# RE: distribution assumption of Eta in NONMEM

From: Jurgen Bulitta <jbulitta>
Date: Fri, 28 May 2010 17:50:23 -0400

Dear Ethan,

There may be two aspects to your question, one is on the
assumptions of the algorithm and software implementation
and one on the use of the models as described by Nick.

To my knowledge, the EM algorithm (e.g. MC-PEM) assumes that the
etas are multivariate normally distributed. As described in Bob's paper [1]
and the underlying theoretical algorithm development work from
Alan Schumitzky [2] and others, the EM algorithm obtains the
maximum likelihood estimates for the population means and the
variance-covariance matrix by calculating the average of the conditional
means and the conditional var-cov matrices of the individual subjects
(see equations 21 and 22 in [1]). These equations assume that the
parameter population density h(theta | mu, Omega) is multivariate
normal. The residual error does not need to follow a normal distribution
(see page E64 in Bob's paper [1]).

Most of the applications of a model are based on simulations
which usually explicitly assume a multivariate normal distribution
(or some transformation thereof). Therefore, it seems fair to say
that for parametric population PK models, most of the inferences
are based on the assumption of a multivariate normal distribution
of the "etas" at one or more stages. We rarely have enough subjects
to assess the appropriateness of this assumption.

You would have to go to a full nonparametric algorithm such as
NPML, NPAG or Bob Leary's new method in Phoenix to not assume
a normal distribution of the "etas".

Best wishes
Juergen

[1] Bauer RJ, Guzy S, Ng C. AAPS J. 2007;9:E60-83.
[2] Schumitzky A . EM algorithms and two stage methods in
pharmacokinetics population analysis. In: D'Argenio DZ , ed. Advanced
Methods of Pharmacokinetic and Pharmacodynamic Systems Analysis.
vol. 2. Boston, MA : Kluwer Academic Publishers ; 1995 :145- 160.

From: owner-nmusers
Behalf Of Nick Holford
Sent: Friday, May 28, 2010 3:51 PM
To: nmusers
Subject: Re: [NMusers] distribution assumption of Eta in NONMEM

For estimation NONMEM estimates one parameter to describe the distribution =
of random effects -- this is the variance (OMEGA) of the distribution. Thus=
it makes no explicit assumption that the distribution is normal. AFAIK any=
distribution has a variance.

For simulation NONMEM assumes all etas are normally distributed. If you use=
OMEGA BLOCK(*) then the distribution is multivariate with covariances but =
still normal.

Nick

Ethan Wu wrote:
I could not find in the NONMEM help guide that explicitly mentioned a norma=
l distribution is assumed, only it was clearly mentioned of assumption of m=
ean of zero.

________________________________
From: Serge Guzy <GUZY
To: Ethan Wu <ethan.wu75
lobomaxnm.com<mailto:nmusers
Sent: Fri, May 28, 2010 1:25:24 PM
Subject: RE: [NMusers] distribution assumption of Eta in NONMEM

As far as I know, this is the assumption in most of the population programs=
like NONMEM, SADAPT, PDX-MC-PEM and SAEM. Therefore when you simulate, ran=
dom values from a normal distribution are generated. However, you have the =
flexibility to use any transformation to create distributions for your mode=
l parameters that will depart from pure normality. For example, CL=theta(=
1)*exp(eta(1)) will generate a log-normal distribution for the clearance a=
lthough the random deviates are all from the normal distribution.
I am not sure how you can simulate data sets if you are using the non param=
etric option that is indeed available in NONMEM.
Serge Guzy; Ph.D
President, CEO, POP_PHARM
www.poppharm.com<http://www.poppharm.com/>

From: owner-nmusers
ilto:owner-nmusers
Sent: Friday, May 28, 2010 9:08 AM
To: nmusers
Subject: [NMusers] distribution assumption of Eta in NONMEM

Dear users,
Is it true NONMEM dose not assume Eta a normal distribution?
If it does not, I wonder what distribution it assumes? I guess this is cr=
itical when we do simulations.
Thanks

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--

Nick Holford, Professor Clinical Pharmacology

Dept Pharmacology & Clinical Pharmacology

University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand

tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53

email: n.holford

http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford

Received on Fri May 28 2010 - 17:50:23 EDT

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