Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 2.2, pp. 64-65

Section 2.2.1. Introduction

H. M. Einspahra* and M. S. Weissb

aPO Box 6483, Lawrenceville, NJ 08648–0483, United States, and bHelmholtz-Zentrum Berlin für Materialien und Energie, Macromolecular Crystallography (HZB-MX), Albert-Einstein-Str. 15, D-12489 Berlin, Germany
Correspondence e-mail:

2.2.1. Introduction

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The genesis of this chapter was a perceived need for a single location in the volume in which consensus definitions could be found for the many statistical indicators of quality or figures of merit that have been developed to monitor a macro­molecular crystallography (MX) experiment and its final outcome, a model. The evolving experiment has generated a rich collection of R values, signal-to-noise indicators, correlation coefficients and other figures of merit. As improvements in data collection, processing and other aspects of the experiment continue, we can expect a continued evolution of new or improved indicators of quality with which to monitor the impact of those improvements.

This chapter, then, attempts to provide a comprehensive list of the indicators of quality currently in use and, for each indicator in the list, a precise definition that conforms to consensus interpretations of the literature and current practice. The authors acknowledge that useful indicators may have been missed in the sweep that produced this list, but hasten to point out that the generation of new indicators is an ongoing process with the newest ones often in obscurity for a period before their utility is recognized and adopted by experimenters. There is also a subset of the indicators in the list whose members stand out as universally accepted and crucial indicators that either every experimental description must include or that every experimenter should be familiar with. A summary of these is given at the end of this article in Table[link]. The authors also acknowledge that there may be competing definitions for some indicators. Where they occur, these competing definitions will be pointed out and a discussion will be provided which, at a minimum, will attempt to clarify the differences, but will also, where possible, attempt to arbitrate or discriminate among opposing views. It should be noted that the scope of this chapter is primarily concerned with the crystallographic experiment. While quality indicators useful for model refinement are given in Sections 2.2.8[link] to 2.2.10[link][link], the validation of the refined model is covered more extensively in Part 21[link] of this volume.

Before proceeding with the quality indicators themselves, it is necessary to discuss a small collection of parameters whose values, typically left to the experimenter to fix, impact on virtually all the indicators of quality we present here. The reason for this wide impact is that they are used to determine which reflections are included in the final data set in question. In their simplest form, the effect is a limit, applied as a cutoff in intensity or resolution, beyond which reflections are excluded from consideration.

When a cutoff is applied based on intensities, the limiting value may be simply a fixed number, for example zero, or it may vary from reflection to reflection, for example some multiple of the standard uncertainty of the reflection intensity. Reflections with intensities below limiting values are excluded. Reflections excluded in this manner are often referred to as unobserved.

When the objective is to estimate electron-density distributions, the exclusion of intensity terms below very low limiting values is unlikely to have any significant negative impact and may offer a positive savings in computation time. This may not be the case, however, when the objective is refinement. Inclusion of reflections of low intensity, even zero, may have important positive results on the quality of the final model. True, some of these intensities may be poorly known or, if less than zero, physically unrealistic, but given the excellent quality of modern data-collection instrumentation and techniques, few intensities, if any, are without some reasonable estimate of their standard uncertainty, so that weighting procedures can be applied to modulate the impact of individual terms in the final result and judgments can be made about exclusion of intensity values that are truly improbable. With the application of proper weighting procedures, there seems to be little justification for exclusion of reflections from consideration based on intensity (except for the truly improbable mentioned above) when computing indicators of quality. As instrumentation and techniques continue to improve, this is a topic that merits continued attention and debate within the MX community in search of consensus best practices for handling weak reflections and for including them in estimators of quality.

