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

[Peter Dalgaard]

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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>   


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