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Δcc12 is a quantity, that detects datasets/frames, that are non-isomorphous. As described in [https://scripts.iucr.org/cgi-bin/paper?zw5005 Assmann and Diederichs (2016)], Δcc12 is calculated with the σ-τ method. This method is a way to calculate the Pearson correlation coefficient for the special case of two sets of values (intensities) that randomly deviate from their true values, but is not influenced by a random number sequence as shown in [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3457925/ Karplus and Diederichs (2012)]. For the σ-τ method CC12 is calculated for all datasets/frames, which will be called CC12_overall (?) and CC12 is calculated for all datasets/frames except for one dataset i, which is omitted from calculations and denoted as CC12_i. The difference of the two quantities is Δcc12.
ΔCC<sub>1/2</sub> is a quantity that detects datasets/frames which are non-isomorphous. As described in [https://scripts.iucr.org/cgi-bin/paper?zw5005 Assmann and Diederichs (2016)], ΔCC<sub>1/2</sub> is calculated with the σ-τ method. This method is a way to calculate the Pearson correlation coefficient for the special case of two sets of values (intensities) that randomly deviate from their true values. The σ-τ method is not influenced by a random number sequence as shown in [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3457925/ Karplus and Diederichs (2012)]. For the σ-τ method CC<sub>1/2</sub> is calculated for all datasets/frames, which will be called CC<sub>1/2_overall</sub> and CC<sub>1/2</sub> is calculated for all datasets/frames except for one dataset i, which is omitted from calculations and denoted as CC<sub>1/2_i</sub>. The difference of the two quantities is ΔCC<sub>1/2</sub>.
: <math>\Delta CC_{1/2}= CC_{1/2 overall}-CC_{1/2 i} </math>


If Δcc12 is > 0 -CC12overall is bigger than CC12i- that means if omitting dataset i from calculations, a lower CC12 results, which is why we want to keep it. Thus it is improving the whole merged dataset. If Δcc12 is < 0, -CC12overall is smaller than CC12i- that means that by omitting dataset i from calculations a higher CC12 results, which is why we want to exclude it from calculations, because it is impairing the whole merged dataset. CC12 is calculated by:
: <math>\Delta CC_{1/2}= CC_{1/2 overall}-CC_{1/2\_i} </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>
If ΔCC<sub>1/2</sub> is > 0 (CC<sub>1/2_overall</sub> is bigger than CC<sub>1/2_i</sub>) it means that by omitting dataset i from calculations a lower CC<sub>1/2</sub> results. As we want to maximize CC<sub>1/2</sub> the dataset is kept for calculations, it is improving the whole merged dataset. If Δ CC<sub>1/2</sub> is < 0 (CC<sub>1/2_overall</sub> is smaller than CC<sub>1/2_i</sub>) it means that by omitting dataset i from calculations a higher CC<sub>1/2</sub> results, which is why we want to exclude it from calculations, because it is impairing the whole merged dataset.


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.


== Implementation ==
== Applications ==


===''' <math>\sigma^2_{y} </math>'''===
The ΔCC<sub>1/2</sub> method is applicable for single frames, SSX data and SFX data. The program [[XDSCC12]] calculates ΔCC<sub>1/2</sub> for the isomorphous and anomalous signal for XDS_ASCII.HKL and XSCALE.HKL files. Exact description of calculation and implementation are found at [[CC1/2]].
 
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} x^2_i - \frac{\left ( \sum^n_{i}x_{i} \right )^2}{ n} \right ) </math>
 
With <math>x_{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.
 
----
 
 
===''' <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 by:
 
<math>\sigma^2_{\epsilon} =  \frac{1}{n-1} \cdot \left ( \sum^n_{i} x^2_i - \frac{\left ( \sum^n_{i}x_{i} \right )^2}{ n} \right )    \backslash \frac{n}{2} </math>
 
With <math>x_{i} </math> , a single observation i of all observations n one reflection. <math>\sigma^2_{\epsilon} </math> is then divided by the factor  <math>\frac{n}{2} </math>, because the variance of the sample mean (the merged observations) is the quantity of interest. As we are considering This is done for all reflections n in a resolution shell.
Gassmann
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