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R-factors: Difference between revisions

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R-factors (edit)

Revision as of 15:03, 26 January 2018

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In the following, all sums over hkl extend only over unique reflections with more than one observation!
In the following, all sums over hkl extend only over unique reflections with more than one observation!
* R<sub>sym</sub> and R<sub>merge</sub> - the formula for both is:
* R<sub>sym</sub> and R<sub>merge</sub> - the formula for both is:
<math>
 
R = \frac{\sum_{hkl} \sum_{j} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}
 
</math>
: <math>
R = \frac{\sum_{hkl} \sum_{j} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}
</math>
 
 
where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection. The formula is due to Arndt, U.W., Crowther, R.A. & Mallet, J.F.W. A computer-linked cathode ray tube microdensitometer for X-ray crystallography. J. Phys. E:Sci. Instr. 1, 510−516 (1968). Any unique reflection with n=2 or more observations enters the sums.
where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection. The formula is due to Arndt, U.W., Crowther, R.A. & Mallet, J.F.W. A computer-linked cathode ray tube microdensitometer for X-ray crystallography. J. Phys. E:Sci. Instr. 1, 510−516 (1968). Any unique reflection with n=2 or more observations enters the sums.


It can be shown that this formula results in higher R-factors when the redundancy is higher (Diederichs and Karplus <ref name="DiKa97">K. Diederichs and P.A. Karplus (1997). Improved R-factors for diffraction data analysis in macromolecular crystallography. Nature Struct. Biol. 4, 269-275 [http://strucbio.biologie.uni-konstanz.de/strucbio/files/nsb-1997.pdf]</ref>). In other words, low-redundancy datasets appear better than high-redundancy ones, which obviously violates the intention of having an indicator of data quality!
It can be shown that this formula results in higher R-factors when the redundancy is higher (Diederichs and Karplus <ref name="DiKa97">K. Diederichs and P.A. Karplus (1997). Improved R-factors for diffraction data analysis in macromolecular crystallography. Nature Struct. Biol. 4, 269-275 [http://strucbio.biologie.uni-konstanz.de/strucbio/files/nsb-1997.pdf]</ref>). In other words, low-redundancy datasets appear better than high-redundancy ones, which obviously violates the intention of having an indicator of data quality!
* Redundancy-independant version of the above:  
* Redundancy-independant version of the above:  
<math>
 
R_{meas} = \frac{\sum_{hkl} \sqrt \frac{n}{n-1} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}
 
</math>
: <math>
R_{meas} = \frac{\sum_{hkl} \sqrt \frac{n}{n-1} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}
</math>
 
 
which unfortunately results in higher (but more realistic) numerical values than R<sub>sym</sub> / R<sub>merge</sub>  
which unfortunately results in higher (but more realistic) numerical values than R<sub>sym</sub> / R<sub>merge</sub>  
(Diederichs and Karplus <ref name="DiKa97"/> ,  
(Diederichs and Karplus <ref name="DiKa97"/> ,  
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for intensities use  
for intensities use  
(Weiss <ref name="We01">M.S. Weiss. Global indicators of X-ray data quality. J. Appl. Cryst. (2001). 34, 130-135 [http://dx.doi.org/10.1107/S0021889800018227]</ref>)
(Weiss <ref name="We01">M.S. Weiss. Global indicators of X-ray data quality. J. Appl. Cryst. (2001). 34, 130-135 [http://dx.doi.org/10.1107/S0021889800018227]</ref>)
<math>
 
R_{p.i.m.} = \frac{\sum_{hkl} \sqrt \frac{1}{n-1} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}
 
</math>
: <math>
R_{p.i.m.} = \frac{\sum_{hkl} \sqrt \frac{1}{n-1} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}
</math>


R<sub>mrgd-I</sub> (defined in Diederichs and Karplus <ref name="DiKa97"/>) only differs by a factor (FIXME: what is the factor? 0.5 or 1.4142 or ?) since it likewise takes the improvement in precision from multiplicity into account. R<sub>split</sub> , which is what the X-FEL community uses, is the same as R<sub>mrgd-I</sub> but that community seems not to be aware of this.  
R<sub>mrgd-I</sub> (defined in Diederichs and Karplus <ref name="DiKa97"/>) only differs by a factor (FIXME: what is the factor? 0.5 or 1.4142 or ?) since it likewise takes the improvement in precision from multiplicity into account. R<sub>split</sub> , which is what the X-FEL community uses, is the same as R<sub>mrgd-I</sub> but that community seems not to be aware of this.  
      
      
Similarly, one should use R<sub>mrgd-F</sub> as a quality indicator for amplitudes <ref name="DiKa97"/>, which may be calculated as:  
Similarly, one should use R<sub>mrgd-F</sub> as a quality indicator for amplitudes <ref name="DiKa97"/>, which may be calculated as:  
<math>
 
 
: <math>
  R_{mrgd-F} = \frac{\sum_{hkl} \sqrt \frac{1}{n-1} \sum_{j=1}^{n} \vert F_{hkl,j}-\langle F_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}F_{hkl,j}}
  R_{mrgd-F} = \frac{\sum_{hkl} \sqrt \frac{1}{n-1} \sum_{j=1}^{n} \vert F_{hkl,j}-\langle F_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}F_{hkl,j}}
</math>
</math>
 
 
with <math>\langle F_{hkl}\rangle</math> defined analogously as <math>\langle I_{hkl}\rangle</math>.
with <math>\langle F_{hkl}\rangle</math> defined analogously as <math>\langle I_{hkl}\rangle</math>.


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We can plot (Diederichs <ref name="Di06">K. Diederichs (2006). Some aspects of quantitative analysis and correction of radiation damage. Acta Cryst D62, 96-101 [http://strucbio.biologie.uni-konstanz.de/strucbio/files/Diederichs_ActaD62_96.pdf]</ref>)
We can plot (Diederichs <ref name="Di06">K. Diederichs (2006). Some aspects of quantitative analysis and correction of radiation damage. Acta Cryst D62, 96-101 [http://strucbio.biologie.uni-konstanz.de/strucbio/files/Diederichs_ActaD62_96.pdf]</ref>)


<math>
 
R_{d} = \frac{\sum_{hkl} \sum_{|i-j|=d} \vert I_{hkl,i} - I_{hkl,j}\vert}{\sum_{hkl} \sum_{|i-j|=d} (I_{hkl,i} + I_{hkl,j})/2}
: <math>
</math>
R_{d} = \frac{\sum_{hkl} \sum_{|i-j|=d} \vert I_{hkl,i} - I_{hkl,j}\vert}{\sum_{hkl} \sum_{|i-j|=d} (I_{hkl,i} + I_{hkl,j})/2}
</math>
 


which gives us the average R-factor of two reflections measured d frames apart. As long as the plot is parallel to the x axis there is no radiation damage. As soon as the plot starts to rise, we see that there's a systematical error contribution due to radiation damage.
which gives us the average R-factor of two reflections measured d frames apart. As long as the plot is parallel to the x axis there is no radiation damage. As soon as the plot starts to rise, we see that there's a systematical error contribution due to radiation damage.
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