Maximum-likelihood methods in powder diffraction refinements

The validity of least-squares procedures commonly used nowadays for the analysis of single-crystal, X- ray and neutron diffraction data is examined. An improved methodology that rests on sound statistical theory is proposed and turns out to be a fruitful way to consider any crystallographic refinement. A maximum-likelihood estimation procedure is developed for Poisson regression models. Measures of the goodness of fit (other than the R factor), generalized residuals and diagnostic plots are described. Confidence regions and intervals are also discussed. A set of measures of the influence of data on the fit and the parameter estimates is obtained for Poisson statistics. Finally, the effect of under or over dispersion of the data randomness with respect to a true Poisson distribution is considered and model-independent estimates of this dispersion are discussed.