RE: Reporting Modeling Results
Hi Leonid,
You are right that BW normalization with a fixed slope (exponent)
shouldn't influence the estimation accuracy of the theta of clearance
directly. But the covariate AGE of course is correlated with body
weight, and a body weight of 10 corresponds roughly to a point somewhat
above the AGE of 1 - close the center of the AGE effect. The whole age
effect is scaled to the TVCL.
The effect of an excentric (1 kg is below even a newborn) basal
clearance will be a different shape of the AGE relationship because of
the non-linear relationship between AGE and BW - the deviation from a
linear relation will be larger for younger children/babies. Perhaps the
exponential form allows this shape change without large differences
between parameters (a bit more shallow, a bit less steep) but has the
effect of inducing parameter correlation and thus uncertainty.
John, would perhaps be so kind to inspect the correlation between the
bootstrapped parameters? That should provide some insight on the cause
of the inflated confidence intervals.
Best regards,
Jeroen
Quoted reply history
-----Original Message-----
From: Leonid Gibiansky [mailto:[EMAIL PROTECTED]
Sent: Friday, 26 October, 2007 5:27
To: Elassaiss - Schaap, J. (Jeroen)
Cc: John Mondick; [email protected]
Subject: Re: [NMusers] Reporting Modeling Results
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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