1-b*exp(-c*t) is negative only if b*exp(-c*t) > 1, which implies
log(b) > c*t, I think. Is this a reasonable circumstance in terms of
the theory that led to the Richards growth curve?
You say this occurs frequently in your data; since b and c are
presumably constants for a given data set, "frequently" must refer to the
varying values of t, which you described as age squared (and
therefore always positive); these questionable values then must occur
for small values of t. It might be interesting to inquire what the
imaginary part of y-hat does as t increases to the point where
log(b) = c*t.
The phenomenon ought to be invariant with respect to the units of y
and t. Is it? If you take t in days^2, do you get equivalent results
to those you get when t is in hours^2 or months^2? And does it matter
if y is in pounds or kilograms?
Also presumably (I'm not familiar with this area) the parameters a, b,
c, d are assumed to be positive numbers; do they turn out to be positive
when estimated by this procedure?
It all sounds rather as though the fitted value of b (> 0) is too
large for the fitted value of c (also > 0). Does that make sense?
-- DFB.
On Thu, 15 Jun 2000, Michael Henderson wrote:
> Hello all,
> I am trying to fit some data using the NLINFIT of MATLAB and using
> SAS. I am trying to fit the well know RICHARDS growth curve.
> It looks like y=a*(1-b*exp(-c*t))^d where we want to estimate the
> parameters a,b,c,and d. Here t is my input and is age squared while y
> is the weight of some animals. I choose my initial parameter starting
> values and they do converge and I get a wonderful looking fit with
> awesome residual plot. My question is this though. The estimate
> MATLAB finds for d is .44 which of course causes my predicted y's to be
> complex numbers when 1-b*exp(-c*t) is negative (very frequent in my
> case). What questions should this bring up. Is it ok to simply use the
> real parts of the numbers. That is what SAS did and when plotting the
> predicted curve to the original data I must say it looks just fine. Let
> me know your thoughts on the use of only the real parts of the complex
> values. Any advice and input will be much appreciated. Thanks ,
> Mike
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