Re: RE: Standard errors of estimates for strictly positive parameters
I think we are back to the discussion of usefulness of SE/RSE provided by Nonmem (search in archives will recover many e-mails on the subject). I am reluctant to disagree with Lewis Sheiner (if this indeed was his citation that was not taken out of context) but SEs are very useful (not perfect but useful) as an indicator of identifiability of the problem and some measure of precision. One cannot do bootstrap on each and every step of the modeling (moreover, bootstrap CI are not that different from asymptotic CI, especially for well-estimated parameters). Profiling is also local, and unlikely to give something significantly different from the Nonmem SE results unless RSEs are very large (and profiling is as time-consuming as bootstrap). So SEs serve as a very useful indicator whether data support the model.
Note that Nonmem SEs are local; they rely on approximation of the OF surface, essentially, give the curvature of this surface, so they do not take into account any constrains. In this sense, form of the model (CL=THETA(1) or CL=log(THETA(2)) just define the rule of extrapolation beyond the locality of the estimate. For practical purposes, SEs of these two representations are related (approximately) as
SE(THETA2)=SE(THETA(1))/THETA(1)
(propagation of error rule). Given SE(THETA(1)) and THETA(1) you can estimate SE(THETA(2)) and simulate accordingly.
Thanks
Leonid
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Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
Quoted reply history
On 2/12/2015 9:04 AM, Bob Leary wrote:
> Dear Aziz -
> The approximate likelihood methods in NONMEM such as FO, FOCE,and LAPLACE
> optimize an objective function than is parameterized internally
> by the Cholesky factor L of Omega, regardless of whether the matrix is diagonal
> (the EM -based methods do something considerably different and work directly
> with Omega rather than
> the Cholesky factor.)
>
> Thus for the approximate likelihood methods, the SE's computed internally by
> $COV from the Hessian or Sandwich or Fisher score methods
> are first computed with respect to these Cholesky parameters , and then the
> corresponding SE's of the full Omega=LL' are computed by a 'propagation of
> errors' approach
> which skews the results, particularly if the SE's are large. Thus in a sense
> regarding your dilemma about whether Model 1 or Model 2 is better with respect
> to applicability of $COV results, one answer is that both are fundamentally
> distorted by the propagation of errors method with respect to the Omega
> elements.
>
> But regarding your fundamental question 'can we trust the output of $COV '-
> all of this makes very little difference. Standard errors computed by $COV are
> inherently dubious - the applicability of the usual asymptotic arguments is
> very questionable for the types/sizes of data sets we often deal with.
> As Lewis Sheiner used to say of these results, 'they are not worth the
> electrons used to compute them'. They are the best we can do for the level
> of computational investment put into them -
> If you want something better, try a bootstrap or profiling method.
>
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