A Self-Validation Technique for Protein Structure Refinement: the Extended Hamilton Test

An extension is proposed for the self-validation Hamilton test [Hamilton (1965). Acta Cryst. 18, 502-510] for crystallographic refinement. The method is based on the statistical F test and evaluates the significance of the R-factor ratio between two refinement protocols. The general case of two refinements carried out with different numbers and types of non-linear restraints is examined. The restraints are considered as extra observations weighted by a coefficient expressing their effective number. There exists a restriction on the weighting coefficients between the two refinements. An empirical method to evaluate the effective number of restraints is provided. The method may allow the detection of unreasonably tight restraints. The expectation value for r.m.s. R_free, given the r.m.s. R, can be estimated. Thus, the significance of the observed drop in R_free can be assessed. Compared to cross-validation using R_free [Brünger (1992). Nature (London), 355, 472-474] self-validation has the advantage that it does not require omission of any experimental data. The significance of the improvement obtained by moving from isotropic to anisotropic description of thermal parameters in the refinement of a protein at 1.5 Å, resolution is used as an example.