# Difference between revisions of "R-factors"

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− | Historically, R-factors were introduced by ... | + | Historically, R-factors were introduced by ... ??? |

== Definitions == | == Definitions == | ||

=== Data quality indicators === | === Data quality indicators === | ||

+ | In the following, all sums over hkl extend only over unique reflections with more than one observation! | ||

* R<sub>sym</sub> and R<sub>merge</sub> : the formula for both is | * R<sub>sym</sub> and R<sub>merge</sub> : the formula for both is | ||

<math> | <math> | ||

Line 9: | Line 10: | ||

<br> | <br> | ||

<br> | <br> | ||

− | where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection | + | where <math>\langle I_{hkl}\rangle</math> is the average of symmetry- (or Friedel-) related observations of a unique reflection |

− | * Redundancy-independant version of the above: | + | * Redundancy-independant version of the above: |

− | * measuring quality of averaged intensities/amplitudes: | + | <math> |

+ | 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> | ||

+ | <br> | ||

+ | <br> | ||

+ | * measuring quality of averaged intensities/amplitudes: | ||

+ | |||

+ | for intensities use | ||

+ | <math> | ||

+ | 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> | ||

+ | <br> | ||

+ | <br> | ||

+ | |||

+ | and similarly for amplitudes: | ||

+ | <math> | ||

+ | 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> | ||

+ | <br> | ||

+ | <br> | ||

+ | |||

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

* R and R<sub>free</sub> : the formula for both is | * R and R<sub>free</sub> : the formula for both is | ||

<math> | <math> | ||

− | R=\frac{\sum_{ | + | R=\frac{\sum_{hkl}\vert F_{hkl}^{obs}-F_{hkl}^{calc}\vert}{\sum_{hkl} F_{hkl}^{obs}} |

</math> | </math> | ||

<br> | <br> | ||

<br> | <br> | ||

− | where <math>F_{hkl}^{ | + | where <math>F_{hkl}^{obs}</math> and <math>F_{hkl}^{calc}</math> have to be scaled w.r.t. each other. R and R<sub>free</sub> differ in the set of reflections they are calculated from: R is calculated for the [[working set]], whereas R<sub>free</sub> is calculated for the [[test set]]. |

== what do R-factors try to measure, and how to interpret their values? == | == what do R-factors try to measure, and how to interpret their values? == | ||

* relative deviation of | * relative deviation of |

## Revision as of 11:42, 15 February 2008

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

- 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]

- 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]

### 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