Paolo - That is correct - if a symmetric matrix has a zero on the diagonal, it cannot be positive definite. At best it is positive semidefinite (all eigenvalues non-negative, but one or more are zero). If there is a negative on the diagonal, then it cannot even be positive semidefinite - there must be at least one negative eigenvalue.
Note that the idea of a 'real part' does not apply - all eigenvalues of a real symmetric matrix are real. From: [email protected] [mailto:[email protected]] On Behalf Of Paolo Denti Sent: Tuesday, October 08, 2013 8:11 AM To: Lindauer, A (Andreas); [email protected] Subject: Re: [NMusers] RE: Simulations with OMEGA BLOCK Hi Andreas, I believe your OMEGA matrix is positive SEMI-definite, because one of the eigeinvalues is 0. I think the definition is that the real part of all eigenvalues must be STRICTLY larger than 0. I am no "matrix talker", but I think that if there is a zero on the diagonal, 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 provided. My apologies for not being very clear in my previous e-mail, I'm not looking for a work around for that particular example, but rather was seeking for ways to rewrite a 'cluttered' OMEGA statement in a more general, machine-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 positive definite. If you fix one of the elements of the diagonal to 0, it goes against 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 BLOCK 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:[email protected]] Sent: Tuesday, October 08, 2013 11:33 AM To: Lindauer, A (Andreas); [email protected]<mailto:[email protected]> 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 smaller blocks. And when you fix an ETA on the diagonal to zero the corresponding covariances have to be zero as well. (i.e. the left-most variables in you BLOCK(8) matrix) 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: [email protected]<mailto:[email protected]> [mailto:[email protected]] Namens Lindauer, A (Andreas) Verzonden: October 8, 2013 10:22 AM Aan: [email protected]<mailto:[email protected]> Onderwerp: [NMusers] Simulations with OMEGA BLOCK Hi NMUSERS, I have a question regarding the use of OMEGA BLOCK statements in simulations 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 not mentioning it at all - nothing worked, except replacing the 0 diagonal element by a very small number. I know that there are certain constrains of using 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 Notice: This e-mail message, together with any attachments, contains information of Merck & Co., Inc. (One Merck Drive, Whitehouse Station, New Jersey, USA 08889), and/or its affiliates Direct contact information for affiliates is available at http://www.merck.com/contact/contacts.html) that may be confidential, proprietary copyrighted and/or legally privileged. It is intended solely for the use of the individual or entity named on this message. If you are not the intended recipient, and have received this message in error, please notify us immediately by reply e-mail and then delete it from your system. ________________________________ De inhoud van dit bericht is vertrouwelijk en alleen bestemd voor de geadresseerde(n). Anderen dan de geadresseerde(n) mogen geen gebruik maken van dit bericht, het niet openbaar maken of op enige wijze verspreiden of vermenigvuldigen. Het UMCG kan niet aansprakelijk gesteld worden voor een incomplete aankomst of vertraging van dit verzonden bericht. The contents of this message are confidential and only intended for the eyes of the addressee(s). Others than the addressee(s) are not allowed to use this message, to make it public or to distribute or multiply this message in any way. The UMCG cannot be held responsible for incomplete reception or delay of this transferred message. Notice: This e-mail message, together with any attachments, contains information of Merck & Co., Inc. (One Merck Drive, Whitehouse Station, New Jersey, USA 08889), and/or its affiliates Direct contact information for affiliates is available at http://www.merck.com/contact/contacts.html) that may be confidential, proprietary copyrighted and/or legally privileged. It is intended solely for the use of the individual or entity named on this message. If you are not the intended recipient, and have received this message in error, please notify us immediately by reply e-mail and then delete it from your system. -- ------------------------------------------------ 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: [email protected]<mailto:[email protected]> ------------------------------------------------ ________________________________ UNIVERSITY OF CAPE TOWN This e-mail is subject to the UCT ICT policies and e-mail disclaimer published on our website at http://www.uct.ac.za/about/policies/emaildisclaimer/ or obtainable from +27 21 650 9111. This e-mail is intended only for the person(s) to whom it is addressed. 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