AW: Parameter Uncertainty and Covariate effects
Dear Mats,
Thanks a lot for your answer.
Taking into account the correlation/covariance of the estimators for
THETA(1)/THETA(2) certainly solves the problem for additive covariate effects
with normally distributed errors in both parameter and effect (e.g.
CL=TVCL+SEX_EFF**SEX).
However, if we assume a multiplicative effect (CL=TVCL*SEX_EFF**SEX) we do not
have an additive structure, thus it is not clear a priori if the covariance,
which measures linear dependence, will take care of the problem properly.
The expression can be transformed to make the relationship linear, but then we
lose the normal distribution (CL = EXP(LOG(TVCL)+SEX*LOG(SEX_EFF)).
In order to check whether accounting for correlation solves the dependency
problem, I ran a small MATLAB script. I think it is not possible to attach
pictures to mails to this list (correct me if I am wrong), so I will attach the
code I used. From the results (which of course are only toy examples, real-life
correlations may look different) it seems that at least in some cases,
accounting for the correlation is not enough (see code below).
But I still think your post gives the answer. If the model were designed in a
way to additively combine the THETAs for parameter and effect, accounting for
correlation will solve the problem. Essentially this is what mu-modeling forces
one to do anyway when using EM methods (In this case, MU_1 = THETA(1) +
SEX*THETA(2); CL=EXP(MU_1) ).
So if one is consequently using mu-modelling and accounting for the
correlation, one should be on the safe side.
Kind Regards,
Sven
Here is the MATLAB code:
Assumed 95%CIs for the parameters
CL = [18.04 ; 21.96]
SE = [0.504 ; 0.896]
Assumed (positive/negative) correlation: 0.5
testmat = [1 .05; .05 .01];
U_gt = chol(testmat)';
testmat2 = [1 -.05; -.05 .01];
U_lt = chol(testmat2)';
U_0 = diag(diag(U_lt));
CL0 = 20;
SE0 = 0.7;
normals = randn(50000,2)';
uncert_gt = U_gt*normals;
uncert_lt = U_lt*normals;
uncert_0 = U_0*normals;
maleStats_gt = CL0+uncert_gt(1,:);
femaleStats_gt = (CL0+uncert_gt(1,:)).*(SE0+uncert_gt(2,:));
maleStats_gt_X = (CL0*SE0+uncert_gt(1,:))./(SE0+uncert_gt(2,:)); %This is just
a posteriori change of reference, to see what would really happen, I think that
one would need to estimate both ways with a real dataset
maleStats_lt = CL0+uncert_lt(1,:);
femaleStats_lt = (CL0+uncert_lt(1,:)).*(SE0+uncert_lt(2,:));
maleStats_lt_X = (CL0*SE0+uncert_lt(1,:))./(SE0+uncert_lt(2,:));
maleStats_0 = CL0+uncert_0(1,:);
femaleStats_0 = (CL0+uncert_0(1,:)).*(SE0+uncert_0(2,:));
maleStats_0_X = (CL0*SE0+uncert_0(1,:))./(SE0+uncert_0(2,:));
subplot(4,1,1); ksdensity(maleStats_gt); title('correlation>0'); hold on;
ksdensity(femaleStats_gt);
ksdensity(maleStats_gt_X); hold off;
legend('male','female','male with female as reference')
subplot(4,1,2); ksdensity(maleStats_lt); title('correlation<0'); hold on;
ksdensity(femaleStats_lt);
ksdensity(maleStats_lt_X); hold off;
legend('male','female','male with female as reference')
subplot(4,1,3); ksdensity(maleStats_0); title('correlation=0'); hold on;
ksdensity(femaleStats_0);
ksdensity(maleStats_0_X); hold off;
legend('male','female','male with female as reference')
subplot(4,1,4); ksdensity(femaleStats_gt); title('Female only'); hold on;
ksdensity(femaleStats_lt);
ksdensity(femaleStats_0); hold off;
legend('>0','<0','=0')
________________________________________________________________________________________________________________________
SVEN STODTMANN, PHD
Pharmacometrician
AbbVie Deutschland GmbH & Co KG
Clinical Pharmacology and Pharmacometrics
Knollstrasse 50
67065 Ludwigshafen am Rhein, Germany
OFFICE +49 621-589-1940
EMAIL [email protected]
abbvie.com
Please note that any views or opinions presented in this email are solely those
of the author and do not necessarily represent those of the Company.
