# 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: ...

### Model quality

## 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