From: H. Olin Spivey <ospivey@bmb-fs1.biochem.okstate.edu>
  To  : Borries Demeler , rasmb@alpha.bbri.org
  Date: Wed, 26 Apr 2000 13:02:49 -0500

Re: NONLIN question

   I don't use Nonlin, so I don't know what the "multi-channel" option is,
but the question gives me the opportunity to plead for an improvement in
most least-squares programs.  Specifically, I am amazed that no
least-squares programs that I have ever seen, other than ours, actually
prints the parameter correlation coefficients (Borries's
"cross-correlations").  Since most programs use the Marquardt algorithm for
minimizations, these parameter correlation coefficients are easily computed
from the error matrix that is an essential component of the minimizer.
Yes, a few programs allow one to see the variance-covariance matrix from
which the parameter correlation coefficients can be calculated.  But who
has the time to waste calculating these when the computer should calculate
and present them (the lower half of the correlation matrix is all that is
needed)?

   To tell me that parameters are "highly correlated" doesn't tell me much.
(It is also an unsupported and hence unrealiable hunch or excuse for errors
of any type).  How much is "high" and which parameters are the most highly
correlated?  The correlation coefficients confirm or exclude suspicions and
identify the parameters that are seriously correlated.  It can also be very
helpful to discover the parameters that are not highly correlated.  Knowing
precisely which parameters are highly correlated is valuable because this
information often allows one to reduce the correlations by redesign of the
experiment or to use simple computational strategies that either reduce the
correlations or reduce their impact on finding an acceptable fit.  (To
learn these stragies is easier than the novice realizes and gives one an
enormously better intuitive grasp of the model.) True, some programs tell
you that the parameters are too correlated to achieve a fit.  This is
deficient info. since: 1) you still don't know (in most cases) which
parameters are causing the most problem and 2) you can often get a fit even
with badly correlated parameters.  This is reflected into the parameter
errors, but not in a way that allows one to easily or reliably identify the
most highly correlated parameters.  The latter information would often
allow you to design an experiment to reduce the overall uncertainties in
the parameters.

   Both Allen Minton and I use parameters (potentially adjustable) for the
fraction of molecules competent to associate in our programs.  Yes, this
parameter will often be intolerably correlated with other parmeters.  My
programs will identify this situation clearly..  At other times, by
appropriate experimental design and/or the simplicity of the system, this
parameter will have very acceptably low correlations with all other
parameters.  Furthermore, it is a simple matter to simulate any
experimental condition including the easily quantifiable standard
deviations in each data point.  A fit to these data than confirm whether
the correlation coefficients are acceptably low or not for each
experimental data set(s).  In this regard, I never write a least-squares
program without writing the companinon simulation program.  They use the
same model equations so why not invest a little extra time to gain this
ability?

   It has been claimed that most least-squares programs don't include the
covariances in the error matrix.  My computer expert disagrees.  It is true
that the standard deviations and covariances are only approximations for
parameters that appear in a nonlinear manner in the model equations.  We
also calculate these parameter errors by more rigorous methods during final
stages of analysis.  In my experience (about 30 years), the parameter
errors from this simpler "linear approximation" are most often quite
accurate for the model equations I have dealt with.

   In summary, please provide the parameter correlation coefficients in
your programs and printout.  They can be extremely helpful and are trivial
to provide.  Finally, I should add that parameter correlations as high as
0.98 are often acceptable and correlations of 0.990 are sometimes
acceptable. Higher correlations are rarely tolerable.  These correlations
can often be reduced by redesign of experimental conditions, but you are
shooting in the dark until you know which parameters are the biggest
offenders.

   Olin
________________________________________________________________________



At 11:09 AM -0500 4/26/00, Borries Demeler wrote:
>Dear RASMB'ers,
>
>I have a question that keeps popping up - I never really thought about it
>much, but maybe someone has already:
>
>Sometimes, when fitting a monomer-dimer fit for example, one can obtain a
>substantially better variance by turning on the multi-channel option for
>LnK2 in Nonlin. I suppose one possible interpretation for this would be
>the presence of a portion of the sample being in incompetent monomer state
>the rest can reversibly associate like a normal M/D system.
>
>What other possible interpretations exist for this symptom? And what
>models have people used to fit that?
>
>I suppose for the case of an incompetent monomer an additional exponential
>term with the monomer MW and an amplitude proportional to the fraction of the
>incpt. monomer would work, where the fraction of the total concentration
>would be an adjustable (floating) parameter. Seems like something quite
>sensitive to cross-correlation.
>
>Anyway, if you have dealt with such a system previously, I'd be interested
>to hear from you.
>
>Thanks in advance for any comments and suggestions, -Borries


H. Olin Spivey                       Phone: (405) 744-6192
Dept. Biochem. & Molec. Biology      Fax:   (405) 744-7799
246 NRC                              Email: OSpivey@Biochem.Okstate.Edu
Oklahoma State University
Stillwater, OK 74078-3035