A very commonly used criterion for the choice of weights is that the variance of the derived estimates be minimal, although other weighting schemes are permissible. The minimum-variance criterion implies that `W` = `V`^{-1}. Regardless of the choice of weights, proper calculation of standard uncertainties and of the goodness of fit in intensity averaging and in least squares will require an estimate of `V`, the variance-covariance matrix of the observations. The frequentist and Bayesian approaches to statistics lead to different interpretations of `V` (see section on Refinement). For the one, `V` is concerned only with the random fluctuations of the measurements whereas, for the other, `V` also incorporates the scientist's belief in the model with which the observations are being analysed.

(19) . . ` E`(`O`_{i}) = ` µ`_{I} /`c`_{i}.

An often used expression for the uncertainty of ` O`_{i} (Abrahams, 1974) is

(20) . . ` u`^{2}(`O`_{i}) = ` E`(`O`_{i}) + ` b`_{i} + ` k``E`(`O`_{i})^{2}

= `µ`_{I} /`c`_{i} + ` b`_{i} + ` k``µ`_{I}^{2}/`c`_{ i}^{2},

where `b`_{i} is the contribution of the background estimated from peak and background count rates, and ` k``E`(`O`_{i})^{2} is the contribution from known and unknown sources of random error. Minimum variance weights ` u`^{-2}(`I`_{i}) = {`c`_{i}`u`(`O`_{ i})}^{-2} needed for the calculation of the average intensity can be obtained by iteratively replacing ` µ`_{I} by an approximate average value <`I`>. In a first step, ` b`_{i} and `k` may be neglected and the average intensity then becomes

(21) . . <`I`> = (`I`_{i}/`c`_{i}) / (1/`c`_{i}) = `O`_{i} / (1/`c`_{i}).

Estimates of the factor `k` in (20) can be obtained from the variations of periodically measured check reflections. Differences among symmetry-equivalent intensities can (and indeed should) also be used to estimate `k`, the value of which then includes at least part of an anisotropic systematic error. A possible procedure consists in adjusting `k` to obtain comparable values for the variance of estimated from the `u`^{2}(`I`_{i}) derived from (20), and estimated from the spread of the `I`_{i} around the average value <`I`>:

(22) . . ` u`^{2}(<`I`>) 1 / ^{n} ` u`^{-2} (`I`_{i})

[ ^{n} `u`^{-2} (`I`_{i}) { `I`_{i} -<`I`> }^{2} ] / [ (`n` - 1) ^{n} ` u`^{-2}(`I`_{i}) ];

`u`(`I`_{i}) = ` c`_{i}`u`(`O`_{i}); ` n` 2.

The average goodness of fit of all symmetry-equivalent sets is then near 1·0. If the number `n` of symmetry-equivalent observations is large, the larger of the two estimates of `u`(<`I`>) obtained with some approximation for ` u`^{2}(`I`_{i}), their average, or the value from the spread alone may be used. Alternative and more convenient procedures, variants and extensions may be proposed, or have already been implemented (see also Blessing, 1987).

The omission of observations is equivalent to assigning zero weights. Again, a bias may be introduced if the criterion for omission is based on `d`_{j}, ` O`_{j} or ` C`_{j}. Thus, if weak reflections are systematically omitted, those with an accidentally low intensity will be preferentially discarded, and those with an accidentally high intensity will have a better chance of being retained. An example of such a bias has been described by Seiler, Schweizer & Dunitz (1984). Omission of observations according to sin/ or reflection number should *not* result in a bias.

Weight modification schemes designed to lessen the effects of systematic errors are necessarily based on the deviates ` d`_{j}. An example is found in Wang & Robertson (1985). They estimate the variances of the structure amplitudes from the distribution of the weighted deviates. The resulting weights are intended to represent all kinds of random errors from different sources, but include also contributions of systematic errors. The **robust/resistant refinement techniques** are another example. The results of a standard least-squares refinement may be considerably influenced by 'outliers', observations that seem so discrepant with calculated values that a blunder of the investigator or a malfunctioning of the equipment is suspected. The naive approach to such observations is to discard them by setting the weights to zero whenever the discrepancy exceeds a certain limit. The robust/resistant methods avoid the discontinuity of this procedure by decreasing the weights gradually with increasing disagreement (Prince, 1982; Nicholson, Prince, Buchanan & Tucker, 1982).

Obtaining an unweighted least-squares fit between observed and calculated |`F`|^{2} is equivalent to obtaining an unweighted least-squares fit between observed and calculated Patterson functions. Similarly, obtaining an unweighted least-squares fit on the |`F`|s is almost equivalent to obtaining an unweighted least-squares fit between observed and calculated electron densities (Wilson, 1976a).

Special weighting schemes permit more accurate fitting of the electron densities at the atomic sites. Thus, Cochran (1948, 1951) showed that refinement on the |`F`|s with weights inversely proportional to the reciprocal of the atomic scattering factor of an atom gives the coordinates of the maximum of its electron density as represented by a Fourier series. In the presence of several types of atoms, weighting with the reciprocal of a mean scattering factor is probably a satisfactory approximation. Similarly, the weighting scheme of Dunitz & Seiler (1973) designed to determine more accurately the coordinates of the atomic centers also emphasizes high-order reflections.

Bernardinelli & Flack (1985) have designed weights that enhance the sensitivity of a refinement to the centrosymmetric or antisymmetric parts of the electron density, and also to the absolute structure. They are useful in resolving ambiguities between centro- and non-centrosymmetric structures and mainly affect weak reflections, the importance of which in such problems has been pointed out by Marsh (1981, 1986). Weak reflections are also critically important in the refinement of superstructures, and weighting schemes are easily designed to enhance their contribution to the normal equations.

© 1989, 1995 International Union of Crystallography

Updated 23rd Sept. 1996