RE: OMEGA HAS A NONZERO BLOCK

From:Kowalski, Ken Subject:RE: [NMusers] OMEGA HAS A NONZERO BLOCK Date:Thursday, October 03, 2002 11:38 AM Hi Nick, We've had this discussion before. I suspect the correlation is being driven to 1 because of limitations of the design (i.e, insufficient information to precisely estimate the correlation but sufficient information to suggest it is non-zero--otherwise NONMEM would have estimated the covariance to be zero). I draw the analogy to estimating a variance component for ka when there is very little information in the absorption phase. With this analogy, NONMEM might estimate the variance component for ka to be 0. We typically do not interpret this to mean that there is no BSV in ka just that the design cannot support the estimation of the BSV in ka. So, what do we do?...We typically constrain the omega for ka to be 0 even though we know that it is probably unrealistic. With regards to the perfect correlation problem, if we fix the covariance in such a way to restrict the correlation to a more reasonable value less than 1 we will take a hit in the MOF as the maximum likelihood estimates of the parameters (including elements of Omega) wants to estimate this correlation as 1...this is the discussion we had before. At that time you changed your recommendation to a Bayesian solution where you specify a prior on this correlation. I can't argue against that approach if one has such a prior. However, I suspect the prior would have to be quite strong (to move the correlation away from 1) as a flat or non-informative prior is going to run into the same perfect correlation problem as maximum likelihood estimation. What if Steve's ill-conditioned Omega just squeaked by NONMEM when he tried to simulate...perhaps rounding down the off-diagonal elements of Omega as you recommended in a previous message? He would have proceeded perhaps not realizing that his model is ill-conditioned/over-parameterized and would have been simulating with near perfect correlation for P1 and P4. NONMEM (or any other nonlinear regression algorithm) can act quirky (e.g., extremely sensitive to starting values) when the model is ill-conditioned. Steve's model may provide a good fit, I just contend that I can get that same fit with 3 fewer elements in Omega. My solution is not altering the fit that Steve obtained with his BLOCK(4) parameterization unless of course he truly did not achieve a global minimum which is possible due to the over-parameterized Omega. If so, my solution could possibly lead to an even lower MOF. However, I have encountered the problem Steve raises on numerous occasions and typically the solution I propose leads to the identical fit without the instability. Steve didn't indicate whether the COV step failed when he fit his BLOCK(4) model...often it will fail with an over-parameterized Omega even though the estimation step converges. The solution I propose removes the ill-conditioning of Omega and can allow the COV step to run without altering the fit. Mats' parameterization is not a solution to Steve's ill-conditioned Omega problem. He merely re-parameterized Omega so that the variances and covariance are estimated with 3 additional thetas (i.e., theta3, theta4 and theta5 in his example) in lieu of the 3 elements in a BLOCK(2) Omega. With Mat's parameterization the correlation between CL and V is THETA(5)^2/(1+THETA(5)^2). With this parameterization one can gain control over restricting the correlation by fixing THETA(5). However, this parameterization expanded to Steve's BLOCK(4) problem will still have the ill-conditioning problem as it's fitting the same model with the same number of elements in Omega...just reparameterized as Thetas. With Mats' parameterization, the perfect correlation would result in THETA(5) going to infinity. In Steve's BLOCK(4) results I calulated the correlation as 0.998 which would suggest that THETA(5)=22.3. If you want to restrict the correlation to some arbitrary value r, this can be obtained by fixing THETA(5)=sqrt(r/(1-r)). Thus, for r=0.8, THETA(5)=2.0. This is considerably smaller than the THETA(5)=22.3 that I estimate for Steve's problem. However, as I indicated in my previous message, if you use all the digits rather than the 3 signif digits that NONMEM reports out I suspect that the correlation is even closer to 1. Bottom line: We need to get rid of the ill-conditioning by simplifying the model. Simply fixing the correlation to some arbitrary value less than 1 so that we don't have a singular Omega (which is what happens when we try to estimate the correlation as 1) is undesirable because we take a hit on the fit (higher MOF). Ken