Statistical descriptors in crystallography
Defects in the modelIt is evident that the results of a refinement can be improved by identifying the presence and origin of systematic error, and by improving the model to satisfy the assumptions of frequentist statistics (Gauss-Markov theorem) more fully. The almost universally observed deviation of the goodness-of-fit value from unity indicates the presence of defects in the model and/or the variance-covariance matrix of the observations (see also Wilson, 1980b). A normal probability plot gives more detailed information on the presence of systematic error.
Identifying the origin of systematic error and improving the model is far more difficult. Beu and his collaborators (Beu, Musil & Whitney, 1962, 1963; Beu & Whitney, 1967; see also Mitra, Ahmed & Das Gupta, 1985; Mandel, 1980) achieved a major improvement in the precise and accurate determination of lattice parameters by careful tests and corrections for remaining systematic errors based on maximum likelihood. They assumed a normal distribution of errors, and their refinement technique was therefore equivalent to least squares (Wilson, 1980b). The modelling of electron density distributions with aspherical-atom formalisms (Stewart, 1976; Hirshfeld, 1977), and of atomic thermal-displacement probability density functions with anharmonic contributions (Johnson & Levy, 1974; Kuhs, 1983) has shown considerable success, but has not had a significant impact on standard crystal-structure determination, which in its present form answers the needs of chemical crystallographers and will continue to be an increasingly automated analytical technique. A thorough study of intensity measurement and data-reduction procedures might indicate more generally applicable improvements.
© 1989, 1995 International Union of Crystallography
Updated 18th Sept. 1996
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