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R-factors

16 bytes added, 12:06, 15 February 2008
Enclosed the equations in dashed boxes to make things clearer - let me know if this doesn't work well!
=== Data quality indicators ===
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: <math> R = \frac{\sum_{hkl} \sum_{j} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}} </math>
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where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection.
It can be shown that this formula results in higher R-factors when the redundancy is higher. 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:
<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>
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which unfortunately results in higher (but more realistic) numerical values than R<sub>sym</sub> / R<sub>merge</sub>
for intensities use
<math> R_{p.i.m.} (or R_{mrgd-I}) = \frac{\sum_{hkl} \sqrt \frac{1}{n} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}} </math>
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and similarly for amplitudes:
<math> R_{mrgd-F} = \frac{\sum_{hkl} \sqrt \frac{1}{n} \sum_{j=1}^{n} \vert F_{hkl,j}-\langle F_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}F_{hkl,j}} </math>
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with <math>\langle F_{hkl}\rangle</math> defined analogously as <math>\langle I_{hkl}\rangle</math>.
=== Model quality indicators ===
* R and 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>
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