In reality the phase circles like those drawn in Fig. 
will not intersect exactly at one
point due to the presence of experimental error in the diffraction data. The errors which affect the phasing
procedure are the following:
Blow and Crick (1959) [16] dealt with only two errors in their mathematical treatment
of error in the IR method. Points (2) and (3) in the above list were considered numerically as one single error.
Two further assumptions were made by Blow and Crick. Firstly they assumed that all errors were Gaussian and
secondly that all the errors in the measurement of the structure factors and in the calculation of the heavy atom
structure factor are contained in the derivative structure factor. This is
a reasonable assumption since for most proteins the native structure factors and derivative structure factors
are approximately co-linear since the heavy atom structure factor is generally small in comparison.
Fig. shows the important parameters for the representation of experimental error in the
phasing procedure. The vector 
represents the native or reference structure and 
stands for
the derivative structure factor. The calculated heavy atom structure factor is labelled 
.
Figure: Phase triangles showing the native structure factor, 
, the derivative structure factor, 
,
the calculated heavy atom structure factor, 
and the lack of closure, 
for two differing
values of 
, the angle between and . Note that the phase of the reference structure
factor 
may at any time be obtained from by 
.
The two errors which are to be considered are the error in the observed measurements, 
and the
combined error in and the lack of isomorphism, 
acting along . Since the errors
are Gaussian we can define the total error in as
The component of the error in acting perpendicular to is neglected at this stage since any
uncertainty in the position of the point 
in Fig. in this direction has very little
effect on the phase . Assuming a Gaussian error distribution we can write the probability distribution of
the phase as
where 
is a normalisation factor applied such that
The relationship between the lack of closure and the phase is given by the cosine
rule applied to the phase triangle in Fig. . We obtain the expression
Using the above equations (Eqs. to ) it is possible to calculate a probability
distribution for each derivative with respect to the reference structure factor and these may be combined
to give a final phase probability curve using
where 
is the normalisation factor for the combined distribution.
Given a set of phase probability curves for each reflection there still remains a choice of which phase to
use in calculating the Fourier transform giving the electron density map. The most probable Fourier would use
coefficients 
where 
is the phase where the distribution 
is at
its maximum. However it is often the case that the phase probability curves are bimodal. Blow and Crick defined
the best Fourier as that Fourier transform which gave the lowest mean square error in the electron density.
This corresponds to using phases corresponding to the centroid of the phase probability distribution curve.
The best Fourier uses coefficients
Introducing 
, the figure of merit for a reflection and 
the centroid phase we write
In practice the phase probability distribution is calculated at regular intervals of say 
or 
so that the components of 
are given by
and
The figure of merit is in fact equal to the mean of the cosine of error in the phase angle. This can be seen if
we define the error as 
and shift the origin such that 
.
m is then given by [18]
When anomalous scattering data are present Blow and Rossmann [17] showed that the probability distribution
for a reflection may be written as the product of the two individual distributions for the Bijvoet pair
reflections 
and 
with corresponding lack of closure values 
and 
such that
where 
now corresponds to the total error in the determination of and . North [88]
later observed that this method of combining the isomorphous and anomalous differences did not account for the
greater inherent accuracy of the anomalous differences since they are measured from the same crystal sample
and are therefore not affected by non-isomorphism or errors in the scaling together of data from different crystals.
Matthews [84] showed that by writing the total lack of closure 
in a different
form the isomorphous and anomalous contributions to the phasing potential could be separated out and thus assigned
separate probability distributions with appropriate variances taking into account the differing accuracy of the
isomorphous and anomalous data. The total lack of closure became
defining 
and 
the isomorphous and anomalous lack of closures.
Fig. shows the situation when Bijvoet pairs have been measured. and 
are defined
as the real and imaginary parts of the calculated heavy atom structure factor and 
is the angle between
and the vector 
. From Fig. we have
and
Figure: Lack of closure errors when anomalous data is present.
If we define 
the we can write
and
Using the sine rule we have
and substituting for 
in Eq. and rearranging gives
We can now write down the joint probability distribution for in terms of the isomorphous and anomalous
lack of closures as
This basic theory is easily applicable to the MAD method since each set of data arising from a measurement at
a different X-ray energy will produce its own isomorphous and anomalous probability distribution. However the
reference data set will give a zero isomorphous contribution to the phasing and will correspondingly have zero
isomorphous lack of closure and a probability distribution which is unity for all .
The overall phase probability distribution is
where the product over 
is taken over all X-ray energies.
Calculation of the probability distributions 
and 
requires an estimation
of the errors 
and 
which include statistical error, errors in the
calculation of the heavy atom position, occupancy and temperature factors and lack of
isomorphism. In theory the error in the anomalous differences should have a zero contribution
from lack of isomorphism. The statistical error in the structure factors is usually
estimated while making the measurements, e.g. from counting statistics. However the second
source of error from the heavy atom is more difficult to estimate. For the isomorphous case
Blow and Crick [16] showed that may be determined for centric reflections and then applied
to the non-centric reflections.
For centric reflections , and are co-linear allowing an estimation of the
overall r.m.s. error 
to be obtained from
For the centrosymmetric case
and thus 
may be estimated.
For anomalous differences the contribution to may be determined by
inspecting centric reflections where the observed Bijvoet difference 
should in theory
be zero. The total r.m.s. deviation from the theory gives an estimation of 
which in turn
allows to be obtained.
