Difference between revisions of "CC1/2"

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== number of reflection pairs ==
 
[[CORRECT.LP]] and XSCALE.LP do not explicitly state the ''number of reflection pairs'' that were used to calculated CC<sub>1/2</sub>.
 
 
However, the number can be calculated from the numbers available, for each resolution shell: there is the NUMBER OF UNIQUE REFLECTIONS (X), the NUMBER OF OBSERVED REFLECTIONS (Y), and the number of COMPARED reflections (Z) - the latter number is the total number of unmerged observations that contributed to the CC<sub>1/2</sub> and the R-value calculations.
 
 
The ''number of reflections pairs'' that were used for the CC<sub>1/2</sub> calculation can therefore be obtained as follows: Y-Z gives the number of unique reflections that have a single observation. The remaining (X-Y+Z) unique reflections have multiple observations, i.e. there were  (X-Y+Z) reflection pairs that went into CC<sub>1/2</sub>.
 
 
 
== why CC<sub>1/2</sub> can be negative ==
 
There is a mathematical reason, explained in §4.1 of [https://cms.uni-konstanz.de/index.php?eID=tx_nawsecuredl&u=0&g=0&t=1475179096&hash=5cf64234a23a794a1894c5408384c57208d7b602&file=fileadmin/biologie/ag-strucbio/pdfs/Assman2016_JApplCryst.pdf Assmann, G., Brehm, W. and Diederichs, K. (2016) Identification of rogue datasets in serial crystallography (2016) J. Appl. Cryst. 49, 1021-1028.]
 
 
  
 
==CC<sub>1/2</sub> calculation==  
 
==CC<sub>1/2</sub> calculation==  
  
CC<sub>1/2</sub>  is calculated by:
+
CC<sub>1/2</sub>  can be calculated with the so-called σ-τ method ([https://cms.uni-konstanz.de/index.php?eID=tx_nawsecuredl&u=0&g=0&t=1475179096&hash=5cf64234a23a794a1894c5408384c57208d7b602&file=fileadmin/biologie/ag-strucbio/pdfs/Assman2016_JApplCryst.pdf Assmann, G., Brehm, W. and Diederichs, K. (2016) Identification of rogue datasets in serial crystallography (2016) J. Appl. Cryst. 49, 1021-1028.]) by:
  
 
: <math>CC_{1/2}=\frac{\sigma^2_{\tau}}{\sigma^2_{\tau}+\sigma^2_{\epsilon}} =\frac{\sigma^2_{y}- \frac{1}{2}\sigma^2_{\epsilon}}{\sigma^2_{y}+ \frac{1}{2}\sigma^2_{\epsilon}} </math>
 
: <math>CC_{1/2}=\frac{\sigma^2_{\tau}}{\sigma^2_{\tau}+\sigma^2_{\epsilon}} =\frac{\sigma^2_{y}- \frac{1}{2}\sigma^2_{\epsilon}}{\sigma^2_{y}+ \frac{1}{2}\sigma^2_{\epsilon}} </math>
  
This requires calculation of <math>\sigma^2_{y} </math>, the variance of the average intensities across the unique reflections of a resolution shell, and <math>\sigma^2_{\epsilon} </math>, the average of all sample variances of the mean across all unique reflections of a resolution shell.  
+
This requires calculation of <math>\sigma^2_{y} </math>, the variance of the average intensities across the unique reflections of a resolution shell, and <math>\sigma^2_{\epsilon} </math>, the average of all sample variances of the averaged (merged) intensities across all unique reflections of a resolution shell.  
  
== Implementation ==
+
== Method ==
  
 
===''' <math>\sigma^2_{\epsilon} </math>'''===
 
===''' <math>\sigma^2_{\epsilon} </math>'''===
  
The average of all sample variances of the mean across all unique reflections of a resolution shell is obtained by calculating the sample variance of the mean for every unique reflection i by:
+
With <math>x_{j,i} </math> , a single observation j of all observations n of one reflection i, the average of all sample variances of the mean across all unique reflections of a resolution shell is obtained by calculating the sample variance of the mean for every unique reflection i by:
  
<math>\sigma^2_{\epsilon i} =  \frac{1}{n-1} \cdot \left ( \sum^n_{j} x^2_{j,i} - \frac{\left ( \sum^n_{j}x_{j,i} \right )^2}{ n} \right )    / \frac{n}{2} </math>
+
<math>\sigma^2_{\epsilon i} =  \frac{1}{n_{i}-1} \cdot \left ( \sum^{n_{i}}_{j} x^2_{j,i} - \frac{\left ( \sum^{n_{i}}_{j}x_{j,i} \right )^2}{n_{i}} \right )    / \frac{n_{i}}{2} </math>
  
