[R] nonlinear model: pseudo-design matrix

[Spencer Graves]

Hi, Murray:

	  When I have problems with nonconvergence of nls, I often
move the problem to "optim(..., hessian=TRUE)".  Even if the larger 
model is overparameterized and the hessian is singular, I optim usually 
returns an answer from which I can then compute 2*log(likelihood ratio). 
  Moreover, the hessian will help me diagnose the problem.  If it were 
my problem today, I'd try the following:

	  (1) If there are substantial differences in the diagonal elements of 
the hessian, it suggests the scaling should be adjusted.  Not too long 
ago, someone else suggested that this could be done within optim via the 
argument control = list(parscale=...).  I have yet to try that, but I 
think it should work fine.

	  (2) If the diagonal elements of the hessian do not differ by more 
than a couple orders of magnitude, then I'd try eigen(fit$hessian, 
symmetric=TRUE).  The relative magnitudes of the eigenvalues will expose 
the effective numer of paramaters that can be estimated, and the 
eigenvectors associated with the smallest eigenvalues can help one 
diagnose the problem.

	  hope this helps.
	  spencer graves

Murray Jorgensen wrote:

> Hi Spencer,
> 
> you were the only one to reply. Yes I am aware of the intrinsic / 
> parameter effects distinction and the advantages of LR tests and 
> profiling over Wald tests based on the local curvature of the 
> loglikelihood surface at the larger of two models being compared. My 
> situation is that I am comparing two nested models both of which have 
> uncomfortably many parameters for the amount of data available. I am 
> able to fit the smaller of the two models but not the larger. In this 
> situation neither the the Wald nor the LR test is available to me but 
> the score test (a.k.a. the Lagrange Multiplier test) is available to me 
> because it is based on the loglikelihood gradient at the smaller model.
> 
> I have been able to carry out the test by extracting
> 
> X <- smaller.nls$m$gradient()
> 
> and obtaining the extra columns of X for the parameters in larger but 
> not in smaller by numerical differentiation. It seems that there should 
> be some way of obtaining the extra columns without recourse to numerical 
> differentiation, though.
> 
> Cheers,  Murray Jorgensen
> 
> Spencer Graves wrote:
> 
>>       There doubtless is a way to extract the gradient information you 
>> desire, but have you considered profiling instead?  Are you familiar 
>> with the distinction between intrinsic and parameter effects 
>> curvature?  In brief, part of the nonlinearities involved in nonlinear 
>> least squares are intrinsic to the problem, and part are due to the 
>> how the problem is parameterized.  If you change the parameterization, 
>> you change the parameter effects curvature, but the intrinsic 
>> curvature remains unchanged.  Roughly 30 years ago, Doug Bates and Don 
>> Watts reanalized a few dozen published nonlinear regression fits, and 
>> found that in all but perhaps one or two, the parameter effects were 
>> dominant and the intrinsic curvature was negligible.  See Bates and 
>> Watts (1988) Nonlinear Regression Analysis and Its Applications 
>> (Wiley) or Seber and Wild (1989) Nonlinear Regression (Wiley).
>>
>>       Bottom line:
>>
>>       1.  You will always get more accurate answers from profiling 
>> than from the Wald "pseudodesign matrix" approach.  Moreover, often 
>> the differences are dramatic.
>>
>>       2.  I just did RSiteSearch("profiling with nls").  The first hit 
>> was 
>> "http://finzi.psych.upenn.edu/R/library/stats/html/profile.nls.html". 
>> If this is not satisfactory, please explain why.
>>
>>       hope this helps.
>>       spencer graves
>>
>> Murray Jorgensen wrote:
>>
>>> Given a nonlinear model formula and a set of values for all the
>>> parameters defining a point in parameter space, is there a neat way to
>>> extract the pseudodesign matrix of the model at the point? That is the
>>> matrix of partial derivatives of the fitted values w.r.t. the parameters
>>> evaluated at the point.
>>>
>>> (I have figured out how to extract the gradient information from an 
>>> nls fitted model using the nlsModel part, but I wish to implement a 
>>> score test, so I need to be able to extract the information at points 
>>> other than the mle.)
>>>
>>> Thanks, Murray Jorgensen
> 
>