Nonlinear regression and goodness-of-fit (a mixed real and theoretical problem)
I'm going to show you a real problem that I have at my research laboratory
and that is a theoretical problem too (statistics). A very interesting problem.
It is well known that when we fit experimental data to a linear model
we can use the R2 (R squared) value in order to compare the goodness-of-fit
of this fitted particular model to the data in front of other linear models.
However, when we fit experimental data to a NONLINEAR model the
R2 is not well defined. In fact, the underlying hypothesis that
SST=SSE+SSR (total variance=explained variance+residual variance)
is not true in this case.
Some authors had proposed to use an R2-like parameter
for nonlinear models: plot experimental values in front
of the predicted values (using a fitted model) and getting
the linear regression (R2) of this plot.
I must highlight that this R2 (from now on R2*) is not the conventional R2
that we usually now.
My experimental data is a phyiscal parameter (i.e.: flow)
in front of time (400 values of time).
I'm trying to fit this data to some nonlinear models (no mather what models).
The fact is that when I apply this R2* criteria, I got
very good R2* in all cases (>90%).
However, if I see the predicted curves (using the
fitted models) I see very clear that some models
does not fit good (despite its R2 is very high).
It is to say, very high R2 but very bad
I have read some technical books, for example:
"Nonlinear regression" (G.A.F. Seber by Wiley Series).
But the authors of those books offer a way of comparing two models
(one model in front of another model each time) by means of contrast
The fact is that I have a lot of models and
a lot of experiments and these kind of
test is very tedious.
I must highlight too that the methods of those books
suppose that the residual data is normaly distributed,
but my residual data does not follow a particular distribution.
Does anyone know a parameter that can
express the goodness-of-fit of a nonlinear model
to a experimental data without the need of comparing models (among them)
by contrasting of hypothesis and wihout the assumption that the residuals are normally distributed ?
I will be very pleased if anyone can help me.
Thanks in advance.