# Logseries

The logseries calculator returns the Kolmogorov-Smirnov test statistic for the comparison of observed rank-abundance data to a fitted log series distribution and the critical values for α equal to 0.01 or 0.05. This calculator can be used in the summary.single(), collect.single(), and rarefaction.single() commands.

$n_i=\frac{\alpha x^i}{i}$

where,

$\alpha=\frac{N\left(1-x\right)}{x}$

$\frac{S_{obs}}{N}=\frac{1-x}{x}\left(-\ln \left(1-x\right)\right)$ is used to find x

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

$N$ = the total number of individuals

$n_i$ = the number of species with i individuals

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.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 numbers in the subsequent columns indicate the number of singletons, doubletons, etc. Here we will calculate the expected number of individuals in each OTU based on the log series distribution:

$\frac{55}{98}=\frac{1-x}{x}\left(-\ln \left(1-x\right)\right)$, x = 0.654 by Excel

$\alpha=\frac{98\left(1-0.65428\right)}{0.65428}=51.783$

$n_i=\frac{\left(51.783\right) x^i}{i}$

Individuals Obs. OTUs Expect OTUs
1 34 33.9
2 13 11.1
3 3 4.8
4 2 2.4
5 0 1.2
6 0 0.7
7 3 0.4
Sum 55 54.5

Next we break the table into octaves based on log2, adding 0.5 to the top boundary for each octave and compare the difference between the observed and expected frequencies:

Octave Upper Bound Obs. OTUs Expect OTUs Cumul. Obs. F0.5 Cumul. Exp. Difference
1 2.5 47 45.0 47 46.5 45.0 1.5358
2 4.5 5 7.2 52 51.5 52.2 0.6712
3 8.5 3 2.3 55 54.5 54.5 0.0303

To determine whether the log series model describes the distribution of individuals among OTUs as we observed, we will use the Kolmogorov-Smirnov test statistic ($D_{max}$). The statistic is the maximum difference between the cumulative observed and expected values plus 0.5 (i.e. 2.04) divided by the total number of species observed (i.e. 55). So for this case the value was 0.0371. To test this statistic we can calculate the critical value for α=0.05 as 0.886√Sobs or 0.1195 and α=0.01 as 1.031√Sobs or 0.1390. Because our calculated value is less than both critical values we have no reason to reject the hypothesis that the observed data follows the geometric distribution.

Running...

mothur > summary.single(calc=geometric)


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

label	logseries	logseries_lci	logseries_hci
unique	0.010417	0.090427	0.105226
0.00	0.010526	0.090902	0.105778
0.01	0.010753	0.091874	0.106910
0.02	0.034520	0.093916	0.109286
0.03	0.023393	0.096671	0.112491
0.04	0.014553	0.098444	0.114556
0.05	0.013699	0.103698	0.120669
0.06	0.015534	0.107443	0.125027
0.07	0.019353	0.109059	0.126907
0.08	0.018977	0.115347	0.134225
0.09	0.017544	0.117354	0.136559
0.10	0.037017	0.119468	0.139020 <---


In this table the data in the column "logseries" are the calculated statistic value, those in column "logseries_lci" are the critical values for α=0.01, and those in column "logseries_hci" are the critical values for α=0.05. These are the same values that we found above for a cutoff of 0.10.