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

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?
  

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.

    

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.

    

What should I be looking for in the output of the nls() routine that 
alerts me to the fact that the Hessian is potentially ill-conditioned?

Glenn

Peter Dalgaard wrote:

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.

    

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.

--
  O__  ---- Peter Dalgaard             ?ster Farimagsgade 5, Entr.B
 c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
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