While an acceptable estimate of the nominal resolution of a diffraction data set is widely considered to be a high-value indicator of data quality, assignment of a value to this limit is typically left to the experimenter and is thus prone to subjectivity. Some guidelines for estimation have emerged. One would set the nominal resolution based on the percentage of weak reflections above the limit so that reflections with intensities above that limit would be included. An example might be the resolution at which 70% of unique reflections have intensities above zero or above some multiple of their standard uncertainties. Another way to estimate the nominal resolution which has gained wide acceptance is based on the overall signal-to-noise value where, in its most popular expression, the limit value is the resolution at which the mean signal-to-noise ratio in the outer resolution shell falls to 2. Each of these estimation methods is susceptible to convolution with limits based on intensity leading to, for example, a limit set as the resolution at which the mean signal-to-noise ratio `of observed reflections' falls to 2. The guidelines for estimating resolution limits – the methods to be applied, the constraining values such as 70% or 2, and the proper integration with limits applied based on intensity – also merit further attention and discussion by the MX community, the goal being definition of consensus best practices. We take this opportunity to suggest that the widely accepted estimate, the resolution at which the mean signal-to-noise in the outer resolution shell falls to 2, calculated without imposition of a cutoff in intensity, has much to recommend it. True, some susceptibility to subjectivity remains in the definition of resolution-shell ranges and limits, but the impact is minimal and hardly worth the effort. On the other hand, modifying the indicator to remove the shells so that the mean signal-to-noise ratio applied to all data above an appropriately adjusted limit value would extinguish that source of subjectivity.

When a cutoff is applied based on nominal resolution, then considerations similar to those for intensity-based cutoffs apply. If the objective is to estimate electron-density distributions, exclusion of large numbers of weak intensity terms beyond a limiting resolution is unlikely to be of any significant impact except, perhaps, a favourable one in computation time. If a sparse population of more intense reflections is also excluded, the effect may also be positive by reducing aberrations that might interfere with interpretation. If the objective is refinement, the benefit is less clear, except possibly in computation time. The cost for that may be the exclusion of a few more intense terms that provide positive guidance to refinement. Another positive effect of a sharply defined limiting resolution might be improved estimation of the spherical interference function where it is needed. It appears that, in practice, the consensus is that, in applying a cutoff, the exclusion of a few intense terms is at worst of negligible impact overall.

Because many imposed limits that exclude data are dependent on estimates of standard uncertainty, procedures for estimation of standard uncertainty need to be considered. As suggested earlier, a diffraction data set without a standard uncertainty for each intensity measurement is certainly the exception in current practice. The methods used by data-processing programs to estimate these standard uncertainties may be difficult, even impossible, to discern, but at core they must all be based on counting statistics. With the continuing trend toward data sets of high multiplicity (also referred to as redundancy), however, estimates of standard uncertainties are available from distributions of replicate measurements about means. Both of these estimates have value and express impacts of somewhat different sources of error. It would seem therefore that, where applicable, the best way to accommodate both would be to calculate weighted average values of individual intensities, where the weights are derived from the standard uncertainties from data processing, and then to estimate the standard uncertainties for the weighted averages by application of standard propagation-of-error procedures. This, finally, is the third area we highlight here that deserves focused discussion within the MX community with the objective of defining consensus best practices.

Little remains to be included in these introductory preliminaries. It should be understood that most of these indicators of quality may be cast in terms of either structure factors or intensities. We draw little attention to this as we define individual indicators, except that, where one of the two possibilities seems to dominate in usage, we tend to define that form alone. It is also true that many of these indicators have counterparts in which individual reflection terms are weighted. In current practice, these forms find little use in application to biological macromolecules and we ignore them here. Finally, many of these indicators may be expressed as fractions, that is, as numbers between one and zero. They are also often expressed in the percentage form, that is, as numbers between one hundred and zero. While we confine our use to the former here, we make no statement of preference.

The remainder of this chapter consists primarily of individual sections that reflect the various steps from the crystallographic data-collection experiment to the refinement of the final model. Relevant quality indicators and definitions are given for each of the steps and the most commonly used are collected in Table[link].

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