# Ace

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Example Calculations

*ace*

These files give the ACE richness estimates as described by Chao and her colleagues (3, 4). MOTHUR calculates the 95% confidence interval using an algorithm for the standard error estimate obtained through a personal communication with Anne Chao.

$N_{rare} = \sum_{i=1}^{10}{in_i}$

$C_{ACE} = 1 - \frac {n_1}{N_{rare}}$

${{\gamma}_{ACE}^2} = max \left[ \frac {S_{rare}}{C_{Ace}} \frac{\sum_{i=1}^{10} i \left ( i-1 \right ) n_i }{N_{rare} \left( N_{rare} - 1 \right )} - 1,0 \right ]$

$S_{ACE} = S_{abund} + \frac {S_{rare}}{C_{ACE}} + \frac{n_1}{C_{ACE}}{{\gamma}_{ACE}^2}$

$var \left( S_{ACE} \right ) {\approx} \sum_{j=1}^{n} \sum_{i=1}^{n} \frac{{\partial}S_{ACE}}{{\partial}n_i} \frac{{\partial}S_{ACE}}{{\partial}n_j}, cov \left( f_i, f_j \right) = f_i \left(1-f_i / S_{ACE} \right ), \mbox{if } i = j, cov\left ( f_i, f_j \right) = -f_i f_j / {S_{ACE}}, \mbox{if } i j$

where,

$n_{i}$ = The number of OTUs with i individuals

$S_{rare}$ = The number of OTUs with 10 or fewer individuals

$S_{abund}$ = The number of OTUs with more than 10 individuals

Returning to the Amazonian dataset at distance 0.03, with the previously described distribution, there are no "abundant" OTUs so $S_{abund}$ is 0 and $S_{rare}$ and $S_{obs}$ are 84 and $N_{rare}$ is 98. Since there are 75 singletons, the coverage, $C_{ACE}$, is 0.235. Calculating the 2 value we obtain the value 0.581. This gives an ${S_{ACE}}$ value of 543.69.

File Samples on the Amazonian Dataset

• .sabund

This file contains data for constructing a rank-abundance plot of the OTU data for each distance level. The first column contains the distance and the second is the number of OTUs observed at that distance. The successive values in the row are the number of OTUs that were found once, twice, etc.

unique	   2	94	2
0	   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	36	15	4	2	1	0	1
0.09	   7	36	12	4	3	0	0	2
0.1	   7	35	12	2	3	0	0	3


• .ace

The first line contains the labels of all the columns. First numsampled which shows the frequency of the observed calculations. The frequency was set to 10, so after each 10 selected the observed is calculated at each of the distances, with a calculation done after all are sampled. The following labels in the first line are the distances at which the calculations were made, the lci (lower bound of confidence interval) and the hci (higher bound of confidence interval). Note: the entire file is not shown below. Each additional line starts with the number of sequences sampled followed by the ${S_{ACE}}$ calculation at the column's distance and the confidence intervals. For instance, at distance 0.01, after 80 samples ace was 1026.67, the lci was 130.41 and the hci was 16963.81.

numsampled	0.01	lci	hci	0.02	lci	hci	0.03	lci	hci
1		0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
10		0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
20		0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
30		0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
40		0.00	0.00	0.00	380.00	56.55	6344.87	780.00	56.82	30860.25
50		1225.00	74.06	55231.49 600.00	73.63	11938.38 600.00	73.63	11938.38
60		1770.00	380.69	9159.34	449.27	192.09	1188.89	870.00	91.45	19769.41
70		1190.00	109.96	30068.86 1241.48 711.79	2203.64	475.04	220.06	1146.61
80		1026.67	130.41	16963.81 1209.01 727.09	2044.92	731.50	322.55	1810.65
90		967.50	152.51	11768.79 1605.86 973.30	2686.86	609.23	305.71	1320.90
98		911.40	171.65	8608.54	1964.42	1196.76	3264.05	543.69	296.30	1079.36


References

3. Chao, A., R. L. Chazdon, R. K. Colwell, and T. J. Shen. 2005. A new statistical approach for assessing similarity of species composition with incidence and abundance data. Ecol. Lett. 8:148-159.

4. Chao, A., W. H. Hwang, Y. C. Chen, and C. Y. Kuo. 2000. Estimating the number of shared species in two communities. Stat. Sinica. 10:227-246.