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Example Calculations

*.jack

These files give the interpolated Jackknife estimate as describe by Burnham and Overton (1).

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

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

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

where,

k = The order of the Jackknife estimate

$n_t$ = 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).

References

1. Burnham, K. P., and W. S. Overton. 1979. Robust estimation of population size when capture probabilities vary among animals. Ecology 60:927-936.