From: nplock@zedat.fu-berlin.de
Subject: [NMusers] restraining eta code
Date: Tue, 28 Mar 2006 11:46:57 +0200 (CEST)

Dear all,

I have a question regarding a model which restrains all individual
parameter estimates to take values between 0 and 1. Im using the
following code:

PHI=LOG(THETA(7)/(1-THETA(7)))
P=EXP(PHI+ETA(4))/(1+EXP(PHI+ETA(4)))

My question is: How do I calculate the coefficient of variation for
THETA(7), if for example NONMEM gives a value of 0.7 for THETA(7) and and
a value of 6.4 for the respective diagonal element of the omega matrix? It
was called to my attention that in this case you cannot just take the
square root of diagonal element?

I would really appreciate your help!
Nele
_______________________________________________________

From: Leonid Gibiansky leonidg@metrumrg.com
Subject: Re: [NMusers] restraining eta code
Date: Tue, 28 Mar 2006 08:23:21 -0500

Nele,
The best way to see what is going on is simulations (e.g., using R or S+). 
If you run this code:

THETA7 = 0.7
ETA4 = rnorm(100000,mean=0,sd=sqrt(6.4))
PHI=log(THETA7/(1-THETA7))
P=exp(PHI+ETA4)/(1+exp(PHI+ETA4))

hist(P)
p95CI = quantile(P,probs=c(0.025,0.975))


you will see that the distribution for P is not normal (at all!), and 95% confidence
intervals are essentially from 0 to 1:

      2.5%      97.5%
0.01641659 0.99714220

Leonid 
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From: nplock@zedat.fu-berlin.de
Subject: Re: [NMusers] restraining eta code
Date: Tue, 28 Mar 2006 15:24:00 +0200 (CEST)

Dear Leonid,

thanks a lot! What about the coefficient of variation? If I calculate it
back from the obtained confidence intervals (THETA7-(97.5%quantile)/2) I
would end up with values between 21 and 28%. So could one say that the
coefficient of variation is approx. 25%, for example?

Nele
_______________________________________________________

From: "Mats Karlsson" mats.karlsson@farmbio.uu.se
Subject: RE: [NMusers] restraining eta code
Date: Tue, 28 Mar 2006 15:25:05 +0200

Hi Nele,

The CV is approximately OMEGA(ETA(4))*THETA(7)*(1-THETA(7)). Where OMEGA is
SD (not variance).

Best regards,
Mats

--
Mats Karlsson, PhD
Professor of Pharmacometrics
Div. of Pharmacokinetics and Drug Therapy
Dept. of Pharmaceutical Biosciences
Faculty of Pharmacy
Uppsala University
Box 591
SE-751 24 Uppsala
Sweden
phone +46 18 471 4105
fax   +46 18 471 4003
mats.karlsson@farmbio.uu.se 
_______________________________________________________

From: Leonid Gibiansky leonidg@metrumrg.com
Subject: Re: [NMusers] restraining eta code
Date: Tue, 28 Mar 2006 08:37:35 -0500

Nele,
I do not think it is appropriate to describe the P distribution in terms of sd, cv etc.
:just look on the histogram, that is U-shaped with the peaks at near 0 and 1. Individual
P can be anywhere between 0 and 1 (and they are more likely be near 0 or 1 than in the
intermediate points). If you need to report variability, use confidence intervals and
mentions that probability cannot be described in terms of the normal-distribution terminology

Leonid

_______________________________________________________

From: "Mats Karlsson" mats.karlsson@farmbio.uu.se
Subject: RE: [NMusers] restraining eta code
Date: Tue, 28 Mar 2006 16:09:10 +0200

Nele,

I didn't look at the size of your SD. When it is that large I totally agree
with Leonid, that CV is not a useful descriptor of variability. Note also
that whenever the typical value is close to 0 or 1, the distribution will be
very skewed.
Finally, a correction, it is the SD(P) that is approximately
OMEGA(ETA(4))*THETA(7)*(1-THETA(7)). 

Best regards,
Mats
--
Mats Karlsson, PhD
Professor of Pharmacometrics
Div. of Pharmacokinetics and Drug Therapy
Dept. of Pharmaceutical Biosciences
Faculty of Pharmacy
Uppsala University
Box 591
SE-751 24 Uppsala
Sweden
phone +46 18 471 4105
fax   +46 18 471 4003
mats.karlsson@farmbio.uu.se
_______________________________________________________