# AW: Parameter Uncertainty and Covariate effects

From: Stodtmann, Sven <sven.stodtmann>
Date: Tue, 12 Jan 2016 08:53:55 +0000

Dear Mats,

Taking into account the correlation/covariance of the estimators for THETA(=
1)/THETA(2) certainly solves the problem for additive covariate effects wit=
h 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 d=
o not have an additive structure, thus it is not clear a priori if the cova=
riance, which measures linear dependence, will take care of the problem pro=
perly.
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 c=
ases, 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, accountin=
g for correlation will solve the problem. Essentially this is what mu-model=
ing forces one to do anyway when using EM methods (In this case, MU_1 = T=
HETA(1) + SEX*THETA(2); CL=EXP(MU_1) ).
So if one is consequently using mu-modelling and accounting for the correla=
tion, 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 sven.stodtmann

abbvie.com

Please note that any views or opinions presented in this email are solely t=
hose of the author and do not necessarily represent those of the Company.
___________________________________________________________________________=
_____________________________________________

-----Ursprüngliche Nachricht-----
Von: Mats Karlsson [mailto:Mats.Karlsson
Gesendet: Tuesday, January 12, 2016 6:02 AM
An: Stodtmann, Sven
Cc: nmusers
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 probl=
em in this respect is that publications never give such values even if it o=
f 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 fin=
d the information.

Best regards,
Mats

Skickat från min iPhone

> 11 jan 2016 kl. 14:28 skrev Stodtmann, Sven <sven.stodtmann
>
> 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, asymptot=
ic) distribution of the estimator (i.e. normal with mean THETA and covarian=
ce according to the NONMEMs \$COR output for the THETAs).
> This approach may in some cases (particularly, when there are a lot of co=
assigning some quite high probability of unphysiological values if one did=
n't have good quality data, strong priors or a very careful parametrization=
of the model (e.g. transforming/bounding parameters, which requires/introd=
uces 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 par=
ameter uncertainty, the female population will have a more variable (uncert=
ain) PK, not just different one. If we phrase the problem differently, usin=
g
> CL = TVCL * SEX_EFF**(1-SEX) , (=
Eq. 2)
> The conclusion would be the other way around (i.e. male PK is more uncert=
ain).
>
> One approach to deal with the second problem could be this:
> In order to remove this (usually unjustified) assumption (the female popu=
lation having a less certain PK compared to the male), one could try to mod=
el 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 parameter=
s 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-populat=
ion gets how much uncertainty exactly instead of evenly splitting it.
>
> How do you deal with uncertainty in the estimates of covariate effects wh=
en 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 sven.stodtmann
>
> abbvie.com
> _________________________________________________________________________=
_______________________________________________
>
> ________________________________
>
> Persönlich haftende Gesellschafterin: AbbVie Komplementär GmbH
> Sitz der persönlich haftenden Gesellschafterin: Wiesbaden - Registerger=
> Geschäftsführer: Dr. Patrick Horber, Thomas Scheidmeir, Dr. Stefan Si=
mianer, William J. Chase
> Vorsitzende des Aufsichtsrats: Dr. Azita Saleki-Gerhardt
>
> This communication may contain information that is proprietary, confident=
ial, or exempt from disclosure. If you are not the intended recipient, plea=
se note that any other dissemination, distribution, use or copying of this =
communication is strictly prohibited. Anyone who receives this message in e=
rror 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 be=
absichtigte Empfänger sind, nehmen Sie bitte zur Kenntnis, dass jede Weit=
ergabe, Verteilung, Verwendung oder Vervielfältigung dieser. Kommunikatio=
n strikt untersagt ist. Jeder, der diese Nachricht fehlerhaft erhält, sol=
lte den Sender unverzüglich telefonisch oder durch Rücksendung der E-Ma=
il benachrichtigen und diese von seinem oder ihrem Computer löschen.
________________________________

Persönlich haftende Gesellschafterin: AbbVie Komplementär GmbH
Sitz der persönlich haftenden Gesellschafterin: Wiesbaden - Registergeric=
Geschäftsführer: Dr. Patrick Horber, Thomas Scheidmeir, Dr. Stefan Simi=
aner, William J. Chase
Vorsitzende des Aufsichtsrats: Dr. Azita Saleki-Gerhardt

This communication may contain information that is proprietary, confidentia=
l, or exempt from disclosure. If you are not the intended recipient, please=
note that any other dissemination, distribution, use or copying of this co=
mmunication is strictly prohibited. Anyone who receives this message in err=
or should notify the sender immediately by telephone or by return e-mail an=
d delete it from his or her computer.

Diese Kommunikation kann Informationen enthalten, die geheim, vertraulich o=
der hinsichtlich der Offenlegung beschränkt sind. Wenn Sie nicht der beab=
sichtigte Empfänger sind, nehmen Sie bitte zur Kenntnis, dass jede Weiter=
gabe, Verteilung, Verwendung oder Vervielfältigung dieser. Kommunikation =
strikt untersagt ist. Jeder, der diese Nachricht fehlerhaft erhält, sollt=
e den Sender unverzüglich telefonisch oder durch Rücksendung der E-Mail=
benachrichtigen und diese von seinem oder ihrem Computer löschen.
Received on Tue Jan 12 2016 - 03:53:55 EST

The NONMEM Users Network is maintained by ICON plc. Requests to subscribe to the network should be sent to: nmusers-request@iconplc.com.

Once subscribed, you may contribute to the discussion by emailing: nmusers@globomaxnm.com.