Re: Condition number

> On Nov 29, 2022, at 7:17 PM, Leonid Gibiansky <[email protected]> > wrote: > > from the manual: > > Iteration -1000000003 indicates that this line contains the condition number > , lowest, highest, Eigen values of the correlation matrix of the variances of > the final parameters. > > > >> On 11/29/2022 7:59 PM, Ken Kowalski wrote: >> Hi Matt, >> I’m pretty sure Stu Beal told me many years ago that NONMEM calculates the >> eigenvalues from the correlation matrix. Maybe Bob Bauer can chime in here? >> Ken >> *From:*Matthew Fidler [mailto:[email protected]] >> *Sent:* Tuesday, November 29, 2022 7:56 PM >> *To:* Ken Kowalski <[email protected]> >> *Cc:* Kyun-Seop Bae <[email protected]>; [email protected]; Jeroen >> Elassaiss-Schaap (PD-value B.V.) <[email protected]> >> *Subject:* Re: [NMusers] Condition number >> Hi Ken, >> I am unsure, since I don't have my NONMEM manual handy. >> I based my understanding on reading about condition numbers in numerical >> analysis, which seemed to use the parameter estimates: >> https://en.wikipedia.org/wiki/Condition_number >> https://en.wikipedia.org/wiki/Condition_number >> If it uses the correlation matrix, it could be less sensitive. >> Matt >> On Tue, Nov 29, 2022 at 6:11 PM Ken Kowalski <[email protected] >> <mailto:[email protected]>> wrote: >> Hi Matt, >> Correct me if I’m wrong but I thought NONMEM calculates the >> condition number based on the correlation matrix of the parameter >> estimates so it is scaled based on the standard errors of the estimates. >> Ken >> *From:*Matthew Fidler [mailto:[email protected] >> <mailto:[email protected]>] >> *Sent:* Tuesday, November 29, 2022 7:04 PM >> *To:* Ken Kowalski <[email protected] >> <mailto:[email protected]>> >> *Cc:* Kyun-Seop Bae <[email protected] >> <mailto:[email protected]>>; [email protected] >> <mailto:[email protected]>; Jeroen Elassaiss-Schaap (PD-value >> B.V.) <[email protected] <mailto:[email protected]>> >> *Subject:* Re: [NMusers] Condition number >> Hi Ken & Kyun-Seop, >> I agree it should be taught, since it is prevalent in the industry, >> and should be looked at as something to investigate further, but no >> hard and fast rule should be applied to if the model is reasonable >> and fit for purpose. That should be done in conjunction with other >> diagnostic plots. >> One thing that has always bothered me about the condition number is >> that it is calculated based on the final parameter estimates, but >> not the scaled parameter estimates. Truly the scaling is supposed >> to help make the gradient on a comparable scale and fix many >> numerical problems here. Hence, if the scaling works as it is >> supposed to, small changes may not affect the colinearity as >> strongly as the calculated condition number suggests. >> This is mainly why I see it as a number to keep in mind instead of a >> hard and fast rule. >> Matt >> On Tue, Nov 29, 2022 at 5:09 PM Ken Kowalski <[email protected] >> <mailto:[email protected]>> wrote: >> Hi Kyun-Seop, >> I would state things a little differently rather than say >> “devalue condition number and multi-collinearity” we should >> treat CN as a diagnostic and rules such as CN>1000 should NOT be >> used as a hard and fast rule to reject a model. I agree with >> Jeroen that we should understand the implications of a high CN >> and the impact multi-collinearity may have on the model >> estimation and that there are other diagnostics such as >> correlations, variance inflation factors (VIF), standard errors, >> CIs, etc. that can also help with our understanding of the >> effects of multi-collinearity and its implications for model >> development. >> That being said, if you have a model with a high CN and the >> model converges with realistic point estimates and reasonable >> standard errors then it may still be reasonable to accept that >> model. However, in this setting I would probably still want to >> re-run the model with different starting values and make sure it >> converges to the same OFV and set of point estimates. >> As the smallest eigenvalue goes to 0 and the CN goes to infinity >> we end up with a singular Hessian matrix (R matrix) so we know >> that at some point a high enough CN will result in convergence >> and COV step failures. Thus, you shouldn’t simply dismiss CN as >> not having any diagnostic value, just don’t apply it in a rule >> such as CN>1000 to blindly reject a model. The CN>1000 rule >> should only be used to call your attention to the potential for >> an issue that warrants further investigation before accepting >> the model or deciding how to alter the model to improve >> stability in the estimation. >> Best, >> Ken >> Kenneth G. Kowalski >> Kowalski PMetrics Consulting, LLC >> Email: [email protected] <mailto:[email protected]> >> Cell: 248-207-5082 >> *From:*[email protected] >> <mailto:[email protected]> >> [mailto:[email protected] >> <mailto:[email protected]>] *On Behalf Of *Kyun-Seop Bae >> *Sent:* Tuesday, November 29, 2022 5:10 PM >> *To:* [email protected] <mailto:[email protected]> >> *Subject:* Fwd: [NMusers] Condition numbera >> Dear All, >> I would like to devalue condition number and multi-collinearity >> in nonlinear regression. >> The reason we consider condition number (or multi-collinearity) >> is that this may cause the following fitting (estimation) problems; >> 1. Fitting failure (fail to converge, fail to minimize) >> 2. Unrealistic point estimates >> 3. Too wide standard errors >> If you do not see the above problems (i.e., no estimation >> problem with modest standard error), you do not need to give >> attention to the condition number. >> I think I saw 10^(n – parameters) criterion in an old version of >> Gabrielsson’s book many years ago (but not in the latest version). >> Best regards, >> Kyun-Seop Bae >> On Tue, 29 Nov 2022 at 22:59, Ayyappa Chaturvedula >> <[email protected] <mailto:[email protected]>> wrote: >> Dear all, >> I am wondering if someone can provide references for the >> condition number thresholds we are seeing (<1000) etc. Also, >> the other way I have seen when I was in graduate school that >> condition number <10^n (n- number of parameters) is OK. >> Personally, I am depending on correlation matrix rather than >> condition number and have seen cases where condition number >> is large (according to 1000 rule but less than 10^n rule) >> but correlation matrix is fine. >> I want to provide these for my teaching purposes and any >> help is greatly appreciated. >> Regards, >> Ayyappa >> >> https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient >> >> Virus-free.www.avast.com >> >> https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient >