Distribution of ratios

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 197-198 | 1 | 2 |

Section 2.1.6.2. Distribution of ratios

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^bSt John's College, Cambridge, England
Correspondence e-mail: ushmueli@post.tau.ac.il

2.1.6.2. Distribution of ratios

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Ratios like $[S_{n, \, m} = J_{n}/K_{m}, \eqno(2.1.6.7)]$ where $[J_{n}]$ is given by equation (2.1.6.1), $[K_{m} = \textstyle\sum\limits_{j = 1}^{m}H_{j}, \eqno(2.1.6.8)]$ and the $[H_{j}]$ 's are the intensities of a set of reflections (which may or may not overlap with those included in $[J_{n}]$ ), are used in correlating intensities measured under different conditions. They arise in correlating reflections on different layer lines from the same or different specimens, in correlating the same reflections from different crystals, in normalizing intensities to the local average or to $[\Sigma]$ , and in certain systematic trial-and-error methods of structure determination (see Rabinovich & Shakked, 1984, and references therein). There are three main cases:

(i) $[G_{i}]$ and $[H_{i}]$ refer to the same reflection; for example, they might be the observed and calculated quantities for the reflection measured under different conditions or for different crystals of the same substance; or
(ii) $[G_{i}]$ and $[H_{i}]$ are unrelated; for example, the observed and calculated values for the reflection for a completely wrong trial structure, of values for entirely different reflections, as in reducing photographic measurements on different layer lines to the same scale; or
(iii) the $[G_{i}]$ 's are a subset of the $[H_{i}]$ 's, so that $[G_{i} = H_{i}]$ for $[i \;\lt\; n]$ and $[m \;\gt\; n]$ .

Aside from the scale factor, in case (i) $[G_{i}]$ and $[H_{i}]$ will differ chiefly through relatively small statistical fluctuations and uncorrected systematic errors, whereas in case (ii) the differences will be relatively large because of the inherent differences in the intensities. Here we are concerned only with cases (ii) and (iii); the practical problems of case (i) are postponed to IT C (2004).

There is little in the crystallographic literature concerning the probability distribution of sums like (2.1.6.1) or ratios like (2.1.6.7); certain results are reviewed by Srinivasan & Parthasarathy (1976, ch. 5), but with a bias toward partially related structures that makes it difficult to apply them to the immediate problem.

In case (ii) ( $[G_{i}]$ and $[H_{i}]$ independent), acentric distribution, Table 2.1.5.1 gives the distribution of the ratio $[u = nY/(mZ) \eqno(2.1.6.9)]$ $[p(u)\;{\rm d}u = \beta_{2}[nY/(mZ)\hbox{; } n, m]\;{\rm d}[nY/(mZ)], \eqno(2.1.6.10)]$ where $[\beta_{2}]$ is a beta distribution of the second kind, Y is given by equation (2.1.6.2) and Z by $[Z = K_{m}/m, \eqno(2.1.6.11)]$ where n is the number of intensities included in the numerator and m is the number in the denominator. The expected value of [Y/Z] is then $[\langle Y/Z \rangle = {{m}\over{m-1}} = 1+{{1}\over{m}}+\ldots \eqno(2.1.6.12)]$ with variance $[\sigma^{2} = {{(n+m-1)m^{2}}\over{(m-1)^{2}(m-2)n}}. \eqno(2.1.6.13)]$ One sees that [Y/Z] is a biased estimate of the scaling factor between two sets of intensities and the bias, of the order of $[m^{-1}]$ , depends only on the number of intensities averaged in the denominator. This may seem odd at first sight, but it becomes plausible when one remembers that the mean of a quantity is an unbiased estimator of itself, but the reciprocal of a mean is not an unbiased estimator of the mean of a reciprocal. The mean exists only if $[m \;\gt\; 1]$ and the variance only for $[m \;\gt\; 2]$ .

In the centric case, the expression for the distribution of the ratio of the two means Y and Z becomes $[p(u)\;{\rm d}u = \beta_{2}[nY/(mZ)\hbox{; } n/2, m/2]\;{\rm d}[nY/(mZ)] \eqno(2.1.6.14)]$ with the expected value of [Y/Z] equal to $[\langle Y/Z \rangle = {{m}\over{m-2}} = 1+{{2}\over{m}}+\ldots \eqno(2.1.6.15)]$ and with its variance equal to $[\sigma^{2} = {{2(n+m-2)m^{2}}\over{(m-2)^{2}(m-4)n}}. \eqno(2.1.6.16)]$ For the same number of reflections, the bias in $[\langle Y/Z \rangle]$ and the variance for the centric distribution are considerably larger than for the acentric. For both distributions the variance of the scaling factor approaches zero when n and m become large. The variances are large for m small, in fact `infinite' if the number of terms averaged in the denominator is sufficiently small. These biases are readily removed by multiplying [Y/Z] by [(m-1)/m] or [(m-2)/m] . Many methods of estimating scaling factors – perhaps most – also introduce bias (Wilson, 1975; Lomer & Wilson, 1975; Wilson, 1976, 1978c) that is not so easily removed. Wilson (1986a) has given reasons for supposing that the bias of the ratio (2.1.6.7) approximates to $[1+{{\sigma^{2}(I)}\over{m\langle I \rangle^{2}}}, \eqno(2.1.6.17)]$ whatever the intensity distribution. Equations (2.1.6.12) and (2.1.6.15) are consistent with this.

References

International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google Scholar

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Rabinovich, D. & Shakked, Z. (1984). A new approach to structure determination of large molecules by multi-dimensional search methods. Acta Cryst. A40, 195–200.Google Scholar

Srinivasan, R. & Parthasarathy, S. (1976). Some statistical applications in X-ray crystallography. Oxford: Pergamon Press. Google Scholar

Wilson, A. J. C. (1975). Effect of neglect of dispersion on apparent scale and temperature factors. In Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 325–332. Copenhagen: Munksgaard.Google Scholar

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International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 197-198