With <math>x_{j,i} </math> , a single observation j of all observations n of one reflection i. <math>\sigma^2_{\epsilon i} </math> is then divided by the factor  <math>\frac{n}{2} </math>, because the variance of the sample mean (intensities of the merged observations) is the quantity of interest. The division by n/2 takes care of providing the variance of the mean (merged) intensity of the half-datasets, as defined in [https://en.wikipedia.org/wiki/Sample_mean_and_covariance#Variance_of_the_sample_mean ]. These "variances of means" are averaged over all unique reflections of the resolution shell:
+
<math>\sigma^2_{\epsilon i} </math> is then divided by the factor  <math>\frac{n}{2} </math>, because the variance of the sample mean (intensities of the merged observations) is the quantity of interest. The division by '''n/2''' takes care of providing the variance of the mean ([https://en.wikipedia.org/wiki/Sample_mean_and_covariance#Variance_of_the_sample_mean ]) (merged) intensity of the '''half'''-datasets, as defined in [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3457925/ Karplus and Diederichs (2012)]. These "variances of means" are averaged over all unique reflections of the resolution shell:
  
 
<math>\sigma^2_{\epsilon}=\sum^N_{i} \sigma^2_{\epsilon i} / N </math>  
 
<math>\sigma^2_{\epsilon}=\sum^N_{i} \sigma^2_{\epsilon i} / N </math>  
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===''' <math>\sigma^2_{y} </math>'''===
 
===''' <math>\sigma^2_{y} </math>'''===
  
The unbiased sample variance from all averaged intensities of all unique reflections is calculated by:  
+
Let N be the number of reflections. With <math>\overline{x}_{i}= \sum^n_{j} x_{j,i}</math> , the unbiased sample variance from all averaged intensities of all unique reflections is calculated by:  
  
 
<math>\sigma^2_{y} = \frac{1}{N-1} \cdot \left ( \sum^N_{i} \overline{x}_{i}^2 - \frac{\left ( \sum^N_{i} \overline{x}_{i} \right )^2}{ N} \right ) </math>
 
<math>\sigma^2_{y} = \frac{1}{N-1} \cdot \left ( \sum^N_{i} \overline{x}_{i}^2 - \frac{\left ( \sum^N_{i} \overline{x}_{i} \right )^2}{ N} \right ) </math>
 
With <math>\overline{x}_{i}= \sum^n_{j} x_{j,i}</math> , average intensity of all observations from all frames/crystals of one unique reflection i. This is done for all reflections N in a resolution shell.
 
  
  
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     0    0    2  7.301E+02  2.405E+01  2  
 
     0    0    2  7.301E+02  2.405E+01  2  
 
</pre>
 
</pre>
<math>\overline{x}_{i} </math> , the average intensity of all observations of this reflection = 669.6999
+
<math>\overline{x}_{1} </math> , the average intensity of all observations of this reflection = 669.6999
  
<math>\sigma^2_{\epsilon i} </math>, the unbiased sample variance of the mean of all observations of this unique reflection i = 20848.2198 (62544.6597/(n/2))
+
<math>\sigma^2_{\epsilon 1} </math>, the unbiased sample variance of the mean of all observations of this unique reflection = 20848.2198 (62544.6597/(n/2))
  
 
   
 
   
Line 72: Line 58:
 
     2    1    1  1.608E+01  2.215E+01  2   
 
     2    1    1  1.608E+01  2.215E+01  2   
 
</pre>
 
</pre>
<math>\overline{x}_{i} </math> , the average intensity of all observations of this reflection = 52.5150  
+
<math>\overline{x}_{2} </math> , the average intensity of all observations of this reflection = 52.5150  
  
<math>\sigma^2_{\epsilon i} </math>, the unbiased sample variance of the mean of all observations of this unique reflection i = 363.3267 (1089.9803/(n/2))
+
<math>\sigma^2_{\epsilon 2} </math>, the unbiased sample variance of the mean of all observations of this unique reflection = 363.3267 (1089.9803/(n/2))
  
  
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As a result of these calculations  CC<sub>1/2</sub> = 0.9458 ((190458.6533-(0.5*10605.7733))/(190458.6533+(0.5*10605.7733))
 
As a result of these calculations  CC<sub>1/2</sub> = 0.9458 ((190458.6533-(0.5*10605.7733))/(190458.6533+(0.5*10605.7733))
  
The described calculation is implemented in [[XDSCC12]] and CC<sub>1/2</sub> and ΔCC<sub>1/2</sub> ([[DeltaCC12]]) can be calculated for XDS_ASCII.HKL and XSCALE.HKL files.
+
The described calculation is implemented in [[XDSCC12]], and CC<sub>1/2</sub> and [[DeltaCC12|ΔCC<sub>1/2</sub>]] can be calculated for XDS_ASCII.HKL and XSCALE.HKL files.
 +
 
