# 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 ^{[1]}. 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} ^{[1]} (M.S. Weiss and R. Hilgenfeld (1997) On the use of the merging R-factor as a quality indicator for X-ray data. J. Appl. Crystallogr. 30, 203-205[2]).

- measuring quality of averaged intensities/amplitudes:

for intensities use (M.S. Weiss. Global indicators of X-ray data quality. J. Appl. Cryst. (2001). 34, 130-135 [3])

```
[math]\displaystyle{
R_{p.i.m.} = \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]
```

[math]\displaystyle{ R_{mrgd-I} }[/math] is similarly defined in Diederichs and Karplus ^{[1]}.

Similarly, one should use R_{mrgd-F} as a quality indicator for amplitudes ^{[1]}, which may be calculated as:

```
[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

#### Relation between R and R_{free} as a function of resolution

References:

- Tickle IJ, Laskowski RA and Moss DS. Rfree and the Rfree Ratio. I. Derivation of Expected Values of Cross-Validation Residuals Used in Macromolecular Least-Squares Refinement. Acta Cryst. (1998). D54, 547-557 [4]

- Tickle IJ, Laskowski RA and Moss DS. Rfree and the Rfree ratio. II. Calculation of the expected values and variances of cross-validation statistics in macromolecular least-squares refinement. Acta Cryst. (2000). D56, 442-450 [5]

- GJ Kleywegt and TA Jones (2002). Homo Crystallographicus - Quo vadis? Structure 10, 465-472. (reprint from http://xray.bmc.uu.se/cgi-bin/gerard/reprint_mailer.pl?pref=65)

- formula from that paper: R_{free} = 1.065*R + 0.036

- plot with empirical data: http://xray.bmc.uu.se/gerard/supmat/rfree2000/rfminusr_vs_resolution.gif

- many more plots: http://xray.bmc.uu.se/gerard/supmat/rfree2000

- harry plotter (java): http://xray.bmc.uu.se/gerard/supmat/rfree2000/plotter.html

## 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

## Notes

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}K. Diederichs and P.A. Karplus (1997). Improved R-factors for diffraction data analysis in macromolecular crystallography. Nature Struct. Biol. 4, 269-275 [1]