Thank you all for your suggestions. The more flexible OMEGA element entries will not be easy to implement in NONMEM's structure, but I will look into it. A couple of things need to be noted.
1) Regarding Matt's suggestions, to enter initial Omega elements in Cholesky format in NM 7.2, $OMEGA CHOLESKY may be used. 2) The traditional methods (FO/FOCE/Laplace) actually vary the omega elements in the Cholesky format for estimation, than transform back in the variance format after each iteration. This assures that as NONMEM arbitrarily searches throughout the Cholesky space (that is, as each cholesky element is arbitrarily varied in search of the minimum OFV), the variance omega is guaranteed to be positive definite. Thus, initially fixing an off-diagonal variance element to a non-zero value within a block of variances cannot be honored by the search algorithm past the first iteration. Additionally, to my best understanding, arbitrarily fixing an off-diagonal variance element to 0 does not correspond to fixing a particular Cholesky off-diagonal element to 0, except for banded matrix patterns, which is already allowed in NONMEM. 3) EM algorithms such as Imp and SAEM update the Omega variance in the variance domain, and also, allow a simple constraint filter to be super-imposed on the resulting matrix after each iteration. In NONMEM, this can be done using constraint.f90, wherein a person may code in fixing any omega element to 0, or any other value, over-riding the EM algorithm's update. This is okay in EM, as the omega elements are not primary search parameters, but are constructed as a consequence of the expectation step, which provide conditional means and variances from which the omega elements are evaluated. User beware, should this constraint cause lack of positive definiteness. The constraint.f90 routine is not used for the traditional update methods, as this would upset their gradient search pattern by providing a dis-continuous intrusion. Robert J. Bauer, Ph.D. Vice President, Pharmacometrics, R&D ICON Development Solutions 7740 Milestone Parkway Suite 150 Hanover, MD 21076 Tel: (215) 616-6428 Mob: (925) 286-0769 Email: robert.ba...@iconplc.com Web: www.iconplc.com -----Original Message----- From: owner-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] On Behalf Of Matt Hutmacher Sent: Wednesday, June 13, 2012 2:35 PM To: 'Nick Holford'; nmusers@globomaxnm.com Subject: RE: [NMusers] omega matrix - friendly suggestion Dear Paolo, Nick, all, A way to ensure a nonnegative definite (NND) variance-covariance matrix is to use a Cholesky decomposition of the matrix (Ref 1). I use this often and it greatly stabilizes estimation such that I can often get better convergence and improved $COV step success for larger OMEGA matrix. It also is great if you sample from the $COV for simulation as an NND $OMEGA will be achieved with each sampling. I currently implement this by hand such that all $OMEGA becomes a Diagonal matrix with 1 FIXED as all of its entries. Note that I have not evaluated whether NONMEM computes the appropriate WRES or CWRES for this parameterization (as I have calculated these myself - Bob may be able to comment on whether this is a concern - ie putting the variances into linear combinations of thetas affecting the residuals). It is not enough to just model variances and correlations; such a parameterization will not guarantee NND OMEGA matrices. With respect to the original question which was to constrain certain covarainces to 0, one might (I have not tried this) be able to do this by writing the covariance matrix as wished, then solving for the Cholesky components, and modeling with the derived Cholesky parameterization. Upon completion, one can then compute the $OMEGA matrix from the Cholesky parameterization. I do not know how easy this would be to generalize in a software application, but if possible, this could perhaps be done behind the scenes as an option (may make standard errors a bit tricky) and would provide a lot of flexibility for analysis (as opposed to hand coding discussed below). There is also a modified Cholesky that can be used, which has the nice feature that one can simply, conveniently remove all the variance components related to a certain random effect (Ref 2). This was recently used in the attached reference to aid in selection of the $OMEGA structure with the fixed effects. Best, Matt Ref.1: Pinhiero, J. and Bates, D. (1996). Unconstrained parameterizations for variance-covariance matrices. Statistics and Computing 6, 289-286. Ref.2: Bondell HD, Krishna A, Ghosh (2010). Joint Variable Selection for Fixed and Random Effects in Linear Mixed-Effects Model. Biometrics 66, 1069-1077. -----Original Message----- From: owner-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] On Behalf Of Nick