Leverage values are obtained from the projection matrix P, which transforms the
differences between observed and calculated observations, ΔY, to the differences
caused by the current set of parameter shifts, AΔX:
P ΔY ¼ A ΔX
ð11Þ
Substituting the expression for ΔX from the normal equations (Eq. 7) yields
A A
T WA
Â
à À1 A
T W ΔY ¼ A ΔX
ð12Þ
By inspection P, the projection matrix is A [A
T WA]
21 A
T W [44, 45]. The
P matrix has a row and column for every observation, but the leverages are the
leading diagonal terms P ii , so the computation and storage can be carried out
efficiently. Its calculation is based on the A and W matrices only – the observations
or residual differences are not used – underlining the fact that leverage is a feature of
the model, not of the values of the observations.
Leverage values of individual X-ray data points tend to have few useful applications: Prince extended his analysis above to compute the improvement to individual
parameter standard uncertainties caused by remeasuring individual X-ray observations. Picking the most useful reflections to remeasure is not immediately compatible
with the way modern diffractometers collect data, since they sweep through large
regions of reciprocal space, collecting multiple parts of the reciprocal lattice at the
same time.
However, leverage finds an application in assessing the appropriate use of
restraints in a structure refinement. In Sect. 3, restraints were defined as additional
observational equations with a user-supplied standard uncertainty, which is used to
compute a weight for the observation in the least squares. As a result of this
mathematical equivalence of X-ray data and restraints, the leverage analysis above
can equally be applied to a restraint to test whether it has (a) no influence,
(b) reasonable influence or (c) far too much influence, on the parameters which it
is restraining.
The result of two refinements of the same structure is shown in Fig. 5, illustrating
how the leverages of restraints change as other data are removed. In this case
30 rigid-bond restraints are applied to 1,2 and 1,3 distances (with standard uncertainties of 0.002 Å
2 and 0.005 Å
2 , respectively). On the left is a ‘normal’ refinement –
the restraints have a very low average leverage, and so we can conclude that the
X-ray data determines the values of the restrained parameters quite satisfactorily. On
the right, only low-angle X-ray data is included, and the restraints now almost
completely determine the values of the displacement parameters with an average
leverage of around 0.8.
Leverage is therefore a useful metric of the influence of each restraint and allows
an analyst to check the source of information determining a parameter before
drawing any conclusions about its value.
Recent Developments in the Refinement and Analysis of Crystal Structures
61
differences between observed and calculated observations, ΔY, to the differences
caused by the current set of parameter shifts, AΔX:
P ΔY ¼ A ΔX
ð11Þ
Substituting the expression for ΔX from the normal equations (Eq. 7) yields
A A
T WA
Â
à À1 A
T W ΔY ¼ A ΔX
ð12Þ
By inspection P, the projection matrix is A [A
T WA]
21 A
T W [44, 45]. The
P matrix has a row and column for every observation, but the leverages are the
leading diagonal terms P ii , so the computation and storage can be carried out
efficiently. Its calculation is based on the A and W matrices only – the observations
or residual differences are not used – underlining the fact that leverage is a feature of
the model, not of the values of the observations.
Leverage values of individual X-ray data points tend to have few useful applications: Prince extended his analysis above to compute the improvement to individual
parameter standard uncertainties caused by remeasuring individual X-ray observations. Picking the most useful reflections to remeasure is not immediately compatible
with the way modern diffractometers collect data, since they sweep through large
regions of reciprocal space, collecting multiple parts of the reciprocal lattice at the
same time.
However, leverage finds an application in assessing the appropriate use of
restraints in a structure refinement. In Sect. 3, restraints were defined as additional
observational equations with a user-supplied standard uncertainty, which is used to
compute a weight for the observation in the least squares. As a result of this
mathematical equivalence of X-ray data and restraints, the leverage analysis above
can equally be applied to a restraint to test whether it has (a) no influence,
(b) reasonable influence or (c) far too much influence, on the parameters which it
is restraining.
The result of two refinements of the same structure is shown in Fig. 5, illustrating
how the leverages of restraints change as other data are removed. In this case
30 rigid-bond restraints are applied to 1,2 and 1,3 distances (with standard uncertainties of 0.002 Å
2 and 0.005 Å
2 , respectively). On the left is a ‘normal’ refinement –
the restraints have a very low average leverage, and so we can conclude that the
X-ray data determines the values of the restrained parameters quite satisfactorily. On
the right, only low-angle X-ray data is included, and the restraints now almost
completely determine the values of the displacement parameters with an average
leverage of around 0.8.
Leverage is therefore a useful metric of the influence of each restraint and allows
an analyst to check the source of information determining a parameter before
drawing any conclusions about its value.
Recent Developments in the Refinement and Analysis of Crystal Structures
61