R-factors: Difference between revisions
(Enclosed the equations in dashed boxes to make things clearer - let me know if this doesn't work well!) |
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=== Data quality indicators === | === Data quality indicators === | ||
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> | * R<sub>sym</sub> and R<sub>merge</sub> - the formula for both is: | ||
<math> | <math> | ||
R = \frac{\sum_{hkl} \sum_{j} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}} | R = \frac{\sum_{hkl} \sum_{j} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}} | ||
</math> | </math> | ||
<br> | <br> | ||
where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection. | where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection. | ||
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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! | 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: | * Redundancy-independant version of the above: | ||
<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}} | 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> | ||
<br> | <br> | ||
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> | ||
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for intensities use | for intensities use | ||
<math> | <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}} | 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> | </math> | ||
<br> | <br> | ||
<br> | <br> | ||
and similarly for amplitudes: | and similarly for amplitudes: | ||
<math> | <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}} | 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> | </math> | ||
<br> | <br> | ||
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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=== Model quality indicators === | === Model quality indicators === | ||
* R and R<sub>free</sub> : the formula for both is | * R and R<sub>free</sub> : the formula for both is | ||
<math> | <math> | ||
R=\frac{\sum_{hkl}\vert F_{hkl}^{obs}-F_{hkl}^{calc}\vert}{\sum_{hkl} F_{hkl}^{obs}} | R=\frac{\sum_{hkl}\vert F_{hkl}^{obs}-F_{hkl}^{calc}\vert}{\sum_{hkl} F_{hkl}^{obs}} | ||
</math> | </math> | ||
<br> | <br> | ||
<br> | <br> | ||
Revision as of 14:06, 15 February 2008
Historically, R-factors were introduced by ... ???
Definitions
Data quality indicators
In the following, all sums over hkl extend only over unique reflections with more than one observation!
- Rsym and Rmerge - the formula for both is:
[math]\displaystyle{
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]\displaystyle{ \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]\displaystyle{
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 Rsym / Rmerge
- measuring quality of averaged intensities/amplitudes:
for intensities use
[math]\displaystyle{
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]
and similarly for amplitudes:
[math]\displaystyle{
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]
with [math]\displaystyle{ \langle F_{hkl}\rangle }[/math] defined analogously as [math]\displaystyle{ \langle I_{hkl}\rangle }[/math].
Model quality indicators
- R and Rfree : the formula for both is
[math]\displaystyle{
R=\frac{\sum_{hkl}\vert F_{hkl}^{obs}-F_{hkl}^{calc}\vert}{\sum_{hkl} F_{hkl}^{obs}}
}[/math]
where [math]\displaystyle{ F_{hkl}^{obs} }[/math] and [math]\displaystyle{ F_{hkl}^{calc} }[/math] have to be scaled w.r.t. each other. R and Rfree differ in the set of reflections they are calculated from: R is calculated for the working set, whereas Rfree 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
what kinds of problems exist with these indicators?
- (Rsym / Rmerge ) should not be used, Rmeas should be used instead (explain why ?)
- R/Rfree and NCS: reflections in work and test set are not independant