[Re: Significance of confidence intervals in the Non-Linear Least Squares Program.]

glenn andrews wrote:

Thanks for the response. I was not very clear in my original request.

What I am asking is if in a non-linear estimation problem using 
nls(), as the condition number of the Hessian matrix becomes larger, 
will the t-values of one or more of the parameters being estimated in 
general become smaller in absolute value -- that is, are low t-values 
a  sign of an ill-conditioned Hessian?
  

In a word: no. Ill-conditioning essentially means that there are one 
or more directions in parameter space along which estimation is 
unstable. Along such directions you get a large SE, but also a large 
variability of the estimate, resulting in t values at least in the 
usual "-2  to +2" range. The large variation may swamp a true effect 
along said direction, though.

Typical nls() ouput:

Formula: y ~ (a + b * log(c * x1^d + (1 - c) * x2^d))

Parameters:
 Estimate Std. Error t value Pr(>|t|) a  0.11918    0.07835   1.521   
0.1403 b -0.34412    0.27683  -1.243   0.2249 c  0.33757    0.13480   
2.504   0.0189 *
d -2.94165    2.25287  -1.306   0.2031
Glenn

Prof Brian Ripley wrote:

 

On Wed, 26 Mar 2008, glenn andrews wrote:

   

I am using the non-linear least squares routine in "R" -- nls.  I 
have a
dataset where the nls routine outputs tight confidence intervals on 
the
2 parameters I am solving for.
      

nls() does not ouptut confidence intervals, so what precisely did 
you do?
I would recommend using confint().

BTW, as in most things in R, nls() is 'a' non-linear least squares 
routine: there are others in other packages.

   

As a check on my results, I used the Python SciPy leastsq module on 
the
same data set and it yields the same answer as "R" for the
coefficients.  However, what was somewhat surprising was the the
condition number of the covariance matrix reported by the SciPy 
leastsq
program = 379.

Is it possible to have what appear to be tight confidence intervals 
that
are reported by nls, while in reality they mean nothing because of the
ill-conditioned covariance matrix?
      

The covariance matrix is not relevant to profile-based confidence 
intervals, and its condition number is scale-dependent whereas the 
estimation process is very much less so.

This is really off-topic here (it is about misunderstandings about 
least-squares estimation), so please take it up with your 
statistical advisor.