Hi Jeroen,
I cannot see any extra covariance introduced by scaling by a fixed
numbers. Moreover, I even not sure that the term "centered" is
applicable to this model. It came from the linear model where the model
Y=ax+b
was called centered if presented in the form
Y=a1*(x-meanX)+b1
(here and below a and b are parameters, THETAs, while x is random)
Indeed, it is more stable in the second, centered form. If you assume
that a1 and b1 are normally distributed without correlation, then
a = a1 and
b = b1-a1*meanX
are correlated, and correlation is higher for large meanX.
In this regard, the model that consist of several equations of the type
Y=b*x^0.75
is equivalent to the model with
Y=b'*(x/10)^0.75,
no extra correlations added.
However, the model
Y=b*x^a
is more like the linear model because it can be presented in the log
scale as
log(Y)=a*log(x) + log(b)
and then it is better be centered:
log(Y)= a1*(log(x)-mean(log(x))) + log(b1)
or equivalently
Y=b1*(x/x0)^a1
where x0 is geometric mean of x.
Thus, these two (examples only!) models need to be centered:
CL=THETA()+(WT-70)*THETA()
or
CL=THETA()*(WT/70)^THETA()
but this one can be used in any shape and form
CL=THETA()*(WT/10)^0.75 = (THETA()/10^0.75) WT^0.75 =~ THETA() WT^0.75
This is just a scheme, not exact derivation/proof, but the bottom line
is that scaling by a constant should not influence the standard errors,
there should be another reason (may be, just insufficient sample size of
the bootstrap process? or bounds on the parameters?) for the
discrepancy. May be we need to stratify it into more than two age
groups, to guarantee sufficient coverage of the entire age range of
interest in each and every bootstrap set.
Leonid
Elassaiss - Schaap, J. (Jeroen) wrote:
Hi Leonid,
The aspect of the results that John describes as being dependent on
scaling are the (bootstrap) confidence intervals, not the parameter
estimates. I actually agree with John's original statements about not
being surprised by the inflated standard errors when the model is not
centered. Removal of normalization induces covariance in the estimation
of affected parameters. This covariance naturally leeds to inflated
standard errors (w/o bootstrapping). And as we all know, high
correlation between parameters can result in matrix singularity and
consequently failing covariance steps, consistent with John's
statements.
Best regards,
Jeroen
J. Elassaiss-Schaap
Scientist PK/PD
Organon NV
PO Box 20, 5340 BH Oss, Netherlands
Phone: + 31 412 66 9320
Fax: + 31 412 66 2506
e-mail: [EMAIL PROTECTED]
-----Original Message-----
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
On Behalf Of Leonid Gibiansky
Sent: Thursday, 25 October, 2007 23:30
To: John Mondick
Cc: [email protected]
Subject: Re: [NMusers] Reporting Modeling Results
Hi John,
The code seems to be good, and the results, in my opinion, should not
depend on scaling. One idea: if you started both sets of problems (with
and without scaling) from the same initial conditions (while the
solutions differ by factor 5 or so), nonmem could have difficulties
finding the correct minimum when started far from optimum. If your
initial conditions were coming from 10.4 kg-normalized solution, then it
could explain wider CI for the non-normalized problem: some of those
runs did not converged or converged to a local minima. If this is true,
you may want to repeat the non-normalized set with the initial
conditions closer to the solution (if you choose to use non-normalized
problem as the final model).
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
John Mondick wrote:
$PK
TVCL = THETA(1)*(WT/10.4)**0.75
BETA = THETA(5)
TCL = THETA(6)
FCL = 1-BETA*EXP(-(AGE-1)*0.693/TCL)
TVCL2 = TVCL*FCL
CL = TVCL2*EXP(ETA(1))
TVV1 = THETA(2)*(WT/10.4)
V1 = TVV1*EXP(ETA(2))
TVV2 = THETA(3)*(WT/10.4)
V2 = TVV2*EXP(ETA(3))
TVQ = THETA(4)*(WT/10.4)**0.75
Q = TVQ
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