From: UMJ <xuyu@UMDNJ.EDU>
  To  : rasmb@alpha.bbri.org
  Date: Thu, 27 Apr 2000 10:38:46 -0400

Re: NONLIN question

In responds to Olin's comments on data analysis:
1) NONLIN provide information about correlation matrix and covariance
matrix. Although you can not get this info from the current Windows'
version
NONLIN directly, it is a simple matter to include this info as a standard
and/or optional output for anyone interested in the statistics of a fit.
The
usefulness of this info is debatable since they are meaningful only when
the
fitting model is correct. Depending on the model(s) used for fitting the
estimations of some parameters are intrinsically correlated and the
correlation coefficients provide no information about the correctness of
the
fitting model.
2) Your 30 years experiences are right: the parameter errors from simpler
"linear approximation" are accurate for the model equations of
sedimentation
equilibrium. This can be demonstrated by evoking the so called
Crème-Rao
lower bound, an approach based on the large number theorem, (the second
part
of my these with David Yphantis was devoted on this subject, which I also
hope will be published soon.) Essentially, it says that for certain 'well
behaved' nonlinear functions (the exponential equation of equilibrium
sedimentation included) the error statistics of least square fitting can
be
approximated by a linear least square regression with probability one
when
the data number is large and when the noise is Gaussian.
Regards
Yujia

----- Original Message -----
From: H. Olin Spivey <ospivey@bmb-fs1.biochem.okstate.edu>
To: Borries Demeler <demeler@bioc09.v19.uthscsa.edu>;
<rasmb@alpha.bbri.org>
Sent: Wednesday, April 26, 2000 2:02 PM
Subject: 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
>