# Jack

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Example Calculations

*.jack

These files give the interpolated Jackknife estimate as describe by Burnham and Overton (1). Since this is a complicated calculation, it makes MOTHUR run much longer. Therefore, if you want the rarefied interpolated Jackknife estimate calculated, you must tell MOTHUR to do so by using the "-j" flag at the command line. If it is not calculated, MOTHUR will still produce the *.r_jack* files, but they will be filled with zeros.

$S_{jack,k} = S_{obs} + \sum_{i=1}^k \left ( -1 \right )^{i+1} {k \choose i} n_i$

$var \left( S_{jack,k} \right ) = \sum_{i=1}^{n_1} \left ( a_{ik} \right )^2 n_i - S_{jack,k}$

$a_{ik} = \langle \left ( -1 \right )^{i+1} {k \choose i} + 1, i = 1...k, 1, i > k$

where,

k = The order of the Jackknife estimate
nt = The number of sequences in the largest OTU

To determine which order of the estimate to use it is necessary to calculate the test statistics, Tk:

$T_k = \frac{S_{jack,k+1} - S_{jack,k}}{\left ( var \left( S_{jack,k+1} - S_{jack,k} | S \right )\right )^2}$

$var \left( S_{jack,k+1} - S_{jack,k} | S \right ) = \frac {S_{obs}}{S_{obs}-1} \left [ \sum_{i=1}^{n_1} \left ( b_{i}^2 n_i \right ) - \frac{\left ( S_{jack,k+1} - S_{jack,k} \right )^2 }{S_{obs}}\right ]$

where,

$b_i = a_{i,k+1}-a_{i,k}$

For each Tk value, calculate its two-sided p-value. Find the first k-value where Pk>0.05 and calculate c and d:

$c = \frac {0.05 - P_{k-1}}{P_k - P_{k-1}}$

$d_i = ca_{i,k} + \left( i-c \right )a_{i,k-1}$

With c and d, calculate the interpolated Sjack and its standard error:

$S_{jack} = \sum_{i=1}^{n_1}d_i n_i$

$se \left ( S_{jack} \right ) = \left ( \sum_{i=1}^{n_1} \left ( d_{i}^2 n_i\right )-S_{jack}\right )^{0.5}$

For the Amazonian dataset, you can calculate the following:

k     Sj,k    var      Tk       Pk
1     159     150     13.91  <0.0001
2     228     450     8.89   <0.0001
3     292     938     5.77   <0.0001
4     350     1700    3.36    0.0008
5     399     2940    1.54    0.1235
6     434     5250

The p-value crosses 0.05 between a order of 4 and 5 and you can calculate a c-value of 0.40 and the interpolated Sjack of 369.64 with 95% confidence interval between 278.98 and 460.30. Note that programs like EstiamteS and various microbial ecology papers present either the first and/or second order Jackknife estimate. This method essentially uses a statistical procedure to determine which order results in the minimum bias (error).