R_free and the R_free Ratio. I. Derivation of Expected Values of Cross-Validation Residuals Used in Macromolecular Least-Squares Refinement

The last five years have seen a large increase in the use of cross validation in the refinement of macromolecular structures using X-ray data. In this technique a test set of reflections is set aside from the working set and the progress of the refinement is monitored by the calculation of a free R factor which is based only on the excluded reflections. This paper gives estimates for the ratio of the free R factor to the R factor calculated from the working set for both unrestrained and restrained refinement. It is assumed that both the X-ray and restraint observations have been weighted correctly and that there is no correlation of errors between the test and working sets. It is also shown that the least-squares weights that minimize the variances of the refined parameters, also approximately minimize the free R factor. The estimated free R-factor ratios are compared with those reported for structures in the Protein Data Bank.