Biology Reference
In-Depth Information
4. Goodness-of-fit; and
5. Uniqueness of the parameters.
Having discussed the first objective in considerable detail with regard to
the weighted least-squares criterion, we move to another optimization
method—maximum likelihood—and the conditions under which the
two are equivalent. We also outline objectives 2 through 5 with regard to
their goals and features essential to the data-fitting analyses.
A. Conditions for Maximum Likelihood
The parameter values determined through least-squares minimization of
the WSSR are estimates, based on the data, for the true parameter values.
It is not unusual, however, for the parameter estimates to be derived
from a different criterion that maximizes the likelihood of the parameter
values. That is, the values sought by this criterion are those that have the
highest probability of being correct based on the data. If the data are
known to satisfy the following set of relatively broad conditions,
the least-squares values for the parameters are also those that guarantee
maximum likelihood:
1. The X
i
values do not contain any measurement errors;
2. The Y
i
values contain measurement errors that follow a bell-shaped,
or Gaussian, distribution with a mean of zero;
3. The fitting function, G, is correct; and
4. The measurement error for each data point is independent of other
measurement errors.
Determining whether the measurement errors satisfy those conditions is
not a trivial task. Figure 8-3, for example, presents a typical situation
illustrating how the least-squares methods can be applied to
experimental data. Let's assume we would like to perform a least-
squares fit of the data to the ligand-binding fitting equations derived in
Chapter 7. The following general problem then becomes apparent:
Actual experimental data are rarely formulated in exactly the correct
form for the algorithms and fitting function to be applied. For example,
in Figure 8-3, the dependent variable (i.e., the y-axis) is not a fractional
saturation, but is in arbitrary units determined by the experimental
protocol (in this case, a drug response). Consequently, either the data or
the fitting equation must be transformed to match the other. The decision
of what to transform and how to transform it should be determined by the
nature of the experimental uncertainties in the data. The idea here is either
to not alter the noise distribution within the data or, in the case where the
existing experimental error distribution is not Gaussian, to perform
the transform so as to make the noise distribution more Gaussian.
