Hello Ken: I am quite unaware that some eigenvalues of a properly positive-definite verified variance-covariance from a pure R matrix would be negative, or that this would even occur for its correlation matrix.
Similarly, if the variance-covariance form is of sandwich form, such as (Rinv)S(Rinv), if there components (R, S) were each verified to be positive definite, then it, and its correlation matrix would necessarily have all positive eigenvalues. I would need to see your NONMEM result file to understand why this would happen. Is the negative eigenvalues very small but negative? (such as 10^-15, or something like that). Robert J. Bauer, Ph.D. Senior Director Pharmacometrics R&D ICON Early Phase 731 Arbor way, suite 100 Blue Bell, PA 19422 Office: (215) 616-6428 Mobile: (925) 286-0769 [email protected]<mailto:[email protected]> www.iconplc.com<http://www.iconplc.com/> From: Ken Kowalski <[email protected]> Sent: Thursday, December 1, 2022 6:59 AM To: Bauer, Robert <[email protected]>; [email protected] Cc: 'Bonate, Peter' <[email protected]> Subject: [EXTERNAL] RE: [NMusers] Condition number Hi Bob, Could it possibly be related to the S matrix and the default sandwich estimator used in estimating the covariance and correlation matrices? Ken From: Ken Kowalski [mailto:[email protected]] Sent: Thursday, December 1, 2022 9:52 AM To: 'Bauer, Robert' <[email protected]>; [email protected] Cc: 'Bonate, Peter' <[email protected]> Subject: RE: [NMusers] Condition number Hey Bob, I get that NONMEM can encounter negative eigenvalues during the R matrix decomposition and inversion step and if it does then the $COV step fails. However, both Pete and I have encountered situations where the R matrix is apparently positive definite since the $COV step runs but NONMEM reports a negative eigenvalue from the correlation matrix from the PRINT=E option. It is very rarely that I have seen this happen but it has happened to me. How can this be if the R matrix is positive definite? Thanks, 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]] On Behalf Of Bauer, Robert Sent: Wednesday, November 30, 2022 1:53 PM To: '[email protected]' <[email protected]<mailto:[email protected]>> Subject: RE: [NMusers] Condition number Hello all: Report of non-positive definiteness or negative eigenvalues, are reported during the analysis of the R matrix (decomposition and inversion), which occurs before the correlation matrix is constructed. Often, this is caused by numerical imprecision. If the R matrix step fails, the $COV step fails to produce a final variance-covariance matrix, and of course, does not produce a correlation matrix. If the R matrix inversion step succeeds, the variance-covariance matrix and its correlation matrix are produced, and the correlation matrix is then assessed for its eigenvalues. So, both the R matrix (first step) and correlation matrix (second step) are decomposed and assessed. Robert J. Bauer, Ph.D. Senior Director Pharmacometrics R&D ICON Early Phase 731 Arbor way, suite 100 Blue Bell, PA 19422 Office: (215) 616-6428 Mobile: (925) 286-0769 [email protected]<mailto:[email protected]> www.iconplc.com<http://www.iconplc.com/> ICON plc made the following annotations. ------------------------------------------------------------------------------ This e-mail transmission may contain confidential or legally privileged information that is intended only for the individual or entity named in the e-mail address. If you are not the intended recipient, you are hereby notified that any disclosure, copying, distribution, or reliance upon the contents of this e-mail is strictly prohibited. If you have received this e-mail transmission in error, please reply to the sender, so that ICON plc can arrange for proper delivery, and then please delete the message. Thank You, ICON plc South County Business Park Leopardstown Dublin 18 Ireland Registered number: 145835 [Image removed by sender.]<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>
