Section 18.4.4.1. Use of F or F²

The X-ray experiment provides two-dimensional diffraction images. These are transformed to integrated but unscaled data, which are transformed to Bragg reflection intensities that are subsequently transformed to structure-factor amplitudes. At each transformation some assumptions are used, and the results will depend on their validity. Invalid assumptions will introduce bias toward these assumptions into the resulting data. Ideally, refinement (or estimation of parameters) should be against data that are as close as possible to the experimental observations, eliminating at least some of the invalid assumptions. Extrapolating this to the extreme, refinement should use the images as observable data, but this poses several severe problems, depending on data quantity and the lack of an appropriate statistical model.

Alternatively, the transformation of data can be improved by revising the assumptions. The intensities are closer to the real experiment than are the structure-factor amplitudes, and use of intensities would reduce the bias. However, there are some difficulties in the implementation of intensity-based likelihood refinement (Pannu & Read, 1996).

Gaussian approximation to intensity-based likelihood (Murshudov et al., 1997) would avoid these difficulties, since a Gaussian distribution of error can be assumed in the intensities but not the amplitudes. However, errors in intensities may not only be the result of counting statistics, but may have additional contributions from factors such as crystal disorder and motion of the molecules in the lattice during data collection.

Nevertheless, the problem of how to treat weak reflections remains. Some of the measured intensities will be negative, as a result of statistical errors of observation, and the proportion of such measurements will be relatively large for weakly diffracting macromolecular structures, especially at atomic resolution. For intensity-based likelihood, this is less important than for the amplitude-based approach. French & Wilson (1978) have given a Bayesian approach for the derivation of structure-factor amplitudes from intensities using Wilson's distribution (Wilson, 1942) as a prior, but there is room for improvement in this approach. Firstly, the assumed Wilson distribution could be upgraded using the scaling techniques suggested by Cowtan & Main (1998) and Blessing (1997), and secondly, information about effects such as pseudosymmetry could be exploited.

Another argument for the use of intensities rather than amplitudes is relevant to least squares where the derivative for amplitude-based refinement with respect to when is equal to zero is singular (Schwarzenbach et al., 1995). This is not the case for intensity-based least squares. In applying maximum likelihood, this problem does not arise (Pannu & Read, 1996; Murshudov et al., 1997).

Finally, while there may be some advantages in refining against F², Fourier syntheses always require structure-factor amplitudes.