The statistics of cross-validation residuals - Theory

R_Gfree and the R_Gfree ratio in restrained refinement

In a restrained refinement, such as is typical in macromolecular crystallography, the ratios derived in the previous two sections would only be applicable if were calculated from a random selection of residuals including both structure amplitude observations and restraints. Since R-factors are traditionally only based on structure amplitudes, the estimation of ratios for restrained refinement requires further analysis.

The number of observations this time is n and f of these are structure amplitudes, the balance consisting of r geometrical, thermal or other restraints which make a contribution to the minimised residual at convergence. From equation (1) we have

where the summation is taken over the restraint observations. Using this equation and equation (2) the summation over the structure amplitude observations can be written

This result and equations (4) and (7) give the following approximation for

Similarly an approximation to can be derived from equations (4), (7) and (14).

Hence an estimate of at the convergence of a correctly weighted restrained refinement with only random uncorrelated errors is

The estimated ratio of the free residual to the included residual is given by

and hence

The statistics of cross-validation residuals - Theory