More about residual values

Abstract

The usual residual values are complemented by expectation values based solely on the experimental data and the number of model parameters. These theoretical R values serve as benchmark values when all of the basic assumptions for a least-squares refinement, i.e. no systematic errors and a fully adequate model capable of describing the data, are fulfilled. The prediction of R values as presented here is applicable to any field where model parameters are fitted to data with known precision. For crystallographic applications, F(2)-based residual benchmark values are given. They depend on the first and second moments of variance, intensity and significance distributions, <σ(2)>, <Io(2)>, <Io(2)/σ(2)>. Possible applications of the theoretical R values are, for example, as a data-quality measure or the detection of systematic deviations between experimental data and model predicted data, although the theoretical R values cannot identify the origin of these systematic deviations. The change in R values due to application of a weighting scheme is quantified with the theoretical R values.

About systematic errors in charge-density studies

“…The most important conclusion from this work is that the requirement of a Gaussian distribution of residuals is in most cases not fulfilled. This was already the case for standard structures (Henn & Schö nleber, 2013) and it is surprisingly Table 4 Influence of correlations introduced by the structure model and of data truncation on 2 S in IAM refinements against type-one consistency data sets with s.u. 's according to equation (22) also the case in high-resolution charge-density studies, which are much more expensive in terms of expenditure of time, know-how and financial support.…”

“…In a former publication theoretical residual values (R values) were introduced (Henn & Schö nleber, 2013). These are also called predicted R values as they give the attainable agreement factors from the experimental data and from the degrees of freedom of a model that is fitted to the data.…”

An alternative to the goodness of fit

“…For a first application of the aGoFs, standard structure determination data were used. The data have been evaluated before; more information about the data sets, for example the parameter values a and b of the weighting scheme, may be found in Henn & Schö nleber (2013). Some numbers are recalculated for the present study and minor changes occurred due to round-off errors or slightly maladjusted parameter values.…”

Statistical tests against systematic errors in data sets based on the equality of residual means and variances from control samples: theory and applications

“…The theoretical R values and the meta residual value, however, both assume a Gaussian distribution of residuals. This assumption holds neither in standard structure determination (Henn & Schö nleber, 2013) nor in charge-density studies (Henn & Meindl, 2014a,b).…”

“…Reciprocal-space methods were also developed, like the theoretical R values (Henn & Schö nleber, 2013;Henn & Meindl, 2014a). These predict the agreement factor from the diffraction data under the assumption of absence of systematic errors and can be applied in a similar way to the free R value (Brü nger, 1992).…”