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

• The PDBe provide a service to plot many different statistical properties in the PDB against other properties. The link is http://www.ebi.ac.uk/pdbe-as/pdbestatistics/PDBeStatistics.jsp . You can see that there is an option of RDiff which is the difference between R and R-Free for all structures that contain both data. Take a look at this first. There is a second parameter which you can set to resolution and this will allow you to draw a plot that you want. This will draw a 3D isometric plot which you can scale, and pick data points to view particular entries.
• formula from Kleywegt and Jones (2002): 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

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)

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. Since the twin related reflections have the same sin(theta)/lambda values they will not be split over the working and reference sets. DATAMAN from the Uppsala Software Factory and XPREP (a program which may be obtained from Bruker) offer this option. The "RFREE SHELL" command in sftools is another way to select thin shells. A disadvantage is the the maps may not be quite as good as when the free R reflections are selected randomly. (FIXME: which Phenix program does 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 . Note that none of the programs mentioned for selecting thin shells will allow you to extend the set of shells to higher resolution if you want to preserve your existing R-free set.