```From:"Austin, Daren J"
Subject:[NMusers] Confidence intervals for Emax model
Date:Thu, February 28, 2002 4:13 am

I have fitted an Emax model to some population data with both NONMEM and
PROC NLMIX in SAS, and was pleased to find that they agree exactly (use
METHOD=1 for NONMEM or method=firo for SAS). I have only one eta parameter
on Emax, which is well estimated and an additive error structure. I would
now to construct 95% confidence intervals. I pressume that because both the
variability on Emax and the residual enter linearly that I can just add
together so that

U95% = theta1*EXP(1.96*SQRT(eta1))*conc/(theta2+conc)+1.96*SQRT(eps1)
L95% = theta1*EXP(-1.96*SQRT(eta1))*conc/(theta2+conc)-1.96*SQRT(eps1)

This assumes that the intra and inter-subject variabilities are uncorrelated
which seems reasonable in this instance (no comedications for example). Of
course had I put variability on the EC50 the approximation would definitel;y
not be valid.

I realise that I could just simulate away and pull out the 2.75thand 97.5th
centiles, but is there a better way?

For those that are interested the SAS code is as follows (I have
deliberately written the code as NONMEM-esque), data is for heart rate,
hence a residual of sqrt(eps) ~ 7bpm

proc nlmixed data=test method=firo; /* method=0 */
parms theta1=20 theta2=60 eta1=0.1 eps=50; /* initial conditions */
Emax=theta1*exp(e1);
EC50=theta2;
pred=Emax*CONC/(EC50+CONC);
model F ~ normal(pred,eps); /* error structure */
random e1 ~ normal(0,eta1) subject=subject;
predict pred out=pred;
run;

At present only additive error is possible in SAS.

Kind regards,

Daren

Dr. Daren J. Austin
Pharmacometrician
GlaxoSmithKline Research & Development
Greenford Road, Greenford, Middlesex UB6 0HE
dja17709@gsk.com
Tel: 7-711 2073 or +44 (0) 20 8966 2073
Fax: 7-711 2123 or +44 (0) 20 8966 2603

*******

From:"Hu, Chuanpu"
Subject:RE: [NMusers] Confidence intervals for Emax model
Date:Fri, March 1, 2002 9:04 am

Dear Daren,

Just my 2 cents. To compute something that is not analytically available, I
suppose nothing more exact can be done if you can "simulate away." If one
variability of eta1 and eps1 dominates the other, your U95% and L95% should
do. If not, you might want to use 2.24 instead of 1.96 to be conservative.

Best regards,
Chuanpu

----------------------------------------------------------------
Chuanpu Hu, Ph.D.
Modeling and Simulation, U.S.
GlaxoSmithKline
Tel: 919-483-8205
Fax: 919-483-6380
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