# Re: var-cov matrix issue?

From: Ethan Wu <ethan.wu75>
Date: Wed, 25 Feb 2009 09:49:19 -0800 (PST)

Hi Bob, I don't have enough math to understand difference of those matrix, =
but,  the final matrix output from nonmem was positive definite in my=
case ________________________________ From: Bob Leary <ble=
ary
25, 2009 8:33:53 AM Subject: RE: [NMusers] var-cov matrix issue? If=
S is singular, then the 'covariance' matrix Rinv * S * Rinv is also singul=
ar, as is the 'inverse coveriance matrix' R*Spseudoinv*R  (the eigenval=
ues of Spseudoinv for the usual Moore Penrose pseudoinverse are the inve=
rse of the eigenvalues of S, except where the S has a zero eigenvalue, in w=
hich case the corresponding eigenvalue of Spseudoinv is also zero.  The e=
igenvectors are the same for S and Spseudoinv).  Thus none of these quant=
ities is really directly suitable for use in simulation if positive defin=
iteness is a requirement. Robert H. Leary, PhD Fellow Pha=
rsight - A Certara(tm) Company 5625 Dillard Dr., Suite 205 Cary, NC 275=
11 Phone/Voice Mail: (919) 852-4625,  Fax: (919) 859-6871 Email: b=
leary
s for the sole use of the intended recipient and may contain confidential=
and proprietary information.  Any disclosure or distribution to third=
parties that is not specifically authorized by the sender is prohibited.=
ly email and destroy all copies of the original message.  -----O=
riginal Message----- From: owner-nmusers
musers
ry 24, 2009 15:59 PM To: Bachman, William Cc: Ethan Wu; justin.wilkins
novartis.com; nmusers
x issue? According to the manual, covariance matrix IS calculated b=
y the default method (Rinv S Rinv) even when S is singular but the inver=
se covariance matrix (R Sinv R) cannot be computed as usual since S is s=
ingular (see below). From the same manual "An error message stating that=
the S matrix is singular indicates strong overparameterization". If som=
e of your OMEGAs are estimated with large error, I would try to remove t=
hose ETAs from the model. Scatter plot matrix of ETAs vs ETAs could be h=
elpful: if some of your ETAs are redundant, you could see strong correla=
tion of the ETAs estimates. -- The  inverse  variance-covaria=
nce  matrix  R*Sinv*R  is  also  output   (labeled  as the =
Inverse Covariance Matrix), where Sinv is the inverse   of the S matrix=
..  If S is judged to be singular, a pseudo-inverse of S   is  used,=
and  since  a  pseudo-inverse  is  not  unique, the inverse=

the   inverse variance-covariance matrix can be used to develop a j=
oint con-   fidence region for the complete set of population parameter=
s.  As  we   usually  develop a confidence region for a very limi=
ted set of popula-   tion parameters, this use of the inverse variance-=
covariance matrix is   somewhat limited. -- --------------------=
------------------ Leonid Gibiansky, Ph.D. President, QuantPharm LLC =
web:    www.quantpharm.com e-mail: LGibiansky at quantpharm.com tel=
:    (301) 767 5566 Bachman, William wrote: > As a clar=
ification, this is not an error.  It is an indication of a > numerical=
condition generated by the matrix algebra.  it says that the > covari=
ance could not be calculated by the default method (possibly due > to il=
l conditioning) so it was calculated by an alternative method.  You > =
could generate standard errors by an alternative method, e.g. bootstrap, =

------------------------------------------ > *From:* owner-nmusers
axnm.com > [mailto:owner-nmusers

vartis.com > *Cc:* nmusers
r-cov matrix issue? > > Hi Justin, only ETA was estimated with high SE=

f > such error message appears > > --------------------------------=
---------------------------------------- > *From:* "justin.wilkins
is.com" <justin.wilkins
ent:* Tuesday, February 24, 2009 1:19:17 PM > *Subject:* Fw: [NMusers] va=
r-cov matrix issue? > > > Dear Ethan, > > Algorithmically sing=
ular matrices are often a sign that that your model > is ill-conditioned=
in some way; I would be careful in how I used the > variance-covariance=
matrix in this scenario, and especially for > simulation. Are there any=
parameters that are being estimated with > particularly high standard e=
rrors? This might suggest overparamaterization. > > Not sure how helpf=
ul this is! > > Best > Justin > *Justin Wilkins > Senior Modeler=
** > Modeling & Simulation (Pharmacology)* > CHBS, WSJ-027.6.076 > No=
vartis Pharma AG > Lichtstrasse 35 > CH-4056 Basel > Switzerland > =
Phone: +41 61 324 6549 > Fax: +41 61 324 3039 > Cell: +41 76 561 0949=

com> > > > > ----- Forwarded by Justin Wilkins/PH/Novartis on 20=
09/02/24 07:15 PM ----- > *Ethan Wu <ethan.wu75
owner-nmusers
> To >     nmusers
Subject >     [NMusers] var-cov matrix issue? > > >   =
> > > > > > Dear all, >  I recently encounter thi=
s error message (below). My objective was to > use nonmem var-cov output=
for approximation of distribution of > parameters for performing a s=
imulation. >  if such error message occur, is the var-cov matrix  sti=
ll OK to use? > -- I know that better way to figure out distribution of p=
arameters is to > do bootstrap, but given limited time I have..... >=
> thanks >  > "0MINIMIZATION SUCCESSFUL > NO. OF FUNCTION E=
VALUATIONS USED:  331 > NO. OF SIG. DIGITS IN FINAL EST.:  3.3 > =
ETABAR IS THE ARITHMETIC MEAN OF THE ETA-ESTIMATES, > AND THE P-VALUE IS=
GIVEN FOR THE NULL HYPOTHESIS THAT THE TRUE MEAN IS 0. >  ETABAR:  0=
..11E-02 > SE:      0.23E-01 >  P VAL.:  0.96E+00 > 0S MATRI=
X ALGORITHMICALLY SINGULAR > 0S MATRIX IS OUTPUT > 0INVERSE COVARIANCE =
MATRIX SET TO RS*R, WHERE S* IS A PSEUDO INVERSE OF S > 1 > " > > =
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Received on Wed Feb 25 2009 - 12:49:19 EST

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