R-free likelihood-based estimates of errors for phases calculated from atomic models

Reasonable assumptions about the statistical properties of errors in an atomic model lead to the probability distributions for the values of structure-factor phases. These distributions contain some generally unknown parameters reflecting how large the model errors are. These parameters must be determined properly to give realistic estimates of phase errors. Maximum-likelihood-based estimates suggested by Lunin & Urzhumtsev [Acta Cryst. (1984), A40, 269-277] are good for models not subjected to refinement but underestimate the errors when being used for refined models. The R-free methodology of Brünger [Nature (London), (1992), 355, 472-474] applied to the likelihood-function calculation allows realistic phase-error estimates to be obtained for both unrefined and refined models. These estimates may be used as an additional indicator in the refinement process.