# Standard errors of estimates for strictly positive parameters

From: Chaouch Aziz <Aziz.Chaouch>
Date: Wed, 11 Feb 2015 16:21:56 +0000

Hi,

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:

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. 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)?

Thanks,

Aziz Chaouch

Received on Wed Feb 11 2015 - 11:21:56 EST

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