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
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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://www.cognigencorp.com/nonmem/nm/99may012003.html

Best regards,
Nele
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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
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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
=====================================================================

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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
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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
=====================================================================

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