# Difference between revisions of "R-factors"

(Enclosed the equations in dashed boxes to make things clearer - let me know if this doesn't work well!) |
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* typical values: ... | * typical values: ... | ||

=== Model quality === | === Model quality === | ||

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+ | ==== Relation between R and R<sub>free</sub> as a function of resolution ==== | ||

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+ | 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 [http://dx.doi.org/10.1107/S0907444997013875] | ||

+ | |||

+ | * 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 [http://dx.doi.org/10.1107/S0907444999016868] | ||

+ | |||

+ | * 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) | ||

+ | - plot: 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? == | == what kinds of problems exist with these indicators? == |

## Revision as of 17:52, 17 February 2008

Historically, R-factors were introduced by ... ???

## Contents

## 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

#### 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 [1]

- 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 [2]

- 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)

- plot: 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