# Sharedchao

The sharedchao calculator returns the shared Chao1 richness estimate for an OTU definition. This calculator can be used in the summary.shared and collect.shared commands. The calculations for the shared Chao1 richness estimator are implemented as described by Chao in the user manual for her program SPADE.

$S_{SharedChao} = \begin{cases} S_{Shared\left ( Obs \right )} + \frac{f_{1+}^{2}}{2f_{2+}} + \frac{f_{+1}^{2}}{2f_{+2}} + \frac{f_{11}^{2}}{4f_{22}}\mbox{, for 2 communities and all denominators}>\mbox{0} \\ S_{Shared\left ( Obs \right )} + \frac{f_{1+}\left(f_{1+}-1\right)}{2\left(f_{2+}+1\right)} + \frac{f_{+1}\left(f_{+1}-1\right)}{2\left(f_{+2}+1\right)} + \frac{f_{11}\left(f_{11}-1\right)}{4\left(f_{22}+1\right)}\mbox{, for 2 communities and any denominator}=\mbox{0} \\ S_{Shared\left ( Obs \right )} + \frac{f_{1++}^{2}}{2f_{2++}} + \frac{f_{+1+}^{2}}{2f_{+2+}} + \frac{f_{++1}^{2}}{2f_{++2}} + \frac{f_{11+}^{2}}{4f_{22+}} + \frac{f_{1+1}^{2}}{4f_{2+2}} + \frac{f_{+11}^{2}}{4f_{+22}} + \frac{f_{111}^{2}}{8f_{222}}\mbox{, for 3 communities and all denominators}>\mbox{0} \\ S_{Shared\left ( Obs \right )} + \frac{f_{1++}\left(f_{1++}-1\right)}{2\left(f_{2++}+1\right)} + \frac{f_{+1+}\left(f_{+1+}-1\right)}{2\left(f_{+2+}+1\right)} + \frac{f_{++1}\left(f_{++1}-1\right)}{2\left(f_{++2}+1\right)} + \frac{f_{11+}\left(f_{11+}-1\right)}{4\left(f_{22+}+1\right)} + ... + \frac{f_{+11}\left(f_{+11}-1\right)}{4\left(f_{+22}+1\right)} + \frac{f_{111}\left(f_{111}-1\right)}{8\left(f_{222}+1\right)} \mbox{, for 3 communities and any denominator}=\mbox{0} \\ S_{Shared\left ( Obs \right )}+ \frac{f_{1+++}^{2}}{2f_{2+++}} + ... + \frac{f_{+++1}^{2}}{2f_{+++2}} + \frac{f_{11++}^{2}}{4f_{22++}} + \frac{f_{1+1+}^{2}}{4f_{2+2+}} + ... + \frac{f_{++11}^{2}}{4f_{++22}} + \frac{f_{111+}^{2}}{8f_{222+}} + \frac{f_{11+1}^{2}}{8f_{22+2}} + ... + \frac{f_{++11}^{2}}{8f_{+222}} + \frac{f_{1111}^{2}}{16f_{2222}}\mbox{, for 4 communities and all denominators}>\mbox{0} \\ S_{Shared\left ( Obs \right )} + \frac{f_{1+++}\left(f_{1+++}-1\right)}{2\left(f_{2+++}+1\right)} + ... + \frac{f_{+++1}\left(f_{+++1}-1\right)}{2\left(f_{+++2}+1\right)} + \frac{f_{11++}\left(f_{11++}-1\right)}{4\left(f_{22++}+1\right)} + ... + \frac{f_{++11}\left(f_{++11}-1\right)}{4\left(f_{++22}+1\right)}+ + \frac{f_{111+}\left(f_{111+}-1\right)}{4\left(f_{222+}+1\right)} + ... + \frac{f_{+111}\left(f_{+111}-1\right)}{8\left(f_{+222}+1\right)}+ \frac{f_{1111}\left(f_{1111}-1\right)}{16\left(f_{2222}+1\right)} \mbox{, for 4 communities and any denominator}=\mbox{0} \\ \mbox{etc.} \end{cases}$

where,

$f_{11}$ = number of shared OTUs with one observed individual in A and B

$f_{1+}, f_{2+}$ = number of shared OTUs with one or two individuals observed in A

$f_{+1}, f_{+2}$ = number of shared OTUs with one or two individuals observed in B

$S_{Shared\left(obs\right)}$ = number of shared OTUs observed in A and B.

mothur is able to estimate the total number of shared OTUs across any number of groupings. Examples calculations for two and three groups are shown below.

