Fixed elements within a block covariance matrix

From:Peter L. Bonate - pbonate@ilexonc.com Subject:[NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 8:50 AM Dear all, I need to generate a covariance matrix of the following: [X11 0 X22 X31 X32 X33] where X12 is fixed to zero and all other elements are non-zero. Is there any easy way to do this? Thanks. Peter L. Bonate, PhD, FCP Director, Pharmacokinetics ILEX Oncology 4545 Horizon Hill Blvd San Antonio, TX 78229 phone: 210-949-8662 fax: 210-949-8219 email: pbonate@ilexonc.com

From: Leonid Gibiansky - lgibiansky@emmes.com Subject: Re: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 9:06 AM sure, just reorder (rename) random effects so that your matrix looks like x11 x13 x33 0 x32 x22 Leonid

From: GIRARD PASCAL - PASCAL.GIRARD@adm.univ-lyon1.fr Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 9:26 AM Hi Peter, Are you meaning you want initialize such a matrix in NONMEM ? If so you can try the following: 1) Renumber your Etas so that you will have the following structure: [X22 X32 X33 0 X31 X11] 2) This should be recognized by NONMEM as a band matrix (see attached (below) Diane Moulds Email for more explanations). 3) Tell us if that works! Cheers, Pascal Subject: Re: Termination due to rounding erros From: "diane r mould" Date: Fri, 23 Mar 2001 20:40:09 +0100 To: "KOWALSKI, KENNETH G. [PHR/1825]" , Dear Ken I will do the best I can :-) In NONMEM, a BAND matrix is when some of the elements in the off diagonal matrix have been set to zero. Doing this effectively fixes these elements to zero, eliminating the need for NONMEM to estimate those values which may not be identifiable, without sacrificing other identifiable elements. So if, for example, you have a one compartment model with CL, V, KA as parameters, if you can identify the covariance terms between CL and V and between CL and Ka, but not between Ka and V, a BAND matrix could be written to describe these relationships. Below is the NONMEM record for this, assuming that ETA(1) describes the variance on V, ETA(2) on CL and ETA(3) on KA. $OMEGA BLOCK(3) .1 .01 .1 0 .01 .1 No further coding is necessary. However, the elements in the BAND matrix must, of course, be symmetrical. Therefore a BAND matrix of $OMEGA BLOCK(3) .1 .01 .1 0 0 .1 will produce an error from NONMEM. You would need to code this arrangement instead as follows: $OMEGA BLOCK(2) .1 .01 .1 $OMEGA .1 I have found the use of the BAND matrix to be quite useful at times, particularly if the model is to be used later for simulation work for example. Best Regards Diane

From: Vladimir Piotrovsky, Ph.D. - VPIOTROV@PRDBE.jnj.com Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 9:27 AM Pete, You can try to change the indices of ETAs to come up with the matrix [X11 X21 X22 0 X32 X33] In this case you can fix the lower left element to zero. Best regards, Vladimir ----------------------------------------------------------------- Vladimir Piotrovsky, Ph.D. Research Fellow, Advanced PK-PD Modeling & Simulation Global Clinical Pharmacokinetics and Clinical Pharmacology (ext. 5463/151) Johnson & Johnson Pharmaceutical Research & Development Turnhoutseweg 30 B-2340 Beerse Belgium

From: "Kowalski, Ken" - Ken.Kowalski@pfizer.com Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 9:38 AM Pete, Leonid beat me to it. The block structure that Leonid gives below is the banded structure. Note that in NONMEM you can specify 0's in a lower triangle portion of the Omega block (i.e., banded structure) and NONMEM will know to fix those elements to zero (i.e., you don't use the FIX option). Ken

From: vgcasabo - vicente.casabo@uv.es Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 11:29 AM Hi, all I do not understand, that if X11 is correlated with X33, and X22 is correlated with X33, X11 and X22 are not correlated. Sorry Vicente G. Casabo

From: Serge Guzy - GUZY@xoma.com Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 12:15 PM You can try using simulation if it is possible. I think that as long as the matrix is positive definite it will work. Sometimes indirect correlation like you suggest (X11 is correlated with X33, and X22 is correlated with X33 ) can exist while there is no direct correlation expressed through the correlation coefficient between X11 and X22 Serge Guzy President POP-PHARM Head of Pharmacometrics and Preclinical Statistics; Xoma Corporation Tel: (w) 510 204 74 76;(Business)510 527 2220

From: "Bonate, Peter" - pbonate@ilexonc.com Subject: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 14:51 Dear all, I need to generate a covariance matrix of the following: [X11 0 X22 X31 X32 X33] where X12 is fixed to zero and all other elements are non-zero. Is there any easy way to do this? Thanks. Peter L. Bonate, PhD, FCP Director, Pharmacokinetics ILEX Oncology 4545 Horizon Hill Blvd San Antonio, TX 78229 phone: 210-949-8662 fax: 210-949-8219 email: pbonate@ilexonc.com

From: Steve Duffull - sduffull@pharmacy.uq.edu.au Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 5:22 PM Vicente You are right - X11 and X22 will be implicitly correlated through the other covariance structure. However in the NONMEM OMEGA statement below they will not be explicitly correlated. Without specifically doing simulations to check this - I would guess that the implicit correlation would hold (to some extent) - even when you fix the explicit covariance to zero in the OMEGA block. I would be interested to know this too, can someone advise me on this? Steve

