Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 7.5, p. 667

Section 7.5.6. Treatment of measured-as-negative (and other weak) intensities

A. J. C. Wilsona

aSt John's College, Cambridge CB2 1TP, England

7.5.6. Treatment of measured-as-negative (and other weak) intensities

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It has been customary in crystallographic computations, but without theoretical justification, to omit all reflections with intensities less than two or three times their standard uncertainties. Hirshfeld & Rabinovich (1973[link]) asserted that the failure to use all reflections, even those for which the subtraction of background has resulted in a negative net intensity, at their measured values will lead to a bias in the parameters resulting from a least-squares refinement. This is, however, inconsistent with the Gauss–Markov theorem (see Section 8.1.2[link] ), which shows that least-squares estimates are unbiased, independent of the weights used, if the observations are unbiased estimates of quantities predicted by a model. Giving some observations zero weight therefore cannot introduce bias. Provided the set of included observations is sufficient to give a nonsingular normal equations matrix, parameter estimates will be unbiased, but inclusion of as many well determined observations as possible will yield the most precise estimates. Requiring that the net intensity be greater than 2σ assures that the value of |F| will be well determined. Furthermore, Prince & Nicholson (1985[link]) showed that, if the net intensity, I, or |F|2 is used as the observed quantity, weak reflections have very little leverage (see Section 8.4.4[link] ), and therefore omitting them cannot have a significant effect on the precision of parameter estimates.

The use of negative values of I or |F|2 is also inconsistent with Bayes's theorem, which implies that a negative value cannot be an unbiased estimate of an inherently non-negative quantity. There are statistical methods for estimating the positive value of |F| that led to a negative value of I. The best known approach is the Bayesian method of French & Wilson (1978[link]), who observe that "Instead of thanking the data for the information that certain structure factor moduli are small, we accuse them of assuming `impossible' negative values. What we should do is combine our knowledge of the non-negativity of the true intensities with the information concerning their magnitudes contained in the data."


French, S. & Wilson, K. (1978). On the treatment of negative intensity observations. Acta Cryst. A34, 517–525.
Hirshfeld, F. L. & Rabinovich, D. (1973). Treating weak reflexions in least-squares calculations. Acta Cryst. A29, 510–513.
Prince, E. & Nicholson, W. L. (1985). The influence of individual reflections on the precision of parameter estimates in least squares refinement. Structure & statistics in crystallography, edited by A. J. C. Wilson, pp. 183–195. Guilderland, NY: Adenine Press.

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