CC1/2: Difference between revisions
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If the standard deviations <math>\sigma_{int} </math> for the single observations are considered as weights for the CC<sub>1/2</sub> calculation, with <math>\sum^{n_{i}}_{j}w_{i}=\sum^{n_{i}}_{j}\frac{1}{\sigma_{int}^2} </math> the unbiased '''weighted''' sample variance of the mean for every unique reflection i is obtained by: | If the standard deviations <math>\sigma_{int\_j,i} </math> for the single observations are considered as weights for the CC<sub>1/2</sub> calculation, with <math>\sum^{n_{i}}_{j}w_{i}=\sum^{n_{i}}_{j}\frac{1}{\sigma_{int\_j,i}^2} </math> the unbiased '''weighted''' sample variance of the mean for every unique reflection i is obtained by: | ||
<math>\sigma^2_{\epsilon i\_w} = \frac{n_{i}}{n_{i}-1} \cdot \left ( \frac{\sum^{n_{i}}_{j}w_{j,i} x^2_{j,i}}{\sum^{n_{i}}_{j}w_{j,i}} -\left ( \frac{ \sum^{n_{i}}_{j}w_{j,i}x_{j,i} }{\sum^{n_{i}}_{j}w_{j,i}}\right )^2 \right ) / \frac{n_{i}}{2} </math> | <math>\sigma^2_{\epsilon i\_w} = \frac{n_{i}}{n_{i}-1} \cdot \left ( \frac{\sum^{n_{i}}_{j}w_{j,i} x^2_{j,i}}{\sum^{n_{i}}_{j}w_{j,i}} -\left ( \frac{ \sum^{n_{i}}_{j}w_{j,i}x_{j,i} }{\sum^{n_{i}}_{j}w_{j,i}}\right )^2 \right ) / \frac{n_{i}}{2} </math> | ||
Revision as of 10:45, 7 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 unbiased 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 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]
If the standard deviations [math]\displaystyle{ \sigma_{int\_j,i} }[/math] for the single observations are considered as weights for the CC1/2 calculation, with [math]\displaystyle{ \sum^{n_{i}}_{j}w_{i}=\sum^{n_{i}}_{j}\frac{1}{\sigma_{int\_j,i}^2} }[/math] the unbiased weighted sample variance of the mean for every unique reflection i is obtained by:
[math]\displaystyle{ \sigma^2_{\epsilon i\_w} = \frac{n_{i}}{n_{i}-1} \cdot \left ( \frac{\sum^{n_{i}}_{j}w_{j,i} x^2_{j,i}}{\sum^{n_{i}}_{j}w_{j,i}} -\left ( \frac{ \sum^{n_{i}}_{j}w_{j,i}x_{j,i} }{\sum^{n_{i}}_{j}w_{j,i}}\right )^2 \right ) / \frac{n_{i}}{2} }[/math]
These " weighted variances of means" are averaged over all unique reflections of the resolution shell:
[math]\displaystyle{ \sigma^2_{\epsilon_w}=\sum^N_{i} \sigma^2_{\epsilon i\_w} / N }[/math]
[math]\displaystyle{ \sigma^2_{y} }[/math]
Let N be the number of reflections in a resolution shell. 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 = 62544.6597/(n/2) = 20848.2198
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 = 1089.9803/(n/2) = 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 = (190458.6533-(0.5*10605.7733))/(190458.6533+(0.5*10605.7733), which results in 0.9458.
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.