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

[Prof Brian Ripley]

On Thu, 27 Mar 2008, 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.

That seems to be about ill-conditioning in the *correlation* matrix.  As I 
pointed out before, the condiition number of the *covariance* matrix 
(which was the original question) is scale-dependent -- uncorrelated 
parameter estimators can have an arbitrarily high condition number of 
their covariance matrix (it is the ratio of the largest to the smallest 
variance).


>> 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.
>>>
>>>
>>
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>>
>
>
> --
>   O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
>  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
> (*) \(*) -- University of Copenhagen   Denmark      Ph:  (+45) 35327918
> ~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)              FAX: (+45) 35327907
>
> ______________________________________________
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>

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
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