SV: Standard errors of estimates for strictly positive parameters
Dear all, an alternative which I try to use on strictly positive parameters is
to estimate on log-scale. Then I think often the assymptitic approximation is
more true and the resulting measures of parameter uncertainty are more reliable.
BW
Magnus
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Från: [email protected] <[email protected]> för Chaouch
Aziz <[email protected]>
Skickat: den 12 februari 2015 12:01:51
Till: Eleveld, DJ; [email protected]
Kopia: [email protected]
Ämne: [NMusers] RE: Standard errors of estimates for strictly positive
parameters
Dear Douglas, dear Pascal,
Thanks a lot for your answers. I guess the main point here is constrained vs
unconstrained optimization as the asymptotic covariance matrix of estimates (as
returned by $COV) is "well defined" only in the latter case. When fitting model
1, one would normally constrain THETA(1) to be positive by using something like:
$THETA
(0, 15, 50) ; TVCL
In this situation I wonder whether it makes sense at all to consider the output
of $COV. It seems model 2 would be here preferable (unconstrained
optimization). If model 1 is fitted without boundary constraints on THETA(1),
the covariance matrix of estimates may have "some" meaning but the optimization
in NONMEM is then likely to crash if it encounters a negative value at some
point, which again speaks somehow in favor of model 2 (unless one is not
interested in the output of $COV).
Now what about $OMEGA? Here NONMEM knows that these are variances and therefore
we do not need to explicitly (i.e. manually) place boundary constraints on the
diagonal elements of the omega matrix. However something must account for it
internally. The covariance matrix of estimates returned by $COV also contains
elements that refer to omega so I'm unsure how these are treated. For diagonal
elements of the omega matrix, does NONMEM optimize log(omega) or omega? Or does
it uses a Cholesky decomposition of the Omega matrix and optimize elements on
that scale? Again, unless the optimization on omega is unconstrained, can we
really trust the output of $COV? Basically the question here is how would you
construct an asymptotic 95% confidence interval for a diagonal element of Omega
(i.e. a variance) based on the information from the covariance matrix of
estimates?
The covariance matrix of estimate is of importance to me because I'm
considering published studies and I do not have access to the data so I cannot
refit the model with an alternative parametrization. Results from $COV (in lst
file when available from the authors) is then the only available piece of
information about the uncertainty of the estimation process.
Kind regards,
Aziz Chaouch
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Quoted reply history
De : Eleveld, DJ [mailto:[email protected]]
Envoyé : mercredi, 11. février 2015 22:26
À : Chaouch Aziz; [email protected]
Objet : RE: Standard errors of estimates for strictly positive parameters
Hi Aziz,
Just some comments off the top of my head in a quite informal way: I'm not
really sure that these are the same problem because they dont start with the
same information in the form of parameter constraints. In model 1 you are
asking the optimizer for the unconstrained maximum likelihood solution for
TVCL. OK, this is reasonable in a lot of situations, but not necessairily in
all situations.
In model 2 you add information by forcing TVCL and CL to be positive. If you
think of the optimal solution as some point in N-dimensional space which has to
be searched for, in model 2 you are saying "dont even look in the space where
TVCL or CL is negative". Even stronger, in model 2 you are also saying "dont
even get close to zero" because the log-normal distribution vanishes towards
zero.
Which solution of these is best for some particular application depends on a
lot of things. One of the things I would think about in this situation is
whether or not my a priori beliefs match with the structual constraints of the
model. Do I really think that the "true" CL could be zero? If yes, then model 2
is hard to defend in that case.
You description of your situation regarding standard errors is a part of the
same thing. When you extrapolate standard errors into low-probability areas you
are checking the boundaries of the probability area. It should not be suprising
that model 1 might tell you that CL is negative since this was part of the
solution space which you allowed. With model 2 your model structure says "dont
even look there"
In short, although these two models might look similar, I think they are really
quite different. This becomes most clear when you consider the low-probability
space.
Sorry for the vauge language.
Warm regards,
Douglas
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De : [email protected] [mailto:[email protected]]
Envoyé : mercredi, 11. février 2015 18:30
À : Chaouch Aziz; [email protected]
Objet : RE: Standard errors of estimates for strictly positive parameters
Dear Aziz,
NM does not return the asymptotic SE of THETA(1) in model 1 on the log-scale.
So I would use model 2.
With best regards / Mit freundlichen Grüßen / Cordialement
Pascal
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From: [email protected] [[email protected]] on behalf of
Chaouch Aziz [[email protected]]
Sent: Wednesday, February 11, 2015 5:21 PM
To: [email protected]
Subject: [NMusers] Standard errors of estimates for strictly positive parameters
Hi,
I'm interested in generating samples from the asymptotic sampling distribution
of population parameter estimates from a published PKPOP model fitted with
NONMEM. By definition, parameter estimates are asymptotically (multivariate)
normally distributed (unconstrained optimization) with mean M and covariance C,
where M is the vector of parameter estimates and C is the covariance matrix of
estimates (returned by $COV and available in the lst file).
Consider the 2 models below:
Model 1:
TVCL = THETA(1)
CL = TVCL*EXP(ETA(1))
Model 2:
TVCL = EXP(THETA(1))
CL = TVCL*EXP(ETA(1))
It is clear that model 1 and model 2 will provide exactly the same fit.
However, although in both cases the standard error of estimates (SE) will refer
to THETA(1), the asymptotic sampling distribution of TVCL will be normal in
model 1 while it will be lognormal in model 2. Therefore if one is interested
in generating random samples from the asymptotic distribution of TVCL, some of
these samples might be negative in model 1 while they'll remain nicely positive
in model 2. The same would happen with bounds of (asymptotic) confidence
intervals: in model 1 the lower bound of a 95% confidence interval for TVCL
might be negative (unrealistic) while it would remain positive in model 2.
This has obviously no impact for point estimates or even confidence intervals
constructed via non-parametric bootstrap since boundary constraints can be
placed on parameters in NONMEM. But what if one is interested in the asymptotic
covariance matrix of estimates returned by $COV? The asymptotic sampling
distribution of parameter estimates is (multivariate) normal only if the
optimization is unconstrained! Doesn't this then speak in favour of model 2
over model 1? Or does NONMEM take care of it and returns the asymptotic SE of
THETA(1) in model 1 on the log-scale (when boundary constraints are placed on
the parameter)?
Thanks,
Aziz Chaouch
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