Pedro Mendes wrote:
>> On 6 May 98 at 10:59, Petr Kuzmic wrote:
>> > and most people -still- use 'formal standard errors' of fitting
> > parameters, instead of full confidence intervals!
[snip]
> But on the topic of confidence intervals, as far as I can see these are
> only easy to obtain if one has used a method like Levenberg-Marquadt. This
> produces, as a "secondary" product, estimates of the co-variances. Many
> other methods of minimisation do not,
Greetings, Pedro and all:
Actually, even if one uses a minimization method without derivatives
(e.g., simplex), once having arrived at the minimum on the least-squares
surface, one can still compute the covariance matrix and compute formal
standard errors. A more serious problem is that these formal standard
errors, no matter which method we use to find the minimum, are grossly
inadequate for nonlinear models of enzyme kinetics.
For example, take a look at a manuscript currently in print, talking
about the kinetics of thrombin and dehydro-thrombin (a chemical mutant)
binding simultaneously and competitively to hirudin:
http://www.biokin.com/papers/thrombin/index.shtmlhttp://www.biokin.com/papers/thrombin/index.pdf
Here we find that the formal standard errors (computed from the
variance/covariance matrix) not even -qualitatively- express the fact
that only a -lower- limiting value could be estimated for both
association rate constants. At 95% confidence level, the upper limit
for these rate constants is infinity (the confidence interval is
open-ended on one side) despite the fact that the formal standard errors
do not indicate that.
[For those that can see GIF files in their email programs, I attach a
GIF file showing the open-ended shape of the confidence interval.]
The numerical results obtained by program DYNAFIT [P. Kuzmic (1996)
Anal. Biochem. 237, 260-273; http://www.biokin.com/dynafit] are as
follows:
PARAMETERS & STANDARD ERRORS
Parameter Initial Fitted Error %Error
k1 0.001 0.001 0.002 200
k2 5e-005 2.2e-005 3e-006 14
k3 0.001 0.00054 0.0011 210
k4 5e-005 3.19e-005 2.9e-006 9.2
CONFIDENCE
INTERVALS
95% Confidence interval for k1
lower upper
Standard error -0.00066 0.0027
Confidence interval 0.00024 infinity
95% Confidence interval for k2
lower upper
Standard error 1.9e-005 2.5e-005
Confidence interval 1.5e-005 3.9e-005
95% Confidence interval for k3
lower upper
Standard error -0.0004 0.0015
Confidence interval 8.3e-005 infinity
95% Confidence interval for k4
lower upper
Standard error 3e-005 3.4e-005
Confidence interval 2.6e-005 4e-005
In other words, the standard errors grossly underestimate the
uncertainty of the association rate constnats. From the standard error
of the rate constant k1, you would have thought that it cannot be larger
than 2.7 microM-1.sec-1, but in fact, at 95% confidence level, its value
could be infinity [ignoring for a moment certain physical limits such as
diffusion].
I think we should all use assymptotic confidence intervals most of the
time, and avoid using standard errors of fitting parameters as much as
possible.
--Petr
_____________________________________________________________
Petr Kuzmic Ph.D. * BioKin Ltd. * Madison, WI 53708-8336, USA
pkuzmic at biokin.com * http://www.biokin.com * 608.256.1269 fax
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