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R factor

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

R = {{\sum | F_{obs} - F_{calc} | } \over {\sum |F_{obs} |}},

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 F2), the R factor based on F2 is used to monitor the progress of refinement:

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

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

R_B = {{\sum | I_{obs} - I_{calc} | } \over {\sum |I_{obs} |}}.

Even for refinement against F2 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 R \ge 0.5 will not respond to attempts at improvement. An early model with R \le  0.4 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 \sim 0.2. Small organic molecules commonly refine to R < 0.05. 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 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\sum w (Y_o - Y_c)^2 (being FF2 or I). The general term for a weighted residual is

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

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 artifacts, and must always be described when reporting R factors.