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R-factors (edit)

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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>
where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection.
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.


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"/> ,  
Weiss and Hilgenfeld <ref name="WeHi97">M.S. Weiss and R. Hilgenfeld (1997) On the use of the merging R-factor as a quality indicator for X-ray data. J. Appl. Crystallogr. 30, 203-205[http://dx.doi.org/10.1107/S0021889897003907]</ref>).
Weiss and Hilgenfeld <ref name="WeHi97">M.S. Weiss and R. Hilgenfeld (1997) On the use of the merging R-factor as a quality indicator for X-ray data. J. Appl. Crystallogr. 30, 203-205[http://dx.doi.org/10.1107/S0021889897003907]</ref>).


==== measuring quality of averaged intensities/amplitudes ====
==== measuring precision of averaged intensities/amplitudes ====


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>


R<sub>mrgd-I</sub> is similarly defined in Diederichs and Karplus <ref name="DiKa97"/>.
 
: <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.  
      
      
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>
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>


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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To my knowledge, the only program that implements this currently (December 2008) is [[xds:XDSSTAT|XDSSTAT]].
To my knowledge, the only program that implements this currently (December 2008) is [[xds:XDSSTAT|XDSSTAT]].
=== Comparing two sets of structure factor amplitudes or intensities ===
The following is symmetric, and suitable for comparing two data sets, or two model amplitudes:
: <math>
R_{scale}=\frac{\sum_{hkl}\vert F_{hkl,i}-F_{hkl,j}\vert}{0.5\sum_{hkl} F_{hkl,i}+F_{hkl,j}}
</math>
for amplitudes, and analogously for intensities.


=== Model quality indicators ===
=== Model quality indicators ===
* R and [[iucr:Free_R_factor|R<sub>free</sub>]] : the formula for both is  
* R and [[iucr:Free_R_factor|R<sub>free</sub>]] : the formula for both is  
<math>
 
R=\frac{\sum_{hkl}\vert F_{hkl}^{obs}-F_{hkl}^{calc}\vert}{\sum_{hkl} F_{hkl}^{obs}}
 
</math>
: <math>
<br>
R=\frac{\sum_{hkl}\vert F_{hkl}^{obs}-F_{hkl}^{calc}\vert}{\sum_{hkl} F_{hkl}^{obs}}
<br>
</math>
 
 
where <math>F_{hkl}^{obs}</math> and <math>F_{hkl}^{calc}</math> have to be scaled w.r.t. each other. R and R<sub>free</sub> differ in the set of reflections they are calculated from: R is calculated for the [[working set]], whereas R<sub>free</sub> is calculated for the [[test set]].
where <math>F_{hkl}^{obs}</math> and <math>F_{hkl}^{calc}</math> have to be scaled w.r.t. each other. R and R<sub>free</sub> differ in the set of reflections they are calculated from: R is calculated for the [[working set]], whereas R<sub>free</sub> is calculated for the [[test set]].
== what do R-factors try to measure, and how to interpret their values? ==
* relative deviation of
=== Data quality ===
* typical values: ...
=== Model quality ===


==== Relation between R and R<sub>free</sub> as a function of resolution ====
==== Relation between R and R<sub>free</sub> as a function of resolution ====
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* Sets of reflections used for calculating R<sub>free</sub> should be maintained throughout a project. This is nicely discussed at http://www.bmsc.washington.edu/people/merritt/xplor/rfree_example.html . Note that none of the programs mentioned for selecting thin shells will allow you to extend the set of shells to higher resolution if you want to preserve your existing R-free set.
* Sets of reflections used for calculating R<sub>free</sub> should be maintained throughout a project. This is nicely discussed at http://www.bmsc.washington.edu/people/merritt/xplor/rfree_example.html . Note that none of the programs mentioned for selecting thin shells will allow you to extend the set of shells to higher resolution if you want to preserve your existing R-free set.


* R-values and twinning: [http://www.science.az/acm/V10,%20N2,%202011,%20pdf/250-261.pdf Garib N. Murshudov (2011) "Some properties of crystallographic reliability index - Rfactor: effect of twinning" Appl. Comput. Math., V.10, N.2, 2011, pp.250-261]
* R-values and twinning: [http://www.ysbl.york.ac.uk/refmac/papers/Rfactor.pdf Garib N. Murshudov (2011) "Some properties of crystallographic reliability index - Rfactor: effect of twinning" Appl. Comput. Math., V.10, N.2, 2011, pp.250-261]. From the paper, the R-value table for random models is:
      twinning  twinning not
      modelled  modelled
twin  0.41      0.49
normal 0.52      0.58
Another paper which investigates the properties of R-values in the presence of twinning is [http://journals.iucr.org/d/issues/2013/07/00/ba5190/index.html P. R. Evans and G. N. Murshudov (2013) "How good are my data and what is the resolution?" Acta Cryst. (2013). D69, 1204-1214]. As the title indicates, this paper discusses at what resolution the data should be cut. One important finding is that a perfect model gives an R value of 42.0% (for a perfect twin, 29.1%) against pure noise. This tells us that a model that gives significantly lower R<sub>free</sub> in the (current) high resolution shell may benefit from including higher resolution data.
* R-values and [[pseudo-translation]]: if you have pseudotranslation you should be aware that if you solve the structure by molecular replacement, starting R factors could be 70-80%.
 
* data R-values are not meaningful at high resolution. This is discussed by [http://strucbio.biologie.uni-konstanz.de//strucbio/files/karplus2012_science.pdf Karplus and Diederichs (2012) "Linking crystallographic data and model quality". ''Science'' '''336''', 1030]


==Notes==
==Notes==
<references/>
<references/>
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