Holford Sent: Wednesday, June 13, 2012 1:20 PM To: nmusers@globomaxnm.com Subject: Re: [NMusers] omega matrix - friendly suggestion Paolo, You may remember this discussion about the history of the correlation feature for OMEGA (and SIGMA) blocks. http://www.cognigencorp.com/nonmem/current/2009-July/1804.html IMHO the only sensible way for a human to write a covariance matrix is using the SD and correlation style introduced in NONMEM 7. If I understand things correctly (this kind of math is not my strong point) it should be impossible to write non-positive definitive matrices provided the correlations are -1<=0<=+1 (although NONMEM might complain at -1 or +1). Can anyone provide any good reason to prefer the variance-covariance parameterization over the SD-correlation parametization? The other issue brought up by Pavel is a different one -- how to specify more simply a zero correlation|covariance between random effects. Monolix has a 0|1 matrix that allows a user to fix any covariance matrix element to zero so presumably the math exists somewhere to allow NONMEM how to understand a matrix to be recognized in this format. Bob (Bauer) -- do you know how to do this? Perhaps you could talk to Marc Lavielle and consider adding something similar to NM-TRAN. I'd prefer the style shown by Pavel to having to write a separate matrix of 0|1 elements as used in the Monolix GUI. Best wishes, Nick On 13/06/2012 5:16 p.m., Paolo Denti wrote: > Dear Pavel and Ken, > I also share the same pain, especially when coding correlations for > between-occasion variability ETAs. This implies reorganizing them in > blocks and renumbering everything. A real chore, and a very > error-prone process. > > Maybe one can think of using the OMEGA in the correlation format, > which should make it easier to write "legitimate" OMEGA matrices. Or > NONMEM can check the positive definiteness of the initial estimate and > complain if necessary (I believe it already does so). > > So, dear NONMEM developers, please count a +1 in the survey for this > feature. :) > > Thank you and ciao, > Paolo > > On 2012/06/12 17:08, Ken Kowalski wrote: >> >> Dear Pavel, >> >> I certainly feel your pain but you have to be careful how you fix >> certain elements in Omega to ensure that you have a valid positive >> definite covariance matrix. The starting values in your $OMEGA block >> do not give rise to a valid covariance matrix. Note in particular >> that the covariance between ETA3 and ETA4 is too large relative to >> the variances for ETA3 and ETA4 such that the correlation is greater >> than 1.0, i.e., >> >> Corr(ETA3,ETA4) = 0.03/[SQRT(0.0166)*SQRT(0.0166)]=1.807 > 1 >> >> You also have the same problem for Corr(ETA1,ETA3) > 1. >> >> Ken >> >> Kenneth G. Kowalski >> >> President & CEO >> >> A2PG - Ann Arbor Pharmacometrics Group, Inc. >> >> 110 Miller Ave., Garden Suite >> >> Ann Arbor, MI 48104 >> >> Work: 734-274-8255 >> >> Cell: 248-207-5082 >> >> Fax: 734-913-0230 >> >> ken.kowal...@a2pg.com <mailto:ken.kowal...@a2pg.com> >> >> www.a2pg.com <http://www.a2pg.com> >> >> *From:*owner-nmus...@globomaxnm.com >> [mailto:owner-nmus...@globomaxnm.com] *On Behalf Of >> *non...@optonline.net >> *Sent:* Tuesday, June 12, 2012 10:24 AM >> *To:* nmusers@globomaxnm.com >> *Subject:* [NMusers] omega matrix - friendly suggestion >> >> Hello NONMEM Community, >> >> Sometimes it takes more time to choose the best omega matrix than to >> develop a PD model. Selecting the omega matric is a tedious, time >> consuming and less creative part of the model development. I hope >> you feel my pain. Will it be helpful to rewrite the NONMEM software >> so that any element of the omega matrix can be fixed to any value? >> It may look like this: >> >> $OMEGA BLOCK (4) 3.60E-02 FIX >> >> 0.01 3.23E-02 >> 0.03 0 FIX 1.66E-02 >> 0.01 0 FIX 0.03 FIX 1.66E-02 >> >> This change can make many NONMEM users happy. >> >> Thanks! >> >> Pavel >> > > -- > ------------------------------------------------ > Paolo Denti, PhD > Junior Lecturer > 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.de...@uct.ac.za > ------------------------------------------------ -- Nick Holford, Professor Clinical Pharmacology First World Conference on Pharmacometrics, 5-7 September 2012 Seoul, Korea http://www.go-wcop.org Dept Pharmacology& Clinical Pharmacology, Bldg 505 Room 202D 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.holf...@auckland.ac.nz http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford ICON plc made the following annotations. ------------------------------------------------------------------------------ This e-mail transmission may contain confidential or legally privileged information that is intended only for the individual or entity named in the e-mail address. 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