R-factors

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!

• R_sym and R_merge - 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 R_sym / R_merge

• 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 R_free : 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 R_free differ in the set of reflections they are calculated from: R is calculated for the working set, whereas R_free 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: ...

what kinds of problems exist with these indicators?

- (R_sym / R_merge ) should not be used, R_meas should be used instead (explain why ?)

- R/R_free and NCS: reflections in work and test set are not independant