R-factors

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Revision as of 12:06, 15 February 2008 by Rogerdodd (talk | contribs) (Enclosed the equations in dashed boxes to make things clearer - let me know if this doesn't work well!)

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]
 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.

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]


which unfortunately results in higher (but more realistic) numerical values than Rsym / Rmerge

  • 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]



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]


with [math]\langle F_{hkl}\rangle[/math] defined analogously as [math]\langle I_{hkl}\rangle[/math].

Model quality indicators

  • R and Rfree : the formula for both is
[math]
 R=\frac{\sum_{hkl}\vert F_{hkl}^{obs}-F_{hkl}^{calc}\vert}{\sum_{hkl} F_{hkl}^{obs}}
 [/math]



where [math]F_{hkl}^{obs}[/math] and [math]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