# 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!

• Rsym and Rmerge - the formula for both is:
$R = \frac{\sum_{hkl} \sum_{j} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}$


where $\langle I_{hkl}\rangle$ 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:
$R_{meas} = \frac{\sum_{hkl} \sqrt \frac{n}{n-1} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}$


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

• measuring quality of averaged intensities/amplitudes:

for intensities use

$R_{p.i.m.} (or R_{mrgd-I}) = \frac{\sum_{hkl} \sqrt \frac{1}{n} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}$


and similarly for amplitudes:

$R_{mrgd-F} = \frac{\sum_{hkl} \sqrt \frac{1}{n} \sum_{j=1}^{n} \vert F_{hkl,j}-\langle F_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}F_{hkl,j}}$


with $\langle F_{hkl}\rangle$ defined analogously as $\langle I_{hkl}\rangle$.

### Model quality indicators

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


where $F_{hkl}^{obs}$ and $F_{hkl}^{calc}$ 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: ...

## 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