The statistics of cross-validation residuals
The statistics of cross-validation residuals
Values of 
are affected by all types of error in the model
and the data. The ratio, however, is independent of random
errors and provides a statistic which can be compared with its
theoretically estimated value and used to detect systematic model or
weighting errors at the convergence of least-squares refinement.
However, achievement of a theoretical value of the ratio is not by
itself proof of the correctness of the model or of the quality of the
refinement. Nevertheless it would still be helpful if refinement
programs printed out the calculated and estimated value of the
ratio using the expressions shown in Table 1. This would
encourage a better understanding of than exists at
present. Calculation of the observed and theoretical values of these
ratios has already been implemented in the refinement program
RESTRAIN (Driessen et al., 1989).
At low resolution the number of data excluded for cross-validation may be small and in these circumstances the precision of free residuals is important. This will be the subject of part II of this work.
The authors would like to thank Professor D W J Cruickshank FRS for his helpful comments on this paper.
The statistics of cross-validation residuals