# Sharedace

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

$S_{A,B ACE} = S_{12 \left ( abund \right )} + \frac {S_{12 \left ( rare \right )}}{c_{12}} + \frac {1}{C_{12}} \left [ f_{\left ( rare \right )1+} {\Gamma}_1 + f_{\left ( rare \right )+1} {\Gamma}_2 + f_{11}{\Gamma}_3 \right ]$

where,

$C_{12} = 1 - \frac {\sum_{i=1}^{S_{12\left ( rare \right )}} {\left \{Y_i I \left ( X_i = 1 \right ) + X_iI \left ( Y_i = 1 \right ) - I \left ( X_i = Y_i = 1 \right ) \right \}}} {T_{11}}$

${\Gamma}_1 = \frac{S_{12 \left (rare \right )} T_{21}}{C_{12} T_{10}T_{11}} - 1$

${\Gamma}_2 = \frac{S_{12 \left (rare \right )} T_{12}}{C_{12} T_{01}T_{11}} - 1$

${\Gamma}_3 = \left[ \frac{S_{12\left( rare \right)}}{C_{12}}\right ]^2 \frac{T_{22}}{T_{10}T_{01}T_{11}} - \frac{S_{12 \left( rare \right)}T_{11}}{C_{12}T_{01}T_{10}}-{\Gamma}_1-{\Gamma}2$

 $T_{10} = \sum_{i=1}^{S_{12\left( rare \right)}} X_i$ $T_{01} = \sum_{i=1}^{S_{12\left( rare \right)}} Y_i$ $T_{11} = \sum_{i=1}^{S_{12\left( rare \right)}} X_i Y_i$ $T_{21} = \sum_{i=1}^{S_{12\left( rare \right)}} X_i \left( X_i - 1 \right) Y_i$ $T_{12} = \sum_{i=1}^{S_{12\left( rare \right)}} X_i \left( Y_i - 1 \right) Y_i$ $T_{22} = \sum_{i=1}^{S_{12\left( rare \right)}} {X_i \left( X_i - 1 \right) Y_i \left( Y_i - 1 \right)}$

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

$f_{\left(rare \right)1+}$ = number of OTUs with one individual found in A and less than or equal to 10 in B.

$f_{\left(rare \right)+1}$ = number of OTUs with one individual found in B and less than or equal to 10 in A.

$S_{12\left(rare\right)}$ = number of shared OTUs where both of the communities are represented by less than or equal to 10 sequences.

$S_{12\left(abund\right)}$ = number of shared OTUs where at least one of the communities is represented by more than 10 sequences.

$S_{12\left(obs\right)}$ = number of shared OTUs in A and B.

$I\left(\right)$ = evaluates to 1 if the contents are true and to 0 if they are false

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)
mothur > make.shared(list=98_lt_phylip_amazon.fn.list, group=amazon.groups, label=0.10)


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

Looking at the table and treating forest as X and pasture Y, we see that all of the shared OTUs are "rare" (i.e. abundances less than 10). Let's start the calculations...

$T_{10} = 1+3+2+1+1+5+1+1+1=16$

$T_{01} = 1+1+5+3+3+2+1+1+1=18$

$T_{11} = \left(1\right)\left(1\right)+\left(3\right)\left(1\right)+\left(2\right)\left(5\right)+\left(1\right)\left(3\right)+\left(1\right)\left(3\right)+\left(5\right)\left(2\right)+\left(1\right)\left(1\right)+\left(1\right)\left(1\right)+\left(1\right)\left(1\right)=33$

$T_{21} =1\left(1-1\right)1+3\left(3-1\right)1+2\left(2-1\right)5+1\left(1-1\right)3+1\left(1-1\right)3+5\left(5-1\right)2+1\left(1-1\right)1+1\left(1-1\right)1+1\left(1-1\right)1 = 56$

$T_{12} =1\left(1-1\right)1+3\left(1-1\right)1+2\left(5-1\right)5+1\left(3-1\right)3+1\left(3-1\right)3+5\left(2-1\right)2+1\left(1-1\right)1+1\left(1-1\right)1+1\left(1-1\right)1 = 62$

$T_{22} =1\left(1-1\right) 1\left(1-1\right)+3\left(3-1\right) 1\left(1-1\right)+2\left(2-1\right) 5\left(5-1\right)+1\left(1-1\right) 3\left(3-1\right)+1\left(1-1\right) 3\left(3-1\right)+5\left(5-1\right) 2\left(2-1\right)+1\left(1-1\right) 1\left(1-1\right)+1\left(1-1\right) 1\left(1-1\right)+1\left(1-1\right) 1\left(1-1\right)=80$

$C_{12} = 1 - \frac {\left(1+1-1\right)+\left(0+3-0\right)+\left(0+0-0\right)+\left(3+0-0\right)+\left(3+0-0\right)+\left(0+0-0\right)+\left(1+1-1\right)+\left(1+1-1\right)+\left(1+1-1\right)} {33} = 0.606061$

${\Gamma}_1 = \frac{9 \left(56\right)}{0.606061 \left(16\right)33} - 1=0.575$

${\Gamma}_2 = \frac{9 \left(62\right)}{0.606061 \left(18\right)33} - 1=0.550$

${\Gamma}_3 = {\left[ \frac{9}{0.606061}\right ]^2}\frac{80}{16\left(18\right)33} - \frac{9\left(33\right)}{0.606061 \left(16\right)18}-0.575-0.550=-0.97031$

$S_{A,B ACE} = 0 + \frac {9}{0.606061} + \frac {1}{0.606061} \left [ 6\left(0.575\right) + 5\left(0.550\right) + 4 \left(-0.97031\right)\right]=18.676$

Running...

mothur > summary.shared(calc=sharedace)


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

label	comparison		sharedace
0.10	forest	pasture		18.675936


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