RE: OMEGA HAS A NONZERO BLOCK

From:"Kowalski, Ken" Subject:RE: [NMusers] OMEGA HAS A NONZERO BLOCK Date:Mon, 7 Oct 2002 09:59:09 -0400 All, Wow, what a flurry of emails on a Friday afternoon! I agree that we are hashing old ground. I disagree with Lewis that the ONLY way to achieve stability is by adding more information. That is THE solution when such additional information exists. So I have no problem with posing a Bayesian solution. Again, I suspect that the correlation from an independent data source must be fairly precisely estimated to provide strong enough prior information to resolve the ill-conditioning problem with the current data/design. But what if no such data exists or the estimate of the correlation is imprecise (a weak prior) such that the ill-conditioned problem can't be resolved with the additional data/information? Fixing the correlation and assuming it is known perfectly or specifying a strong prior arbitrarily (i.e., not based on existing independent data) does not sit well with me (unless your God, bring me the data). The alternative approach to achieve stability is to reduce the dimensionality of the problem (i.e., the current model is over-parameterized). That is, simplify the model that can adequately describe the data in hand. In otherwords, in the absence of additional data/information, you gotta live with what you've got! I still like my analogy of the zero variance component estimate. Why is it that some of you are willing to fix a variance component to zero for say Ka or V given the limitations of the design/data but are not willing to fix a correlation to 1 given such limitations? Isn't it just as unreasonable to assume that Ka or V is EXACTLY the same in ALL individuals in the population as it is to assume that if I know an individual's CL then I know his V because of the perfect correlation? My proposed solution to Steve's ill-conditioned Omega was merely proposing a simpler form of Steve's model to achieve the same fit he obtained. Steve claims that my solution to his problem is a red-herring but I am not convinced. I challenge Steve to fit the model I propose and report back on the MOF for his ill-conditioned model and my proposed solution...I'll be very much surprized if the MOF's differ by more than what can be explained by rounding errors. However, I do acknowledge Leonid's point that we can't necessarily trust the results from an ill-conditioned Omega to find the direction that can remove the ill-conditioning. Thus, some form of testing of the individual elements of Omega may have some benefit in finding a more parsimonious Omega. If this can be obtained by banding and fixing an element(s) to zero, so be it. Steve, can you report the MOF's for these other Omega structures as well? If banding with only one element restricted to zero (i.e., estimating 9 elements in Omega) gets rid of your ill-conditioning then I suspect that the MOF will be lower than what you obtained with your full BLOCK(4) ill-conditioned Omega because I claim I can get rid of the ill-conditioning without loss in MOF with just 7 elements in Omega. My approach to building Omega is to fit the fullest Omega that can be supported by the data. In Pete's simulations with a correlation of 0.92 where this was reliably estimated (supported by the data) I wouldn't propose fixing it to 1 (of course the MOF will be higher as there is sufficient data to estimate it different from 1). A condition number of 763 is not that large and I wouldn't consider the Omega ill-conditioned (a condition number greater than 10^3 is generally considered moderately ill-conditioned and a condition number greater than 10^5 is considered severe...Steve's problem had a condition number >10^6). I only propose fixing the correlation to 1 when NONMEM estimates it on the boundary such that the model is extremely unstable. Usually when this occurs the COV step will fail. Call me an empiricist if you'd like, but show me the science that say's the correlation is exactly 0.5. Ken