From: Sam Liao - sliao@pharmaxresearch.com Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 8:42 PM Hi Steve, Vicente: I believe what you are saying may hold, but not always. In the case where X11 and X33 are positively correlated, while X22 and X33 are negatively correlated (X3 goes up while X2 goes down), X11 and X22 may not be correlated. This is just my observation in some of the models I had before. Best regards, Sam Liao, Ph.D. PharMax Research 199 Pierce Street, Suite 817, Somerset, NJ 08873 phone: 201-9882043 efax: 1-720-2946783

From: Serge Guzy - GUZY@xoma.com Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/13/2004 10:04 PM I made many simulation exercises where indirect correlations exist while direct correlations were not existent. As long as the matrix is positive definite, this can happen. With 3 dimensions it is easy to make it positive definite. Once we begin with high dimensionality, then arbitrarily fixing one correlation to a certain value(not specifically 0) can make it difficult to be positive definite. Serge

From: vgcasabo - vicente.casabo@uv.es Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/14/2004 4:01 AM Hi all Por fin. I understand. Is possible that if cov1 and cov2 are two physiological variables not known or hidden, and the following dependence exists: Eta1=a1*cov1+e1 Eta2=a2*cov2+e2 Eta3=a3*cov1+a4*cov2+e3 e1, e2 and e3 are independent errors Thus, when dependence of parameters and cov1 and cov2 are not modeled, eta1 and eta3 are correlated, eta2 and et3 are correlated but eta1 and eta2 are not. Vicente G. Casabo

From: Jakob Ribbing - Jakob.Ribbing@farmbio.uu.se Subject: RE: [NMusers] Fixed elements within a block covariance matrix Date: 1/14/2004 4:12 AM Dear all, If one views correlation between two etas as a result of a mutual covariate not currently included in the covariate model, one can easily come up with examples where only one of three off diagonals truly is 0. In the example with correlation between individual CL, V and Ka, a measure of body size could very well act as a covariate on both CL and V, just as geno- or phenotype of a drug transporter may act on both CL and Ka. By this mechanism, as long as there is no correlation between body size and genotype/phenotype of the transporter, individual etas may be correlated for CL and V, as well as for CL and Ka, without any correlation between individual V and Ka. Notice that this example holds even if two different measures of body size (WT, WT^0.75, LBW, BSA, et c) are appropriate for CL and V, respectively. It is also possible that the off-diagonal non-zero correlations are all positive. A positive correlation between CL and Ka may not be plausible from a perspective of physiology or evolution in this example but if it was (or if the example was Q, V and Ka), we could very well have all non-zero correlations between etas being positive. In this example, the off-diagonal etas would not have been necessary, had we only included a proper covariate model. However, my point is not that off-diagonal etas are a result of bad covariate modelling. The true covariate model is far more complex than this example, consisting of covariates which are unknown and cannot be measured. Jakob Ribbing _______________________________________________________

From: Leonid Gibiansky - lgibiansky@emmes.com Subject: Re: [NMusers] Fixed elements within a block covariance matrix Date: 1/14/2004 9:44 AM One can check correlation structure via simulation, but I am not quite sure why should one do it. I guess, we can trust NONMEM to simulate according to the variance-covariance model, and then, for the OMEGA matrix X11 X13 X33 0 X23 X22 Eta1 and Eta2 will not be correlated in any way (implicit or explicit). If simulations show correlation (X12 not equal to zero), then one need to increase the sample size of the simulations. Correlation coefficient X21 should decay to zero with the sample size, if the model is implemented correctly. Example Eta1=a1*cov1+e1 Eta2=a2*cov2+e2 Eta3=a3*cov1+a4*cov2+e3 is not quite correct unless means of cov1 and cov2 are equal to zero (since means of etas should be zero). But the idea holds. If Eta1=a11*e1 Eta2=a22*e2 Eta3=a13*e1+a23*e2+a33*e3 where e1, e2, and e3 are independent [then mean(ei*ej) = 0 if i not equal j] with variance 1 [ mean(ei*ei) = 1 ] then X11=mean(Eta12)=a112 X12=mean(Eta1*Eta2) = 0 (since e1 and e2 are independent) X13=mean(Eta1*Eta3) = a11*a13 X22 = a222 X23 = a22*a23 X33 = a132+a232+a332 Leonid

From: qi - liuqi@ufl.edu Subject: Re: [NMusers] Fixed elements within a block covariance matrix Date: 1/14/2004 10:09 AM Dear all, Here is a simple example: Suppose X and Y are independent standard normal variable n(0,1), let U=X+Y. It could be shown that U is correlated with both X and Y. Another example: Suppose X and Y are independent standard normal variable n(0,1), let U=X+Y and V=X-Y. t could be shown that U and V are independent, but both U and V are correlated with X. Use X, Y and U in the first case, or use U, V and X in the second case to construct the variance-covariance matrix, you will get the mentioned interesting matrix. You can do a simple simulation to see the relationship between the variables (it can be done easily in R or Splus). Hope this helps. Qi Liu PO BOX 100494 Department of Pharmaceutics College of Pharmacy University of Florida Gainesville, FL32610 Tel: (352)8463257