R factor

Definition

The term R factor in crystallography is commonly taken to refer to the 'conventional' R factor

[math]R = {{\sum | F_{obs} - F_{calc} | } \over {\sum |F_{obs} |}}[/math],

a measure of agreement between the amplitudes of the structure factors calculated from a crystallographic model and those from the original X-ray diffraction data. The R factor is calculated during each cycle of least-squares structure refinement to assess progress. The final R factor is one measure of model quality.

More generally, a variety of R factors may be determined to measure analogous residuals during least-squares optimization procedures. Where the refinement attempts to minimize the deviates of the squares of the structure factors (refinement against [math]F^2[/math]), the R factor based on [math]F^2[/math] is used to monitor the progress of refinement:

[math]R(F^2) = {{\sum | F^2_{obs} - F^2_{calc} | } \over {\sum |F^2_{obs} |}}[/math].

Likewise, refinement against I can be tracked using the Bragg R factor

[math]R_B = {{\sum | I_{obs} - I_{calc} | } \over {\sum |I_{obs} |}}[/math].

Even for refinement against [math]F^2[/math] or I, the 'conventional' R factor may be calculated and quoted as a measure of model quality, in order to compare the resulting quality of models calculated at different times and with different refinement strategies.

The R factor is sometimes described as the discrepancy index.

R factor as a measure of structure quality

Theoretical values of R range from zero (perfect agreement of calculated and observed intensities) to about 0.6 for a set of measured intensities compared against a set of random intensities. R factors greater than 0.5 indicate very poor agreement between observed and calculated intensities, and many models with [math]R \ge 0.5[/math] will not respond to attempts at improvement. An early model with [math]R \le 0.4[/math] can usually be improved during refinement. A desirable target R factor for a protein model refined with data to 2.5 Å is considered to be [math]\sim 0.2[/math]. Small organic molecules commonly refine to [math]R \lt 0.05[/math]. However, the R factor must always be treated with caution, as an indicator of precision and not accuracy. Partially incorrect structures have been reported with R values below 0.1; many imprecise but essentially correct structures have been reported with higher R values.

Weighted R factors

In practice, weighted R factors are more often used to track least-squares refinement, since the functions minimized are weighted according to estimates of the precision of the measured quantity Y: [math]\sum w (Y_o - Y_c)^2[/math] (Y being F, [math]F^2[/math] or I). The general term for a weighted residual is

[math]wR = ({{\sum |w |Y_o - Y_c|^2|}\over{\sum |wY^2_o}|})^{1/2}[/math]

The sum is usually computed over all reflections measured in the experiment. However, occasionally reflections are omitted from the calculation, either because they are believed to result from a systematic experimental error or are recorded with an intensity small compared with background noise. Any such selection may introduce statistical artefacts, and must always be described when reporting R factors.