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Difference between revisions of "Jack"

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'''''*.jack'''''  
 
'''''*.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.
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These files give the interpolated Jackknife estimate as describe by Burnham and Overton (1).  
  
  
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where,  
 
where,  
k = The order of the Jackknife estimate
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nt = The number of sequences in the largest OTU  
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k = The order of the Jackknife estimate
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<math>n_t</math> = 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:
 
To determine which order of the estimate to use it is necessary to calculate the test statistics, Tk:
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where,
 
where,
  
    <math>b_i = a_{i,k+1}-a_{i,k}</math>
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<math>b_i = a_{i,k+1}-a_{i,k}</math>
 
 
 
 
  
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second order Jackknife estimate.  This method essentially uses a statistical procedure to  
 
second order Jackknife estimate.  This method essentially uses a statistical procedure to  
 
determine which order results in the minimum bias (error).
 
determine which order results in the minimum bias (error).
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 +
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'''References'''
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 +
1. Burnham, K. P., and W. S. Overton. 1979. Robust estimation of population size when
 +
capture probabilities vary among animals. Ecology 60:927-936.

Revision as of 15:40, 14 January 2009

Validate output by making calculations by hand

Example Calculations

*.jack

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


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


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


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


where,

k = The order of the Jackknife estimate <math>n_t</math> = 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:


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


<math>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 ]</math>


where,

<math>b_i = a_{i,k+1}-a_{i,k}</math>


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


<math>c = \frac {0.05 - P_{k-1}}{P_k - P_{k-1}}</math>


<math>d_i = ca_{i,k} + \left( i-c \right )a_{i,k-1}</math>


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


<math>S_{jack} = \sum_{i=1}^{n_1}d_i n_i</math>


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


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.