# RE: RE: Simulations with OMEGA BLOCK

From: Bob Leary <Bob.Leary>
Date: Tue, 8 Oct 2013 08:40:28 -0400

Paolo -
That is correct - if a symmetric matrix has a zero on the diagonal, it cann=
ot be positive definite. At best it is positive semidefinite (all eigenvalu=
es non-negative, but one or more are zero).
If there is a negative on the diagonal, then it cannot even be positive sem=
idefinite - there must be at least one negative eigenvalue.

Note that the idea of a 'real part' does not apply - all eigenvalues of a r=
eal symmetric matrix are real.

From: owner-nmusers
Behalf Of Paolo Denti
Sent: Tuesday, October 08, 2013 8:11 AM
To: Lindauer, A (Andreas); nmusers
Subject: Re: [NMusers] RE: Simulations with OMEGA BLOCK

Hi Andreas,
I believe your OMEGA matrix is positive SEMI-definite, because one of the e=
igeinvalues is 0.
I think the definition is that the real part of all eigenvalues must be STR=
ICTLY larger than 0.
I am no "matrix talker", but I think that if there is a zero on the diagona=
l, the matrix can't be positive definite (strictly).
I hope this helps, or that some gurus will shed some light.
Ciao,
Paolo

On 2013/10/08 12:40, Lindauer, A (Andreas) wrote:
Hi Douglas,

Please note that this is all about simulations - I'm not trying to estimate=
36 elements of random effects.
The BLOCK(7) suggestion works, just as did the first example that I provide=
d. My apologies for not being very clear in my previous e-mail, I'm not loo=
king for a work around for that particular example, but rather was seeking =
for ways to rewrite a 'cluttered' OMEGA statement in a more general, machin=
e-readable format as a triangular matrix.

Hi Paolo,

Thanks for your response. You said:

Also, the block matrix is either all zeros and FIXED, or it must be positiv=
e definite. If you fix one of the elements of the diagonal to 0, it goes ag=
ainst these rules.

Well, the matrix that I showed as an example is in fact positive definite. =
Or is it that NONMEM just checks if there is any zero diagonal element in B=
LOCK and returns an error without actually checking if the matrix is truly =
non-positive definite? This is how I would interpret the explanations that =
are given in the NONMEM help.

Thanks again, Andreas.

From: Eleveld, DJ [mailto:d.j.eleveld
Sent: Tuesday, October 08, 2013 11:33 AM
To: Lindauer, A (Andreas); nmusers
.com>
Subject: RE: Simulations with OMEGA BLOCK

Hi Andreas,

You cant fix part of a block in NONMEM, you have to fix the whole block. So=
the trick is to construct the covaiance matrix structure you want out of s=
maller blocks.
And when you fix an ETA on the diagonal to zero the corresponding covarianc=
es have to be zero as well. (i.e. the left-most variables in you BLOCK(8) m=
atrix)
So what I think you want for your full-matrix is something like:

\$OMEGA
0 FIX ; IIV_CL2
\$OMEGA BLOCK(7)
0.1 ; IIV_V2
0 0.1 ; IIV_F1
0 0 0.01 ; IIV_KA
0 0 0 0.01 ; IOV_KA
0 0 0 0 0.01
0 0 0 0 0 0.01
0 0 0 0 0 0 0.01

I hope you have LOTS of data since a BLOCK(7) marix has LOTS of paramaters =
to estimate.

You are also combining IIV and IOV variances in a single matrix.
Does it make sense to expect the IIV_KA and IOV_KA to be correlated?
I cant imagine how this is supposed to work, but admittedly I havent given =
it all that much thought.
It just looks fishy to me. I cant seem to understand what behavior you are =
trying to capture in this kind of covariance structure.

warm regards,

Douglas Eleveld

________________________________
Van: owner-nmusers
lto:owner-nmusers
Verzonden: October 8, 2013 10:22 AM
Aan: nmusers
Onderwerp: [NMusers] Simulations with OMEGA BLOCK
Hi NMUSERS,

I have a question regarding the use of OMEGA BLOCK statements in simulation=
s when one (or more) elements of the matrix are 0.

When I use the following lines to describe the OMEGA structure and run the =
simulation everything works well:

\$OMEGA
0 FIX ; IIV_CL2
0.1 ; IIV_V2
0.1 ; IIV_F1
0.01 ; IIV_KA
\$OMEGA BLOCK(1) 0.01 ; IOV_KA
\$OMEGA BLOCK(1) SAME
\$OMEGA BLOCK(1) SAME
\$OMEGA BLOCK(1) SAME

However, rewriting the above as a full matrix gives me an error message:

\$OMEGA BLOCK(8)
0 FIX
0 0.1
0 0 0.1
0 0 0 0.01
0 0 0 0 0.01
0 0 0 0 0 0.01
0 0 0 0 0 0 0.01
0 0 0 0 0 0 0 0.01

NM-TRAN MESSAGES

AN ERROR WAS FOUND IN THE CONTROL STATEMENTS.

AN ERROR WAS FOUND ON LINE 75 AT THE APPROXIMATE POSITION NOTED:
0 0 0 0 0 0 0 0.01

224 A VARIANCE IS ZERO, BUT THE BLOCK IS NOT FIXED TO ZERO.

I tried numerous different ways of placing the term FIX in the block, or no=
t mentioning it at all - nothing worked, except replacing the 0 diagonal el=
ement by a very small number. I know that there are certain constrains of u=
sing 0 values in an OMEGA BLOCK (band symmetric form), but I always thought=
this was only relevant for estimation.
Has anyone come across a similar issue when simulating?

Best regards, Andreas.

Andreas Lindauer, Ph.D.
Associate Principal Scientist, Clinical PKPD
Pharmacokinetics, Pharmacodynamics, and Drug Metabolism
Merck & Co. / MSD

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------------------------------------------------

Paolo Denti, PhD

Pharmacometrics Group

Division of Clinical Pharmacology

Department of Medicine

University of Cape Town

K45 Old Main Building

Groote Schuur Hospital

Observatory, Cape Town

7925 South Africa

phone: +27 21 404 7719

fax: +27 21 448 1989

email: paolo.denti

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Received on Tue Oct 08 2013 - 08:40:28 EDT

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