where,

LCI = Lower bound of confidence interval

UCI = Upper bound of confidence interval

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 Chao1 estimator is then calculated using the values found in the third (the number of singletons)and fourth (the number of doubletons) columns. For example, chao for an OTU definition of 0.10 would be calculated as:

$S_{chao1} = 55 + \frac{34\left (33 \right )}{2\left (14 \right )} = 95.07$

$var\left ( S_{chao1} \right ) = \frac{34\left (33 \right )^2}{2 \left(14 \right)} + \frac{34\left (2 \left(34 \right)-1 \right )^2}{4\left (14 \right )^2} + \frac{34^2\left ( 13 \right ) \left (33 \right )^2}{ 4\left (14 \right )^4} = 341.2494$

$C = exp \left (1.96 \sqrt{\ln \left ( 1 + \frac{341.2494}{\left (95.07 - 55\right )^2 }\right )} \right ) = 2.3641655$

$LCI_{95%} = 55 + \frac {95.07 - 55}{2.3641655} = 71.9$

$UCI_{95%} = 55 + 2.3641655\left ( {95.07 - 55} \right ) = 149.7$

Running...

mothur > summary.single(calc=chao)


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

label	Chao		Chao_lci	Chao_hci
unique	1553.000000	658.490667	3870.016393
0.00	1141.500000	522.185603	2658.668444
0.01	731.000000	376.708692	1527.725165
0.02	1251.000000	533.812860	3124.532743
0.03	480.428571	262.946610	962.226261
0.04	315.600000	192.960027	572.578658
0.05	179.071429	123.996099	293.627620
0.06	143.200000	103.292208	228.234805
0.07	121.647059	91.793695	186.052404
0.08	92.055556	73.303947	135.389388
0.09	96.666667	74.012067	149.489903
0.10	95.071429	71.949979	149.732827 <---


These are the same values that we found above for a cutoff of 0.10.