 +
==Fortran 95 code that assumes that all unique reflections have the same number of observations==
 +
 
 +
<pre>
 +
sumibar=0
 +
sumibar2=0
 +
sumsig2eps=0
 +
DO i=1,nref
 +
  xbar=SUM(iobs(:,i))/nobs
 +
  sumibar=sumi+xbar
 +
  sumibar2=sumibar2+xbar**2
 +
  sumsig2eps=sumeps + (SUM(iobs(:,i)**2)-xbar**2*nobs)/(nobs-1)/(nobs/2)
 +
END DO
 +
sig2y=(sumibar2-sumibar**2/nref)/(nref-1)
 +
sig2eps=sumsig2eps/nref
 +
print *,(sig2y-0.5*sig2eps)/(sig2y+0.5*sig2eps)
 +
</pre>
 +
 
 +
 
 +
== number of reflection pairs ==
 +
[[CORRECT.LP]] and XSCALE.LP do not explicitly state the ''number of reflection pairs'' that were used to calculated CC<sub>1/2</sub>.
 +
 
 +
However, the number can be calculated from the numbers available, for each resolution shell: there is the NUMBER OF UNIQUE REFLECTIONS (X), the NUMBER OF OBSERVED REFLECTIONS (Y), and the number of COMPARED reflections (Z) - the latter number is the total number of unmerged observations that contributed to the CC<sub>1/2</sub> and the R-value calculations.
 +
 
 +
The ''number of reflections pairs'' that were used for the CC<sub>1/2</sub> calculation can therefore be obtained as follows: Y-Z gives the number of unique reflections that have a single observation. The remaining (X-Y+Z) unique reflections have multiple observations, i.e. there were  (X-Y+Z) reflection pairs that went into CC<sub>1/2</sub>.
 +
 
 +
 
 +
== why CC<sub>1/2</sub> can be negative ==
 +
There is a mathematical reason, explained in §4.1 of [https://cms.uni-konstanz.de/index.php?eID=tx_nawsecuredl&u=0&g=0&t=1475179096&hash=5cf64234a23a794a1894c5408384c57208d7b602&file=fileadmin/biologie/ag-strucbio/pdfs/Assman2016_JApplCryst.pdf Assmann, G., Brehm, W. and Diederichs, K. (2016) Identification of rogue datasets in serial crystallography (2016) J. Appl. Cryst. 49, 1021-1028.]

Revision as of 10:00, 6 September 2018

CC1/2 calculation

CC1/2 can be calculated with the so-called σ-τ method (Assmann, G., Brehm, W. and Diederichs, K. (2016) Identification of rogue datasets in serial crystallography (2016) J. Appl. Cryst. 49, 1021-1028.) by:

[math]\displaystyle{ CC_{1/2}=\frac{\sigma^2_{\tau}}{\sigma^2_{\tau}+\sigma^2_{\epsilon}} =\frac{\sigma^2_{y}- \frac{1}{2}\sigma^2_{\epsilon}}{\sigma^2_{y}+ \frac{1}{2}\sigma^2_{\epsilon}} }[/math]

This requires calculation of [math]\displaystyle{ \sigma^2_{y} }[/math], the variance of the average intensities across the unique reflections of a resolution shell, and [math]\displaystyle{ \sigma^2_{\epsilon} }[/math], the average of all sample variances of the averaged (merged) intensities across all unique reflections of a resolution shell.

Method

[math]\displaystyle{ \sigma^2_{\epsilon} }[/math]

With [math]\displaystyle{ x_{j,i} }[/math] , a single observation j of all observations n of one reflection i, the average of all sample variances of the mean across all unique reflections of a resolution shell is obtained by calculating the sample variance of the mean for every unique reflection i by:

[math]\displaystyle{ \sigma^2_{\epsilon i} = \frac{1}{n_{i}-1} \cdot \left ( \sum^{n_{i}}_{j} x^2_{j,i} - \frac{\left ( \sum^{n_{i}}_{j}x_{j,i} \right )^2}{n_{i}} \right ) / \frac{n_{i}}{2} }[/math]

[math]\displaystyle{ \sigma^2_{\epsilon i} }[/math] is then divided by the factor [math]\displaystyle{ \frac{n}{2} }[/math], because the variance of the sample mean (intensities of the merged observations) is the quantity of interest. The division by n/2 takes care of providing the variance of the mean ([1]) (merged) intensity of the half-datasets, as defined in Karplus and Diederichs (2012). These "variances of means" are averaged over all unique reflections of the resolution shell:

