Hi Folks,
Please allow me a few biased reflections/opinions on the numeRology of the
R-value (not R-factor, because it is neither a factor itself nor does it
factor in anything but ill-posed reviewer's critique. Historically the term
originated from small molecule crystallography, but it is only a
'Residual-value')
a) The R-value itself - based on the linear residuals and of apparent
intuitive meaning - is statistically peculiar to say the least. I could not
find it in any common statistics text. So doing proper statistics with R
becomes difficult.
b) rules of thumb (as much as they conveniently obviate the need for
detailed explanations, satisfy student's desire for quick answers, and
allow superficial review of manuscripts) become less valuable if they have a
case-dependent large variance, topped with an unknown parent distribution.
Combined with an odd statistic, that has great potential for misguidance and
unnecessarily lost sleep.
c) Ian has (once again) explained that for example the Rf-R depends on the
exact knowledge of the restraints and their individual weighting, which we
generally do not have. Caution is advised.
d) The answer which model is better - which is actually what you want to
know - becomes a question of model selection or hypothesis testing, which,
given the obscurity of R cannot be derived with some nice plug-in method. As
Ian said the models to be compared must also be based on the same and
identical data.
e) One measure available that is statistically at least defensible is the
log-likelihood. So what you can do is form a log-likelihood ratio (or Bayes
factor (there is the darn factor again, it’s a ratio)) and see where this
falls - and the answers are pretty soft and, probably because of that,
correspondingly realistic. This also makes - based on statistics alone -
deciding between different overall parameterizations difficult.
http://en.wikipedia.org/wiki/Bayes_factor
f) so having said that, what really remains is that the model that fits the
primary evidence (minimally biased electron density) best and is at the same
time physically meaningful, is the best model, i. e., all plausibly
accountable electron density (and not more) is modeled. You can convince
yourself of this by taking the most interesting part of the model out (say a
ligand or a binding pocket) and look at the R-values or do a model selection
test - the result will be indecisive. Poof goes the global rule of thumb.
g) in other words: global measures in general are entirely inadequate to
judge local model quality (noted many times over already by Jones, Kleywegt,
others, in the dark ages of crystallography when poorly restrained
crystallographers used to passionately whack each other over the head with
unfree R-values).
Best, BR
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Bernhard Rupp, Hofkristallrat a.D.
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