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CC1/2: Difference between revisions
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==CC<sub>1/2</sub> calculation== | ==CC<sub>1/2</sub> calculation== | ||
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 | 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. 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> | ||
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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>w_{j,i}=\frac{1}{\sigma_{int\_j,i}^2} </math> the unbiased '''weighted''' sample variance of the mean for every unique reflection i is | 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>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> | <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> | ||
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<math>s^2_{\epsilon_w}=\sum^N_{i} s^2_{\epsilon i\_w} / N </math> | <math>s^2_{\epsilon_w}=\sum^N_{i} s^2_{\epsilon i\_w} / N </math> | ||
It should be noted that it is not straightforward to define the correct way to calculate a weighted variance (and the weighted variance of the mean). The formula <math>s^2_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 )</math> is - after some manipulation - the same as that found at [https://stats.stackexchange.com/questions/6534/how-do-i-calculate-a-weighted-standard-deviation-in-excel],[https://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/weightsd.pdf]. | It should be noted that it is not straightforward to define the correct way to calculate a weighted variance (and the weighted variance of the mean). The formula <math>s^2_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 )</math> is - after some manipulation - the same as that found at [https://stats.stackexchange.com/questions/6534/how-do-i-calculate-a-weighted-standard-deviation-in-excel],[https://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/weightsd.pdf]. Other ways of calculating the weighted variance of the mean ([https://en.wikipedia.org/wiki/Weighted_arithmetic_mean],[https://www.gnu.org/software/gsl/manual/html_node/Weighted-Samples.html]) should be considered. | ||
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If the standard deviations <math>\sigma_{int\_j,i} </math> for the single observations are | If the standard deviations <math>\sigma_{int\_j,i} </math> for the single observations are used for weighting, <math>s^2_{y}</math> is obtained from | ||
<math>\overline{x}_{i_w} = \frac{\sum^n_{j} w_{j,i} x_{j,i}} {\sum^n_{j}w_{j,i}}</math>: | <math>\overline{x}_{i_w} = \frac{\sum^n_{j} w_{j,i} x_{j,i}} {\sum^n_{j}w_{j,i}}</math>: | ||
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== Implementation == | == Implementation == | ||
This way of calculating CC<sub>1/2</sub> is implemented in [[XDSCC12]] and in [http://www.desy.de/~twhite/crystfel/manual-partialator.html partialator] of [http://www.desy.de/~twhite/crystfel/relnotes-0.8.0 CrystFEL]. | This way of calculating CC<sub>1/2</sub> is implemented in [[XDSCC12]] and in [http://www.desy.de/~twhite/crystfel/manual-partialator.html partialator] of [http://www.desy.de/~twhite/crystfel/relnotes-0.8.0 CrystFEL]. | ||
== See also == | |||
[https://www.youtube.com/watch?v=LirxJIcQ6T0 CC* - Linking crystallographic model and data quality.] Video recorded at SBGrid/NE-CAT workshop 2014; the PDF is [https://strucbio.biologie.uni-konstanz.de/pub/CC%20-%20Linking%20crystallographic%20model%20and%20data%20quality.pdf here]. The sound is poor. | |||