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R-factors: Difference between revisions
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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> | |||
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> | |||
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> | |||
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> | |||
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> | |||
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> | |||
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. | ||