CC1/2: Difference between revisions

==CC_1/2 calculation==

CC_1/2 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. 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>

== Method ==

With <math>x_{j,i} </math> , a single observation j of all n observations 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>s^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>s^2_{\epsilon i} </math> is 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>s^2_{\epsilon}=\sum^N_{i} s^2_{\epsilon i} / N </math>  

If the standard deviations <math>\sigma_{int\_j,i} </math> for the single observations are considered as weights for the CC_1/2 calculation, with <math>w_{j,i}=\frac{1}{\sigma_{int\_j,i}^2} </math>, one way to obtain the unbiased '''weighted''' sample variance of the half-dataset mean for every unique reflection i is:  

<math>s^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>s^2_{\epsilon_w}=\sum^N_{i} s^2_{\epsilon i\_w} / N </math>  

Let N be the number of reflections in a resolution shell. 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>s^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>\overline{x}_{i_w} = \frac{\sum^n_{j} w_{j,i}  x_{j,i}} {\sum^n_{j}w_{j,i}}</math>:

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

     h    k    l      int    σ(int)   

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

<math>\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

     h    k    l      int    σ(int)   

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

As a result of these calculations  CC_1/2 = (190458.6533-(0.5*10605.7733))/(190458.6533+(0.5*10605.7733)), which results in 0.9458.