Difference between revisions of "R-factors"

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Historically, R-factors were introduced by ...  
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Historically, R-factors were introduced by ... ???
  
 
== Definitions ==
 
== Definitions ==
 
=== Data quality indicators ===
 
=== 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
 
* R<sub>sym</sub> and R<sub>merge</sub> : the formula for both is
 
<math>
 
<math>
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<br>
 
<br>
 
<br>
 
<br>
where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection, and the first summation is over all unique reflections with more than one observation.
+
where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection
* Redundancy-independant version of the above: R<sub>meas</sub>
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* Redundancy-independant version of the above:  
* measuring quality of averaged intensities/amplitudes: R<sub>p.i.m.</sub> and R<sub>mrgd-F</sub>
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<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>
 +
<br>
 +
<br>
 +
* measuring quality of averaged intensities/amplitudes:
 +
 
 +
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>
 +
<br>
 +
<br>
 +
 
 +
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>
 +
<br>
 +
<br>
 +
 
  
 
=== 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_{unique}}\vert F_{hkl}^{(obs)}-F_{hkl}^{(calc)}\vert}{\sum_{hkl_{unique}} 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>
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? ==
 
== what do R-factors try to measure, and how to interpret their values? ==
 
* relative deviation of
 
* relative deviation of

Revision as of 11:42, 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

  • 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]

  • 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]


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