R-factors

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 (Diederichs and Karplus ). 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 (Diederichs and Karplus , Weiss and Hilgenfeld ).

measuring quality of averaged intensities/amplitudes

for intensities use (Weiss )

[math]\displaystyle{ 
 R_{p.i.m.} = \frac{\sum_{hkl} \sqrt \frac{1}{n-1} \sum_{j=1}^{n} \vert I_{hkl,j}-\langle I_{hkl}\rangle\vert}{\sum_{hkl} \sum_{j}I_{hkl,j}}
  }[/math]

R_mrgd-I is similarly defined in Diederichs and Karplus .

Similarly, one should use R_mrgd-F as a quality indicator for amplitudes , which may be calculated as:

[math]\displaystyle{ 
 R_{mrgd-F} = \frac{\sum_{hkl} \sqrt \frac{1}{n-1} \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].

In the sums above, the summation omits those reflections with just one observation.

measuring radiation damage

We can plot (Diederichs )

[math]\displaystyle{ R_{d} = \frac{\sum_{hkl} \sum_{|i-j|=d} \vert I_{hkl,i} - I_{hkl,j}\vert}{\sum_{hkl} \sum_{|i-j|=d} (I_{hkl,i} + I_{hkl,j})/2} }[/math]

which gives us the average R-factor of two reflections measured d frames apart. As long as the plot is parallel to the x axis there is no radiation damage. As soon as the plot starts to rise, we see that there's a systematical error contribution due to radiation damage.

Strong wiggles at very high d are irrelevant as only few reflections contribute.

To my knowledge, the only program that implements this currently (December 2008) is XDSSTAT.

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 [5]

• 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 [6]

• 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 to judge data quality, R_meas should be used instead. The reason is that the former depend on multiplicity, whereas the latter doesn't.

• R/R_free and NCS: reflections in work and test set are not independent if chosen randomly. It is better to choose the test set reflections in thin resolution shells (FIXME: references and programs for this). A paper investigating this thoroughly is Fabiola, F., A. Korostelev, et al. (2006). "Bias in cross-validated free R factors: mitigation of the effects of non-crystallographic symmetry." Acta Cryst. D 62: 227-38.

• Sets of reflections used for calculating R_free should be maintained throughout a project. This is nicely discussed at http://www.bmsc.washington.edu/people/merritt/xplor/rfree_example.html .