________________________________________________________________________________________________________________________
-----Ursprüngliche Nachricht-----
Von: Mats Karlsson [mailto:[email protected]]
Gesendet: Tuesday, January 12, 2016 6:02 AM
An: Stodtmann, Sven
Cc: [email protected]
Betreff: Re: [NMusers] Parameter Uncertainty and Covariate effects
Dear Sven
If you don't assume the covariance between THETA(1) and THETA(2) to be zero but
use the estimated covariance value, you do let the data speak. A problem in
this respect is that publications never give such values even if it of course
is possible. With online access to model code and output (as with the DDMoRe
repository (repository.ddmore.eu)) it will be more likely to find the
information.
Best regards,
Mats
Skickat från min iPhone
> 11 jan 2016 kl. 14:28 skrev Stodtmann, Sven <[email protected]>:
>
> Dear All,
>
> In order to account for uncertainty in estimated parameters when running a
> simulation, a natural approach would be running multiple simulations for
> different parameter vectors which are drawn from the (theoretical,
> asymptotic) distribution of the estimator (i.e. normal with mean THETA and
> covariance according to the NONMEMs $COR output for the THETAs).
> This approach may in some cases (particularly, when there are a lot of
> covariate effects estimated) lead to very broad parameter distributions, even
> assigning some quite high probability of unphysiological values if one didn't
> have good quality data, strong priors or a very careful parametrization of
> the model (e.g. transforming/bounding parameters, which requires/introduces
> prior knowledge as well).
>
> Another problem connected with parameter uncertainty on covariate effects is
> the following. Say we model
> TVCL = THETA(1)
> SEX_EFF = THETA(2)
> CL = TVCL * SEX_EFF**SEX, (Eq.
> 1)
> where male is coded as SEX=0, female as SEX=1.
> In this case, when using the above mentioned technique to account for
> parameter uncertainty, the female population will have a more variable
> (uncertain) PK, not just different one. If we phrase the problem differently,
> using
> CL = TVCL * SEX_EFF**(1-SEX) , (Eq. 2)
> The conclusion would be the other way around (i.e. male PK is more uncertain).
>
> One approach to deal with the second problem could be this:
> In order to remove this (usually unjustified) assumption (the female
> population having a less certain PK compared to the male), one could try to
> model the same covariate effect as follows:
> TVCL = THETA(1)
> SQRT_SEX_EFF = THETA(2)
> CL = TVCL * SQRT_SEX_EFF**SEX / SQRT_SEX_EFF**(1-SEX)
> In this case TVCL would already include "half" of the effect (on the log
> scale; the "new" TVCL would be TVCL*SQRT(SEX_EFF) in terms of the parameters
> used in Eq.1).
> With this approach, both sub-populations, male and female get "some part" of
> the uncertainty effect.
> Of course it would be even nicer to let the data decide which sub-population
> gets how much uncertainty exactly instead of evenly splitting it.
>
> How do you deal with uncertainty in the estimates of covariate effects when
> it comes to simulations/predictions?
>
> Kind Regards,
> ________________________________________________________________________________________________________________________
> SVEN STODTMANN, PHD
> Pharmacometrician
>
> AbbVie Deutschland GmbH & Co KG
> Clinical Pharmacology and Pharmacometrics
> Knollstrasse 50
> 67065 Ludwigshafen am Rhein, Germany
> OFFICE +49 621-589-1940
> EMAIL [email protected]
>
> abbvie.com
> ________________________________________________________________________________________________________________________
>
> ________________________________
>
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Sitz der Gesellschaft: Wiesbaden - Registergericht: AG Wiesbaden HRA 9790
Persönlich haftende Gesellschafterin: AbbVie Komplementär GmbH
Sitz der persönlich haftenden Gesellschafterin: Wiesbaden - Registergericht: AG
Wiesbaden HRB 26371
Geschäftsführer: Dr. Patrick Horber, Thomas Scheidmeir, Dr. Stefan Simianer,
William J. Chase
Vorsitzende des Aufsichtsrats: Dr. Azita Saleki-Gerhardt
This communication may contain information that is proprietary, confidential,
or exempt from disclosure. If you are not the intended recipient, please note
that any other dissemination, distribution, use or copying of this
communication is strictly prohibited. Anyone who receives this message in error
should notify the sender immediately by telephone or by return e-mail and
delete it from his or her computer.
Diese Kommunikation kann Informationen enthalten, die geheim, vertraulich oder
hinsichtlich der Offenlegung beschränkt sind. Wenn Sie nicht der beabsichtigte
Empfänger sind, nehmen Sie bitte zur Kenntnis, dass jede Weitergabe,
Verteilung, Verwendung oder Vervielfältigung dieser. Kommunikation strikt
untersagt ist. Jeder, der diese Nachricht fehlerhaft erhält, sollte den Sender
unverzüglich telefonisch oder durch Rücksendung der E-Mail benachrichtigen und
diese von seinem oder ihrem Computer löschen.