Covariance: Matrix=S or Matrix=R

From: "Xu, Zhenhua (Mike) [CNTUS]" ZXu5@CNTUS.JNJ.COM Subject: [NMusers] Covariance: Matrix=S or Matrix=R Date: Tue, 13 Sep 2005 18:47:58 -0400 Dear NM Users: Could somebody in the NM Users Group elaborate the difference between Martrix=R and Matrix=S during the Covarince step? Are there any disavantages/advantages for using Matrix=S since the default Matrix=R? Thanks a lot. Zhenhua Xu

From: "Nele Plock" nplock@zedat.fu-berlin.de Subject: Re: [NMusers] Covariance: Matrix=S or Matrix=R Date: Wed, 14 Sep 2005 09:43:53 +0200 You will find a very helpful explanation on: http://huxley.phor.com/nonmem/nm/99may012003.html Best regards, Nele

From: "Nele Plock" nplock@zedat.fu-berlin.de Subject: Re[2]: [NMusers] Covariance: Matrix=S or Matrix=R Date: Wed, 14 Sep 2005 16:57:26 +0200 Michael, If the R-matrix fails this might indicate that you have not reached a global minimum but a saddle point. It could also indicate that the model is overparameterised. Try using a more simple model (in either structural or statistical aspects). S-matrix usually runs easier, but this does not mean you can always trust it. However, Im not an expert on this, so I cannot tell you whether your estimates represent the best fit. If you shared some of your data and control file with the group, maybe somebody else might be able to help you. Nele Plock

From: "Stephen Duffull" sduffull@pharmacy.uq.edu.au Subject: RE: Re[2]: [NMusers] Covariance: Matrix=S or Matrix=R Date: Thu, 15 Sep 2005 09:05:31 +1000 Hi I think we can be a little more definitive. NONMEM uses a local search strategy and therefore you can never tell if it has reached a global minimum based on assessment of the R or S matrix. The NONMEM repository has plenty of discussions on the relevance of estimated standard errors (which I won't bring up here). Steve ==================================================================== Stephen Duffull School of Pharmacy, University of Queensland, Brisbane 4072, Australia Tel +61 7 3365 8808, Fax +61 7 3365 1688, Email: sduffull@pharmacy.uq.edu.au www http://www.uq.edu.au/pharmacy/index.html?page=31309 Design: http://www.uq.edu.au/pharmacy/sduffull/POPT.htm MCMC: http://www.uq.edu.au/pharmacy/sduffull/MCMC_eg.htm University Provider Number: 00025B

From: mark.e.sale@gsk.com Subject: RE: Re[2]: [NMusers] Covariance: Matrix=S or Matrix=R Date: Wed, 14 Sep 2005 20:03:50 -0400 Steve et al. To continue beating this horse ... A saddle point is to be distinguished from a local minima. All local search algorithms have the potential for local mimima - period. A saddle point is different. A saddle point is a point at which the first derivative of the OBJ wrt parameters is (close to) zero - i.e., the function is flat, locally, but one dimension if curving up and one dimension is curing down. Minima and saddle points can be distinguished by using the second derivative (is the surface curving up or down). If the second derivative (Hessian) is poorly defined, you can't be certain that the flatness isn't due to being at the top of a peak (curving down - a maxima) in one dimension vs being a the bottom (curving up - a minima) in another. My understanding (for what it is) is that modern non-linear regression algorithms are pretty robust to not getting stuck in saddle points - of course depending on how well defined the surface is. If the surface is flat as far as the algorithm can see, it has a hard time telling if this is maxima or a minima. But, again, this is a known problem for non-linear regression and great effort has be applied to getting modern algorithms (which NONMEM actually uses) to address it robustly. There are non-linear regression-like algorithms that are (more) robust to local minima. They are complex, inefficient and rarely used. Other algorithms that are robust to local minima include the convexity stuff from the USC group, and I suppose MCMC could be included as well, seems to me it should not have a problem with local minima, but I'm not sure. I think the text is unclear, the R and S matrix tell you nothing about whether this is a local or global minima, only if it is a minima (or either kind) or a saddle point. I think the work global should be ignored. Mark Sale M.D. Global Director, Research Modeling and Simulation GlaxoSmithKline 919-483-1808 Mobile 919-522-6668

From: "Stephen Duffull" sduffull@pharmacy.uq.edu.au Subject: RE: Re[2]: [NMusers] Covariance: Matrix=S or Matrix=R Date: Fri, 16 Sep 2005 08:49:23 +1000 Hi Mark Thanks for your email. You make your argument very clear and conceptually I agree completely. You wrote in your last comment: >> I think the text is unclear, the R and S matrix tell you >> nothing about whether this is a local or global minima, only >> if it is a minima (or either kind) or a saddle point. To relate this to the original question which I believe related to the idea that the R matrix is more sensitive to saddle points compared to the S matrix. Now without my NONMEM manuals at hand I risk making some frightful mistakes (but someone will correct me, I'm sure). The R matrix (I think) most closely relates to the Hessian. The S matrix is an asymptotic result for the R matrix (I am sure someone who has a manual with them will comment here). For all intents and purposes S should be pretty similar to R and therefore also to the "sandwich" matrix reported by NONMEM in $COV which is akin to the weighted average of the R and S matrices. Therefore I am not convinced that there is much between the R and S matrices in terms of what they tell you, which is similar to what you have suggested above, other than the R matrix is more difficult to compute. i.e. I am not convinced that R tells you more about saddle points than S. Does anyone have any more specific beliefs about R and S? Steve ==================================================================== Stephen Duffull School of Pharmacy, University of Queensland, Brisbane 4072, Australia Tel +61 7 3365 8808, Fax +61 7 3365 1688, Email: sduffull@pharmacy.uq.edu.au www http://www.uq.edu.au/pharmacy/index.html?page=31309 Design: http://www.uq.edu.au/pharmacy/sduffull/POPT.htm MCMC: http://www.uq.edu.au/pharmacy/sduffull/MCMC_eg.htm University Provider Number: 00025B ===================================================================== _______________________________________________________