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

From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>
Date: Thu, 27 Mar 2008 23:47:58 +0100

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.
>>
>>
>
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>

-- 
   O__  ---- Peter Dalgaard             ุster 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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Received on Thu 27 Mar 2008 - 23:02:19 GMT

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