Testing for remanent systematic error

Parrish, W.; Wilson, A. J. C.; Langford, J. I.

doi:10.1107/97809553602060000596

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

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International Tables for Crystallography (2006). Vol. C. ch. 5.2, p. 498

Section 5.2.9. Testing for remanent systematic error

W. Parrish,^a ^‡ A. J. C. Wilson^b ^‡ and J. I. Langford^c

^a IBM Almaden Research Center, San Jose, CA, USA,^bSt John's College, Cambridge CB2 1TP, England, and ^cSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England

5.2.9. Testing for remanent systematic error

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Since about 1930, it has been claimed that the lattice parameters of cubic substances could be measured within one part in 50 000. Precision (that is, reproducibility of measurements by one technique within one laboratory) of this order is achieved, but accuracy (agreement between determinations by different techniques or by the same technique in different laboratories) is lower. The IUCr lattice-parameter project (Parrish, 1960) showed a standard deviation of 1 in 30 000 in inter-laboratory comparisons, with some outlying values differing from the mean by one or two parts in 10 000. At that time, therefore, precision was considerably better than accuracy (absence of significant remanent systematic error). Testing for remanent systematic error is thus valuable as an occasional test of methodology, though not undertaken as routine. The principle is outlined here, and more details are given in Chapters 8.4 and 8.5 .

When refinement of parameters is performed by least squares , weighted in accordance with the reciprocal of the estimated variance, the expected value of the weighted sum of squares is $[\langle S\rangle=n-p, \eqno (5.2.9.1)]$ where n is the number of terms summed and p is the number of parameters determined. The standard deviation of the sum S is expected to be $[\sigma_S=[2(n-p)]{}^{1/2} \eqno (5.2.9.2)]$ approximately (Wilson, 1980), so that if the actual value of S exceeds $[\eqalignno{ \langle S\rangle+k\sigma_S &= n-p+k\sigma_S \cr &= n-p+k[2(n-p)]{}^{1/2} &(5.2.9.3)}]$ (where k = 2 or 3), one can reasonably conclude that there are defects in the model (remanent systematic errors). If S is less than this value, one can reasonably conclude that any defects in the model (systematic errors) are at worst of the same order of magnitude as the statistical fluctuations; the sensitivity of the test increases rather slowly with n − p. The method was advocated by Beu and his collaborators (Beu, Musil & Whitney, 1962, 1963; Beu, 1964; Beu & Whitney, 1967; Langford, Pike & Beu, 1964; see also Mitra, Ahmed & Das Gupta, 1985) because tests of the hypothesis `no remaining systematic error' based on likelihood were available; they assumed a normal distribution of errors, possibly without realizing, and certainly without emphasizing, that the method was then equivalent to least squares . Their application of the method to testing for remanent systematic error in lattice-parameter determination was successful: the aberrations of the counter diffractometer were found to be adequately accounted for: additional aberrations were found for the Bond method (see Chapter 5.3 ); Boom (1966) used it in testing the accuracy of the Debye–Scherrer method.

In statistical literature, the weighted sum of squares S is often called the scaled deviance, and $[E=[S-(n-p)]/[2(n-p)]{}^{1/2} \eqno (5.2.9.4)]$ is called the excess. The test for the absence of significant systematic error is then that the excess should be less than three.

References

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International Tables for Crystallography (2006). Vol. C. ch. 5.2, p. 498