Re: Condition number

On 29-11-2022 17:25, Ayyappa Chaturvedula wrote: > Hi Ken, > > You are correct, the 10^n rule is in the context of individual level modeling. > > Thank you Pete for chiming in, I learned the difference you mention from your book too. > > Regards, > Ayyappa > > On Tue, Nov 29, 2022 at 10:19 AM Ken Kowalski < [email protected] > wrote: > > I have seen models with a successful COV step and CN > 10^5 but I > certainly have not seen COV steps run with a CN > 10^20. Thus, > the CN > 10^n has got to break down when n is large. Does > Gabrielsson and Weiner discuss this rule in the context of simple > nonlinear regression of individual subject (or animal) curves or > do they also propose this rule in the context of population models > with nonlinear mixed effects. I suspect it was only proposed for > the former. > > Not to rehash old ground, but a successful COV step does not imply > that a model is stable even if none of the pairwise correlations > are extreme if CN is very large. > > *From:*Ayyappa Chaturvedula [mailto:[email protected]] > *Sent:* Tuesday, November 29, 2022 11:07 AM > *To:* Ken Kowalski <[email protected]> > *Cc:* [email protected] > *Subject:* Re: [NMusers] Condition number > > Hi Ken, > > Thank you again. But, I have seen models with 10^5 and above > with no issues with covariance step and correlations not reaching > 0.95 but some with moderate levels. It will be interesting to > know other experiences. > > The 10^n rule is from the PK-PD Data analysis, Gabrielsson and > Weiner, Edition 3, page 313. I read this book most of my grad > school days. > > Regards, > > Ayyappa > > On Tue, Nov 29, 2022 at 9:35 AM Ken Kowalski > <[email protected]> wrote: > > Hi Ayyappa, > > I have not seen this rule but it strikes me as being too > liberal to apply in pharmacometrics where n can be very large > for the models we fit. If we have a structural model with say > n=4 or 5 parameters and then also investigate covariate > effects on these parameters it would not be unusual to have a > covariate model with n=20+ fixed effects parameters. I doubt > we can get the COV step to run such that we can observe a CN > >10^20. > > I have not seen CN criteria indexed by n. The classifications > of collinearity that I've seen based on CN are: > > Moderate: 100 <= CN < 1000 > High: 1000 <= CN < 10,000 > Extreme: CN >= 10,000 > > Ken > > -----Original Message----- > From: Ayyappa Chaturvedula [mailto:[email protected]] > Sent: Tuesday, November 29, 2022 10:20 AM > To: Ken Kowalski <[email protected]> > Cc: [email protected] > Subject: Re: [NMusers] Condition number > > Thank you, Ken. It is very reassuring. > > I have also seen a discussion on other forums that Condition > number as a function of dimension of problem (n). I am seeing > contradiction between 10^n and a static >1000 approach. I am > curious if someone can also comment on this and 10^n rule? > > Regards, > Ayyappa > > > On Nov 29, 2022, at 9:04 AM, Ken Kowalski > <[email protected]> wrote: > > > > Hi Ayyappa, > > > > I think the condition number was first proposed as a > statistic to > > diagnose multicollinearity in multiple linear regression > analyses > > based on an eigenvalue analysis of the X'X matrix. You can > probably > > search the statistical literature and multiple linear > regression > > textbooks to find various rules for the condition number as > well as > > other statistics related to the eigenvalue analysis. For > the CN<1000 > > rule I typically reference the following textbook: > > > > Montgomery and Peck (1982). Introduction to Linear > Regression Analysis. > > Wiley, NY (pp. 301-302). > > > > The condition number is good at detecting model instability > but it is > > not very good for identifying the source. Inspecting the > correlation > > matrix for extreme pairwise correlations is better suited > for identifying the source of > > the instability when it only involves a couple of > parameters. It becomes > > more challenging to identify the source of the instability > > (multicollinearity) when the CN>1000 but none of the pairwise > > correlations are extreme |corr|>0.95. Although when CN>1000 > often we > > will find several pairwise correlations that are moderately > high > > |corr|>0.7 but it may be hard to uncover a pattern or source > of the > > instability without trying alternative models that may > eliminate one > > or more of the parameters associated with these moderate to > high correlations. > > > > Best, > > > > Ken > > > > Kenneth G. Kowalski > > Kowalski PMetrics Consulting, LLC > > Email: [email protected] > > Cell: 248-207-5082 > > > > > > > > -----Original Message----- > > From: [email protected] > > [mailto:[email protected]] On Behalf Of Ayyappa > > Chaturvedula > > Sent: Tuesday, November 29, 2022 8:52 AM > > To: [email protected] > > Subject: [NMusers] Condition number > > > > 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 > > > > > > -- > > This email has been checked for viruses by Avast antivirus > software. > > www.avast.com http://www.avast.com > > -- This email has been checked for viruses by Avast antivirus > > software. > www.avast.com http://www.avast.com > > 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 > > <#m_4632534978889413432_DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2>