# Thetan

The thetan calculator returns the community Jaccard index of Smith (aka the Θn of Yue) describing the dissimilarity between the structure of two communities. This calculator can be used in the summary.shared and collect.shared commands.

$D_{\Theta_N}=1-\frac{\left(\sum_{i=1}^{S_{12}} a_i\right)\left(\sum_{i=1}^{S_{12}} b_i\right)}{\left(\sum_{i=1}^{S_{12}} a_i\right)+\left(\sum_{i=1}^{S_{12}} b_i\right)-\left(\sum_{i=1}^{S_{12}} a_i\right)\left(\sum_{i=1}^{S_{12}} b_i\right)}$

where,

$S_{12}$ = the number of shared OTUs between communities A and B

$a_i$ = the relative abundance of OTU i in community A

$b_i$ = the relative abundance of OTU i in community B

Open the file 98_lt_phylip_amazon.fn.sabund 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 (A) pasture (B) ai bi
1 1 0
2 1 0
3 1 0
4 1 1 0.0204 0.0204
5 1 0
6 1 0
7 3 1 0.0612 0.0204
8 3 0
9 2 0
10 2 5 0.0408 0.1020
11 1 0
12 1 0
13 3 0
14 2 0
15 1 0
16 1 3 0.0204 0.0612
17 1 0
18 1 0
19 2 0
20 1 3 0.0204 0.0612
21 1 0
22 2 0
23 5 2 0.1020 0.0408
24 1 1 0.0204 0.0204
25 1 0
26 1 1 0.0204 0.0204
27 1 0
28 2 0
29 1 0
30 1 0
31 1 0
32 1 0
33 1 1 0.0204 0.0204
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 49 49 0.3265 0.3673

Using these sums to evaluate C we get:

$C_{\Theta_N}=1-\frac{\left(0.3265\right)\left(0.3673\right)}{\left(0.3265\right)+\left(0.3673\right)-\left(0.3265\right)\left(0.3673\right)}=0.7910$

Running...

mothur > summary.shared(calc=thetan)


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

label	comparison		ThetaN
0.10	forest	pasture		0.791001


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