Biology Reference
In-Depth Information
the magnitude of these parameter correlations because they are
associated with the difficulties encountered by any data-fitting
procedure.
The cross-correlation coefficient for the i-th and k-th parameter can be
evaluated from the elements of the inverse of the P
T
P matrix that was
already evaluated by the Gauss-Newton least-squares parameter
estimation procedure, namely,
Þ
1
ik
ð
P
T
P
Cross Correlation
ik
¼
q
ð
i
6¼
k
:
(8-50)
Þ
1
ii
Þ
1
kk
P
T
P
ð
P
T
P
These cross-correlation coefficients have a range of
1, with zero being
optimal. As the cross-correlation approaches
1, the fitting
procedure becomes increasingly more difficult and the results more
questionable, because the P
T
P matrix is becoming nearly singular and
cannot easily be inverted for use in Eq. (8-45). For practical purposes, if
the magnitudes of the cross-correlation coefficients are less than
þ
1or
0.97,
the least-squares procedure can usually function adequately. However,
0.97 should not be considered an absolute threshold with everything
acceptable below
0.97 and everything unacceptable outside this range.
All fitting procedures get progressively worse as the magnitude of the
cross-correlations increase toward 1.
C. Precision of the Model Parameters
Finding estimates of the precision of the estimated parameters is also of
paramount importance because this allows investigators to test the
significance of their results. For example, consider an experiment and
subsequent analysis that determines the molecular weight of
hemoglobin to be 67,000 daltons. In reality, this information tells us
nothing new about hemoglobin, because virtually all proteins have a
molecular weight of 67,000
50,000 daltons. If all we know is that the
molecular weight of hemoglobin is approximately 67,000 daltons, then
all we can say about hemoglobin is that it appears to be a typical protein.
However, if we know the molecular weight of hemoglobin is 67,000
1,000 daltons, we have a lot more useful information. For example,
because we also know that hemoglobin contains one iron atom per
16,700
500 daltons, we can easily conclude that the hemoglobin
molecule contains four irons and thus four oxygen-binding sites.
Conversely, if our estimate of the molecular weight of hemoglobin is
67,000
50,000 daltons, then we would have to conclude that the
hemoglobin molecule contains 4
3 irons and thus 4
3 oxygen-
binding sites.
The most common but least accurate approach is to use the asymptotic
standard errors, which assume the fitting equation is linear and are
calculated as follows:
