Peiming – Thanks – you are right. The only constraint that the diagonal 3-eta parameterization parameterization places is that the covariance term of the block 2-eta Omega is non-negative.
From: [email protected] [mailto:[email protected]] On Behalf Of Peiming Ma Sent: Monday, November 25, 2013 8:55 PM To: 'nmusers' Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume parameters Dear Bob, The 3-eta parameterization really is mathematically equivalent to a 2-eta parameterization that has a non-negative covariance term. Here the 3-eta form is just two linear combinations of normal random variables, which form a bivariate normal with non-negative covariance. No other restrictions are there. Regards, Peiming From: [email protected]<mailto:[email protected]> [mailto:[email protected]] On Behalf Of Bob Leary Sent: Tuesday, November 26, 2013 5:09 AM To: Nick Holford; 'nmusers' Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume parameters Nick – I defer to you and the undoubtedly many other readers who know far more about pharmacokinetic theory than I do as to which particular formulation is more appropriate from a PK theoretic point of view. I was merely trying to note (and as I point out below, incorrectly) that something like the 2-eta formulation CL=THETA(1)*EXP(ETA(1)) V=THETA(2)*EXP(ETA(2)) Where ETA(1) and ETA(2) have a full 2 by 2 block correlation matrix so that correlation between ETA(1) and ETA(2) is Handled by an OMEGA(1,2) parameter Is ‘mathematically equivalent’ to a 3-eta formulation with a 3 by 3 diagonal Omega (ETA(1), ETA(2), ETA(3) independent) FF1=EXP(ETA(3)) CL=THETA(1)*EXP(ETA(1))/FF1 V=THETA(2)*EXP(ETA(2))/FF1 (The fact that FF1 formally looks like a bioavailability is irrelevant here, since I was not really intending to make any specific comments or recommendations with respect to how best to deal with bioavailabilities) Now that I look at it a bit more closely, the formulations actually are not at all mathematically equivalent (the 2 by 2 block formulation is much more General than the 3 by 3 diagonal formulation, even though they have the same number of parameters). While all 3 by 3 diagonal Omegas have Equivalent 2 by 2 block Omegas, the reverse is clearly not true. This is most easily seen in in the second 3 by 3 diagonal formulation where CL=THETA(1)*EXP(ETA(1)-ETA(3)), and V=THETA(2)*EXP(ETA(2)-ETA(3)), so cov(log CL, log V) = var(ETA(3)) >0. Thus in the second diagonal 3-eta formulation, the log CL-log V correlation must be positive (or at least non-negative), while there is no such restriction on the full block 2-eta formulation. So in fact the 2-eta block formulation is more general. I think it is even worse than this – there appear to be some regions of the block 2 eta parameter space that do not have equivalents in the diagonal 3-eta space even when the correlations are positive. (For example, if log CL and log V are highly correlated, then the variance of ETA(3) must be very large relative to the variance of ETA(2) and ETA(1) in the 3-eta formulation. But this means the variance of ETA(1) and ETA(2) in an equivalent two eta formulation must be relatively similar and roughly equal to the variance of ETA(3) in the 3-eta formulation. So without working out the details, I think there are regions of the block 2-eta space corresponding to highly correlated log CL and log V but with very different log CL and log V variances that are unattainable in the 3-eta formulations. So in fact the second 3 eta diagonal formulation is fundamentally different and less general than the first 2eta block formulation. But this just means that if CL and V are correlated only thru the F11 bioavailability like mechanism posited in the 3-eta formulation, there are restrictions as to what the corresponding 2 by 2 full block omega matrix can looks like. This leaves open the interesting point – run it both ways, and then see if the 2 by 2 and 3 by 3 methods produce compatible Omegas. If not, then this might provide some evidence that the coupling is more complicated than just that posited in the 3 by 3 diagonal model But in any event, the EM methods are not well suited to the second case, and will be inefficient relative to the first case if indeed they work at all (which may depend on the particular implementation) One problem is that the EM update of THETA(1) in the second case depends on the means for the various subjects of the posterior distributions of both ETA(1) and ETA(3) – most EM implementations usually have one or possibly several fixed effects coupled to a single random effect, and the update of that fixed effect, at least in the simple mu-modeled case, comes from a simple linear regression of the associated fixed effects on the posterior means of the single random effect. The fact that now there are multiple random effects paired with a single fixed effect is unusual and may not in fact be handled (I am not sure what NONMEM IMPEM will do with this; I am pretty sure that the analogous Phoenix NLME QRPEM will reject it). Bob From: [email protected]<mailto:[email protected]> [mailto:[email protected]] On Behalf Of Nick Holford Sent: Monday, November 25, 2013 1:43 PM To: 'nmusers' Subject: Re: [NMusers] Getting rid of correlation issues between CL and volume parameters Bob, You use an estimation method justification for choosing between estimating the covariance of CL and V and estimating the variance of F. An alternative view is to apply a fixed effect assumption based on pharmacokinetic theory. The fixed effect assumption is that some of the variation in CL and V is due to differences in bioavailability and other factors such as linear plasma protein binding and differences in the actual amount of drug in the oral formulation. This fixed effect assumption is described in the model by the variance of F. It is quite plausible to imagine that there is still some covariance between CL and V that is not related to the differences in F. For example, if you did not know the subject's weights and therefore could not account for the correlated effects of weight on CL and V. The estimation of the variance of F would only partly account for this because of the non-linear correlation of weight with CL and V. Another non-linear correlation would occur if plasma protein binding was non-linear in the range of measured total concentrations. In such case one might propose trying to estimate the covariance of CL and V as well as including F as a fixed effect and estimating the variance of F. Do you think that SAEM or IMP would be able to come up with a reasonable estimate of the covariance of CL and V? 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