# Shannon

The shannon calculator returns the Shannon diversity index for an OTU definition. This calculator can be used in the summary.single, collect.single, and rarefaction.single commands.

$H_{shannon} = - \sum_{i=1}^{S_{obs}} \frac{n_i}{N} ln \frac{n_i}{N}$

$var\left ( H_{Shannon} \right ) = \frac {\sum_{i=1}^{S_{obs}} \frac{n_i}{N} \left ( ln \frac{n_i}{N} \right )^2 - H_{shannon}^{2}}{N} + \frac{S_{obs} - 1}{2N^{2}}$

$LCI_{95%}=H_{shannon}-1.96\sqrt{var\left ( H_{shannon} \right )}$

$UCI_{95%}=H_{shannon}+1.96\sqrt{var\left ( H_{shannon} \right )}$

where,

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

$n_i$ = the number of individuals in OTU i

$N$ = the total number of individuals in the community

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 third column indicates the number of OTUs with only one indivdiual, the fourth the number of OTUs with two individuals, etc. The Shannon index is then calculated using the values found in the subsequent columns. For demonstration we will calculate the Shannon index for an OTU definition of 0.03:

$H_{shannon} = -\left(75\left(\frac{1}{98}\right)\ln\left(\frac{1}{98}\right)+6\left(\frac{2}{98}\right)\ln\left(\frac{2}{98}\right)+1\left(\frac{3}{98}\right)\ln\left(\frac{3}{98}\right)+2\left(\frac{4}{98}\right)\ln\left(\frac{4}{98}\right)\right) = 4.3533$

$var\left(H_{shannon}\right)=\frac{75\frac{1}{98}\left(\ln\frac{1}{98}\right)^2+6\frac{2}{98}\left(\ln\frac{2}{98}\right)^2+1\frac{3}{98}\left(\ln\frac{3}{98}\right)^2+2\frac{4}{98}\left(\ln\frac{4}{98}\right)^2-\left(4.35\right)^2}{98}+\frac{83}{2\left(98\right)^2} = 0.006644$

$LCI_{95%}= 4.3533-1.96\sqrt{0.006644}=4.19$

$UCI_{95%}= 4.3533 +1.96\sqrt{0.006644}=4.51$

Running...

mothur > summary.single(sabund=98_lt_phylip_amazon.fn.sabund, calc=shannon)


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

label	Shannon		Shannon_lci	Shannon_hci
unique	4.556676	4.415476	4.697875
0.00	4.542530	4.400817	4.684243
0.01	4.514238	4.371670	4.656806
0.02	4.432830	4.279100	4.586561
0.03	4.353294	4.196449	4.510139<---
0.04	4.310857	4.154934	4.466780
0.05	4.187011	4.032469	4.341554
0.06	4.093330	3.936160	4.250500
0.07	4.065038	3.910282	4.219793
0.08	3.909320	3.745866	4.072774
0.09	3.838295	3.665956	4.010634
0.10	3.770031	3.590406	3.949656


These are the same values that we found above for a cutoff of 0.03.