### Two group example

Open the file 98_lt_phylip_amazon.fn.shared 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.shared file will contain the following two lines:

0.10	forest	55	1	1	1	1	1	1	3	3	2	2	1	1	3	2	1	1	1	1	2	1	1	2	5	1	1	1	1	2	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0.10	pasture	55	0	0	0	1	1	0	1	0	0	5	0	0	0	0	0	2	0	0	0	3	0	0	2	1	0	1	0	0	0	0	0	0	1	2	1	1	1	1	1	7	1	1	2	1	1	1	1	1	1	1	1	1	2	1	1


This indicates that the label for the OTU definition was 0.10. The first line is from the forest sample and the second is from the pasture sample. There are a total of 55 OTUs between the two communities. Writing the data in a more presentable manner we see:

index forest pasture S12 f1+ f+1 f2+ f+2 f11
1 1 0
2 1 0
3 1 0
4 1 1 X X X X
5 1 0
6 1 0
7 3 1 X X
8 3 0
9 2 0
10 2 5 X X
11 1 0
12 1 0
13 3 0
14 2 0
15 1 0
16 1 3 X X
17 1 0
18 1 0
19 2 0
20 1 3 X X
21 1 0
22 2 0
23 5 2 X X
24 1 1 X X X X
25 1 0
26 1 1 X X X X
27 1 0
28 2 0
29 1 0
30 1 0
31 1 0
32 1 0
33 1 1 X X X X
34 0 2
35 0 1
36 0 1
37 0 1
38 0 1
39 0 1
40 0 7
41 0 1
42 0 1
43 0 2
44 0 1
45 0 1
46 0 1
47 0 1
48 0 1
49 0 1
50 0 1
51 0 1
52 0 1
53 0 2
54 0 1
55 0 1
Total 33 31 9 6 5 1 1 4

Using these sums to evaluate $S_{SharedChao}$ we get:

$S_{SharedChao} = 9 + \frac{6\left(5\right)}{2\left(2\right)} + \frac{5\left(4\right)}{2\left(2\right)} + \frac{4\left(3\right)}{4\left(1\right)} = 24.5$

Running...

mothur > summary.shared(calc=sharedchao)


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

label	comparison		sharedchao
0.10	forest	pasture		24.500000


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

### Three group example

Open the file esophagus.fn.shared generated using the Esophagus dataset with the following commands:

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


The esophagus.fn.shared file will contain the following three lines:

0.10	B	51	2	66	0	0	27	1	37	5	1	2	8	2	0	0	0	0	0	0	0	23	3	0	1	2	1	0	0	6	1	0	0	1	0	5	2	0	0	0	0	2	0	0	0	0	0	2	1	1	1	1	1
0.10	C	51	14	84	4	4	22	2	42	13	10	13	7	5	2	4	3	2	8	4	3	2	2	1	1	2	6	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0.10	D	51	5	147	0	3	10	1	18	3	0	0	1	1	2	1	0	2	0	0	0	2	5	0	1	6	2	0	0	3	6	2	2	3	3	2	1	1	1	1	1	1	1	1	1	3	1	1	0	0	0	0	0


This indicates that the label for the OTU definition was 0.10. The first line is from patient B, the second from patient C, and the third from patient D. There are a total of 51 OTUs between the three communities and there are 15 OTUs shared between all three patients. Writing the data in a more presentable manner we see:

OTU B C D shared f1++ f+1+ f++1 f11+ f1+1 f+11 f111 f2++ f+2+ f++2 f22+ f2+2 f+22 f222
1 2 14 5 X X
2 66 84 147 X
3 27 22 10 X
4 1 2 1 X X X X X
5 37 42 18 X
6 5 13 3 X
7 8 7 1 X X
8 2 5 1 X X X
9 23 2 2 X X X X
10 3 2 5 X X
11 1 1 1 X X X X X X X X
12 2 2 6 X X X X
13 1 6 2 X X X
14 6 1 3 X X
15 1 1 6 X X X X
16 0 4 0
17 0 4 3
18 1 10 0
19 2 13 0
20 0 2 2
21 0 4 1
22 0 3 0
23 0 2 2
24 0 8 0
25 0 4 0
26 0 3 0
27 0 1 0
28 0 1 0
29 0 1 0
30 0 0 2
31 0 0 2
32 1 0 3
33 0 0 3
34 5 0 2
35 2 0 1
36 0 0 1
37 0 0 1
38 0 0 1
39 0 0 1
40 2 0 1
41 0 0 1
42 0 0 1
43 0 0 1
44 0 0 3
45 0 0 1
46 2 0 1
47 1 0 0
48 1 0 0
49 1 0 0
50 1 0 0
51 1 0 0
Total 205 264 245 15 4 3 4 2 2 1 1 3 4 2 1 0 1 0

Using these sums to evaluate $S_{SharedChao}$ we get:

$S_{SharedChao} = 15 + \frac{4\left(3\right)}{2\left(4\right)} + \frac{3\left(2\right)}{2\left(5\right)} + \frac{4\left(3\right)}{2\left(3\right)} + \frac{2\left(1\right)}{4\left(2\right)} + \frac{2\left(1\right)}{4\left(1\right)} + \frac{1\left(0\right)}{4\left(2\right)} + \frac{1\left(0\right)}{8\left(1\right)} = 19.85$

Running...

mothur > summary.shared()


...and opening esophagus.unique.fn.sharedmultiple.summary gives:

label	comparison	sharedsobs	sharedchao
0.10	B-C-D		15.000000	19.850000


These are the same values that we found above for a cutoff of 0.10. The file esophagus.unique.fn.shared.summary will contain the pairwise comparisons between patients B, C, and D.

This function is ram dependent, so as your number of groups grows the memory needed to do the combined group estimate grows exponentially. To avoid this problem, try comparing less groups by using the groups parameter to select the groups you would like to compare.