[math]\displaystyle{ \sigma^2_{\epsilon}=\sum^N_{i} \sigma^2_{\epsilon i} / N }[/math]



[math]\displaystyle{ \sigma^2_{y} }[/math]

Let N be the number of reflections. With [math]\displaystyle{ \overline{x}_{i}= \sum^n_{j} x_{j,i} }[/math] , the unbiased sample variance from all averaged intensities of all unique reflections is calculated by:

[math]\displaystyle{ \sigma^2_{y} = \frac{1}{N-1} \cdot \left ( \sum^N_{i} \overline{x}_{i}^2 - \frac{\left ( \sum^N_{i} \overline{x}_{i} \right )^2}{ N} \right ) }[/math]


Example

An example is shown for a very simplified data file (unmerged ASCII.HKL).

First reflection with 6 observations:
     h     k     l       int     σ(int)  
     2     0     0  9.156E+02  3.686E+00   1 
     0     2     0  5.584E+02  3.093E+00   1 
     0     0     2  6.301E+02  2.405E+01   1  
     2     0     0  9.256E+02  3.686E+00   2 
     0     2     0  2.584E+02  3.093E+00   2 
     0     0     2  7.301E+02  2.405E+01   2 

[math]\displaystyle{ \overline{x}_{1} }[/math] , the average intensity of all observations of this reflection = 669.6999

[math]\displaystyle{ \sigma^2_{\epsilon 1} }[/math], the unbiased sample variance of the mean of all observations of this unique reflection = 20848.2198 (62544.6597/(n/2))


Second reflection with 6 observations:
     h     k     l       int     σ(int)  
     1     1     2  2.395E+01  8.932E+01   1  
     1     2     1  9.065E+01  7.407E+00   1  
     2     1     1  5.981E+01  9.125E+00   1  
     1     1     2  3.395E+01  8.932E+01   2  
     1     2     1  9.065E+01  7.407E+00   2  
     2     1     1  1.608E+01  2.215E+01   2  

[math]\displaystyle{ \overline{x}_{2} }[/math] , the average intensity of all observations of this reflection = 52.5150

[math]\displaystyle{ \sigma^2_{\epsilon 2} }[/math], the unbiased sample variance of the mean of all observations of this unique reflection = 363.3267 (1089.9803/(n/2))


[math]\displaystyle{ \sigma^2_{\epsilon} }[/math] , the average of all the [math]\displaystyle{ \sigma^2_{\epsilon i} }[/math] = 10605.7733

[math]\displaystyle{ \sigma^2_{y} }[/math], the variance of all the averaged intensities = 190458.6533

As a result of these calculations CC1/2 = 0.9458 ((190458.6533-(0.5*10605.7733))/(190458.6533+(0.5*10605.7733))

The described calculation is implemented in XDSCC12, and CC1/2 and ΔCC1/2 can be calculated for XDS_ASCII.HKL and XSCALE.HKL files.

Fortran 95 code that assumes that all unique reflections have the same number of observations

sumibar=0
sumibar2=0
sumsig2eps=0
DO i=1,nref
  xbar=SUM(iobs(:,i))/nobs
  sumibar=sumi+xbar
  sumibar2=sumibar2+xbar**2
  sumsig2eps=sumeps + (SUM(iobs(:,i)**2)-xbar**2*nobs)/(nobs-1)/(nobs/2)
END DO
sig2y=(sumibar2-sumibar**2/nref)/(nref-1)
sig2eps=sumsig2eps/nref
print *,(sig2y-0.5*sig2eps)/(sig2y+0.5*sig2eps)


number of reflection pairs

CORRECT.LP and XSCALE.LP do not explicitly state the number of reflection pairs that were used to calculated CC1/2.

However, the number can be calculated from the numbers available, for each resolution shell: there is the NUMBER OF UNIQUE REFLECTIONS (X), the NUMBER OF OBSERVED REFLECTIONS (Y), and the number of COMPARED reflections (Z) - the latter number is the total number of unmerged observations that contributed to the CC1/2 and the R-value calculations.

The number of reflections pairs that were used for the CC1/2 calculation can therefore be obtained as follows: Y-Z gives the number of unique reflections that have a single observation. The remaining (X-Y+Z) unique reflections have multiple observations, i.e. there were (X-Y+Z) reflection pairs that went into CC1/2.


why CC1/2 can be negative

There is a mathematical reason, explained in §4.1 of Assmann, G., Brehm, W. and Diederichs, K. (2016) Identification of rogue datasets in serial crystallography (2016) J. Appl. Cryst. 49, 1021-1028.