Chemistry Reference
In-Depth Information
1.2 The Limited Use of R-Values
There are a variety of statistical descriptors and quality measures available in
crystallography. The probably most important single concept in use is the compari-
son or agreement between measured and predicted entities such as intensities or
moduli of structure factors. Several definitions are used differing in the entity they
refer to (intensities or structure factors) and in the weights assigned to individual
terms. These are called
R
-values, where the “
R
” abbreviates “residuals” - the part of
the data, that is left over, or which remains undescribed by the model. As the
residuals may be positive and negative, the absolute residuals or the residuals
squared are considered. These are summed up to the absolute residual sum, giving
a positive number or zero.
P
hkl
jj
F
obs
jj
F
calc
jj
R
1
¼
P
hkl
j
;
(1)
F
obs
j
s
P
hkl
w
2
jj
F
obs
jj
F
calc
jj
wR
1
¼
;
(2)
P
hkl
w
2
j
F
obs
j
s
P
hkl
ð
2
I
obs
I
calc
Þ
R
2
¼
;
(3)
P
hkl
ð
2
I
obs
Þ
s
P
hkl
w
2
ð
I
obs
I
calc
Þ
wR
2
¼
P
hkl
w
:
(4)
2
ð
I
obs
Þ
Also, the “Goodness of Fit,”
GoF
, and the
R
free
use similar residual sums. One
might think of these
R
-values in terms of a distance of the actual model from the
experimental data. Perfect agreement would result in each
R
-value being zero. But
perfect agreement is not expected, as the experimental data unavoidably contain
noise [
2
]. Furthermore, there may be a large number or even infinite many density
models with the same
R
-value (even if the
R
-value is zero!), as the phase informa-
tion is lost in these measures. And finally, the reconstruction process implies fitting
of a model to the data. For the fitting procedure, a choice must be made, which
entity (which
R
-value) is used for minimizing the distance between the experimen-
tal data and the model. Note that different choices lead to different “best models”.
1.2.1 Smaller R-Value: Better Model?
It has been emphasized above that the resulting model depends on the entities
considered (intensities or structure factors) for the comparison and, furthermore, in
which way measured and calculated entities are compared (unit- or other weights).
