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RE: Standard errors of estimates for strictly positive parameters

From: Eleveld, DJ <d.j.eleveld>
Date: Wed, 11 Feb 2015 21:26:08 +0000

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 th=
e 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 yo=
u 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 al=
so saying “dont even get close to zero” because the log-normal distribu=
tion 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 th=
e same thing. When you extrapolate standard errors into low-probability are=
as you are checking the boundaries of the probability area. It should not b=
e 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 struc=
ture says “dont even look there”

In short, although these two models might look similar, I think they are re=
ally quite different. This becomes most clear when you consider the low-pro=
bability space.

Sorry for the vauge language.

Warm regards,


From: owner-nmusers
 of Chaouch Aziz [Aziz.Chaouch
Sent: Wednesday, February 11, 2015 5:21 PM
To: nmusers
Subject: [NMusers] Standard errors of estimates for strictly positive param=


I'm interested in generating samples from the asymptotic sampling distribut=
ion of population parameter estimates from a published PKPOP model fitted w=
ith NONMEM. By definition, parameter estimates are asymptotically (multivar=
iate) normally distributed (unconstrained optimization) with mean M and cov=
ariance C, where M is the vector of parameter estimates and C is the covari=
ance matrix of estimates (returned by $COV and available in the lst file).

Consider the 2 models below:

Model 1:



Model 2:



It is clear that model 1 and model 2 will provide exactly the same fit. How=
ever, although in both cases the standard error of estimates (SE) will refe=
r 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 inter=
ested in generating random samples from the asymptotic distribution of TVCL=
, some of these samples might be negative in model 1 while they'll remain n=
icely positive in model 2. The same would happen with bounds of (asymptotic=
) confidence intervals: in model 1 the lower bound of a 95% confidence inte=
rval for TVCL might be negative (unrealistic) while it would remain positiv=
e in model 2.

This has obviously no impact for point estimates or even confidence interva=
ls constructed via non-parametric bootstrap since boundary constraints can =
be placed on parameters in NONMEM. But what if one is interested in the asy=
mptotic covariance matrix of estimates returned by $COV? The asymptotic sam=
pling distribution of parameter estimates is (multivariate) normal only if =
the optimization is unconstrained! Doesn't this then speak in favour of mod=
el 2 over model 1? Or does NONMEM take care of it and returns the asymptoti=
c SE of THETA(1) in model 1 on the log-scale (when boundary constraints are=
 placed on the parameter)?


Aziz Chaouch

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Received on Wed Feb 11 2015 - 16:26:08 EST

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