Dear all, an alternative which I try to use on strictly positive parameters is to estimate on log-scale. Then I think often the assymptitic approximation is more true and the resulting measures of parameter uncertainty are more reliable. BW Magnus
________________________________________ Från: owner-nmus...@globomaxnm.com <owner-nmus...@globomaxnm.com> för Chaouch Aziz <aziz.chao...@chuv.ch> Skickat: den 12 februari 2015 12:01:51 Till: Eleveld, DJ; pascal.gir...@merckgroup.com Kopia: nmusers@globomaxnm.com Ämne: [NMusers] RE: Standard errors of estimates for strictly positive parameters Dear Douglas, dear Pascal, Thanks a lot for your answers. I guess the main point here is constrained vs unconstrained optimization as the asymptotic covariance matrix of estimates (as returned by $COV) is "well defined" only in the latter case. When fitting model 1, one would normally constrain THETA(1) to be positive by using something like: $THETA (0, 15, 50) ; TVCL In this situation I wonder whether it makes sense at all to consider the output of $COV. It seems model 2 would be here preferable (unconstrained optimization). If model 1 is fitted without boundary constraints on THETA(1), the covariance matrix of estimates may have "some" meaning but the optimization in NONMEM is then likely to crash if it encounters a negative value at some point, which again speaks somehow in favor of model 2 (unless one is not interested in the output of $COV). Now what about $OMEGA? Here NONMEM knows that these are variances and therefore we do not need to explicitly (i.e. manually) place boundary constraints on the diagonal elements of the omega matrix. However something must account for it internally. The covariance matrix of estimates returned by $COV also contains elements that refer to omega so I'm unsure how these are treated. For diagonal elements of the omega matrix, does NONMEM optimize log(omega) or omega? Or does it uses a Cholesky decomposition of the Omega matrix and optimize elements on that scale? Again, unless the optimization on omega is unconstrained, can we really trust the output of $COV? Basically the question here is how would you construct an asymptotic 95% confidence interval for a diagonal element of Omega (i.e. a variance) based on the information from the covariance matrix of estimates? The covariance matrix of estimate is of importance to me because I'm considering published studies and I do not have access to the data so I cannot refit the model with an alternative parametrization. Results from $COV (in lst file when available from the authors) is then the only available piece of information about the uncertainty of the estimation process. Kind regards, Aziz Chaouch ________________________________________ De : Eleveld, DJ [mailto:d.j.elev...@umcg.nl] Envoyé : mercredi, 11. février 2015 22:26 À : Chaouch Aziz; nmusers@globomaxnm.com Objet : RE: Standard errors of estimates for strictly positive parameters Hi Aziz, Just some comments off the top of my head in a quite informal way: I'm not really sure that these are the same problem because they dont start with the same information in the form of parameter constraints. In model 1 you are asking the optimizer for the unconstrained maximum likelihood solution for TVCL. OK, this is reasonable in a lot of situations, but not necessairily in all situations. In model 2 you add information by forcing TVCL and CL to be positive. If you think of the optimal solution as some point in N-dimensional space which has to be searched for, in model 2 you are saying "dont even look in the space where TVCL or CL is negative". Even stronger, in model 2 you are also saying "dont even get close to zero" because the log-normal distribution vanishes towards zero. Which solution of these is best for some particular application depends on a lot of things. One of the things I would think about in this situation is whether or not my a priori beliefs match with the structual constraints of the model. Do I really think that the "true" CL could be zero? If yes, then model 2 is hard to defend in that case. You description of your situation regarding standard errors is a part of the same thing. When you extrapolate standard errors into low-probability areas you are checking the boundaries of the probability area. It should not be suprising that model 1 might tell you that CL is negative since this was part of the solution space which you allowed. With model 2 your model structure says "dont even look there" In short, although these two models might look similar, I think they are really quite different. This becomes most clear when you consider the low-probability space. Sorry for the vauge language. Warm regards, Douglas ________________________________________ De : pascal.gir...@merckgroup.com [mailto:pascal.gir...@merckgroup.com] Envoyé : mercredi, 11. février 2015 18:30 À : Chaouch Aziz; nmusers@globomaxnm.com Objet : RE: Standard errors of estimates for strictly positive parameters Dear Aziz, NM does not return the asymptotic SE of THETA(1) in model 1 on the log-scale. So I would use model 2. With best regards / Mit freundlichen Grüßen / Cordialement Pascal ________________________________________ From: owner-nmus...@globomaxnm.com [owner-nmus...@globomaxnm.com] on behalf of Chaouch Aziz [aziz.chao...@chuv.ch] Sent: Wednesday, February 11, 2015 5:21 PM To: nmusers@globomaxnm.com Subject: [NMusers] Standard errors of estimates for strictly positive parameters Hi, I'm interested in generating samples from the asymptotic sampling distribution of population parameter estimates from a published PKPOP model fitted with NONMEM. By definition, parameter estimates are asymptotically (multivariate) normally distributed (unconstrained optimization) with mean M and covariance C, where M is the vector of parameter estimates and C is the covariance matrix of estimates (returned by $COV and available in the lst file). Consider the 2 models below: Model 1: TVCL = THETA(1) CL = TVCL*EXP(ETA(1)) Model 2: TVCL = EXP(THETA(1)) CL = TVCL*EXP(ETA(1)) It is clear that model 1 and model 2 will provide exactly the same fit. However, although in both cases the standard error of estimates (SE) will refer to THETA(1), the asymptotic sampling distribution of TVCL will be normal in model 1 while it will be lognormal in model 2. Therefore if one is interested in generating random samples from the asymptotic distribution of TVCL, some of these samples might be negative in model 1 while they'll remain nicely positive in model 2. The same would happen with bounds of (asymptotic) confidence intervals: in model 1 the lower bound of a 95% confidence interval for TVCL might be negative (unrealistic) while it would remain positive in model 2. This has obviously no impact for point estimates or even confidence intervals constructed via non-parametric bootstrap since boundary constraints can be placed on parameters in NONMEM. But what if one is interested in the asymptotic covariance matrix of estimates returned by $COV? The asymptotic sampling distribution of parameter estimates is (multivariate) normal only if the optimization is unconstrained! Doesn't this then speak in favour of model 2 over model 1? Or does NONMEM take care of it and returns the asymptotic SE of THETA(1) in model 1 on the log-scale (when boundary constraints are placed on the parameter)? Thanks, Aziz Chaouch ________________________________ De inhoud van dit bericht is vertrouwelijk en alleen bestemd voor de geadresseerde(n). Anderen dan de geadresseerde(n) mogen geen gebruik maken van dit bericht, het niet openbaar maken of op enige wijze verspreiden of vermenigvuldigen. Het UMCG kan niet aansprakelijk gesteld worden voor een incomplete aankomst of vertraging van dit verzonden bericht. The contents of this message are confidential and only intended for the eyes of the addressee(s). Others than the addressee(s) are not allowed to use this message, to make it public or to distribute or multiply this message in any way. The UMCG cannot be held responsible for incomplete reception or delay of this transferred message. ________________________________ Confidentiality Notice: This message is private and may contain confidential and proprietary information. If you have received this message in error, please notify us and remove it from your system and note that you must not copy, distribute or take any action in reliance on it. Any unauthorized use or disclosure of the contents of this message is not permitted and may be unlawful.
