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RE: var-cov matrix issue?

From: Bob Leary <bleary>
Date: Wed, 25 Feb 2009 08:33:53 -0500

If S is singular, then the 'covariance' matrix Rinv * S * Rinv is also =
singular,
as is the 'inverse coveriance matrix' R*Spseudoinv*R (the eigenvalues =
of
Spseudoinv for the usual Moore Penrose pseudoinverse are the inverse of =
the eigenvalues of S, except where the S has a zero eigenvalue, in which =
case the corresponding eigenvalue of Spseudoinv is also zero. The =
eigenvectors are the same for S and Spseudoinv). Thus none of these =
quantities is really directly suitable for
use in simulation if positive definiteness is a requirement.

 

Robert H. Leary, PhD
Fellow

Pharsight - A Certara(tm) Company
5625 Dillard Dr., Suite 205
Cary, NC 27511

Phone/Voice Mail: (919) 852-4625, Fax: (919) 859-6871
Email: bleary

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-----Original Message-----
From: owner-nmusers
[mailto:owner-nmusers
Sent: Tuesday, February 24, 2009 15:59 PM
To: Bachman, William
Cc: Ethan Wu; justin.wilkins
Subject: Re: [NMusers] var-cov matrix issue?


According to the manual, covariance matrix IS calculated by the default
method (Rinv S Rinv) even when S is singular but the inverse covariance
matrix (R Sinv R) cannot be computed as usual since S is singular (see
below). From the same manual "An error message stating that the S matrix =

is singular indicates strong overparameterization". If some of your
OMEGAs are estimated with large error, I would try to remove those ETAs
from the model. Scatter plot matrix of ETAs vs ETAs could be helpful: if =

some of your ETAs are redundant, you could see strong correlation of the =

ETAs estimates.

--
The inverse variance-covariance 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
  variance-covariance matrix is really not unique. In either case, the
  inverse variance-covariance matrix can be used to develop a joint con-
  fidence region for the complete set of population parameters. As we
  usually develop a confidence region for a very limited 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 clarification, this is not an error. It is an indication of a
> numerical condition generated by the matrix algebra. it says that the =

> covariance could not be calculated by the default method (possibly due =

> to ill conditioning) so it was calculated by an alternative method. =
You
> could generate standard errors by an alternative method, e.g. =
bootstrap,
> and compare them to those produced by NONMEM to make your decision to
> trust or not trust the values.
>
> =
------------------------------------------------------------------------
> *From:* owner-nmusers
> [mailto:owner-nmusers
> *Sent:* Tuesday, February 24, 2009 2:09 PM
> *To:* justin.wilkins
> *Cc:* nmusers
> *Subject:* Re: [NMusers] var-cov matrix issue?
>
> Hi Justin, only ETA was estimated with high SE
> but, again, I guess it came back to the question: how trustful it is =
if
> such error message appears
>
> =
------------------------------------------------------------------------
> *From:* "justin.wilkins
> *To:* ethan.wu75
> *Sent:* Tuesday, February 24, 2009 1:19:17 PM
> *Subject:* Fw: [NMusers] var-cov matrix issue?
>
>
> Dear Ethan,
>
> Algorithmically singular 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 errors? This might suggest =
overparamaterization.
>
> Not sure how helpful this is!
>
> Best
> Justin
> *Justin Wilkins
> Senior Modeler**
> Modeling & Simulation (Pharmacology)*
> CHBS, WSJ-027.6.076
> Novartis Pharma AG
> Lichtstrasse 35
> CH-4056 Basel
> Switzerland
> Phone: +41 61 324 6549
> Fax: +41 61 324 3039
> Cell: +41 76 561 0949
> Email : _justin.wilkins
<mailto:justin.wilkins
>
>
>
> ----- Forwarded by Justin Wilkins/PH/Novartis on 2009/02/24 07:15 PM =
-----
> *Ethan Wu <ethan.wu75
> Sent by: owner-nmusers
>
> 2009/02/24 07:12 PM
>
>
> To
> nmusers
> cc
>
> Subject
> [NMusers] var-cov matrix issue?
>
>
>
>
>
>
>
>
> Dear all,
> I recently encounter this error message (below). My objective was to
> use nonmem var-cov output for approximation of distribution of
> parameters for performing a simulation.
> if such error message occur, is the var-cov matrix still OK to use?
> -- I know that better way to figure out distribution of parameters is =
to
> do bootstrap, but given limited time I have.....
>
> thanks
>
> "0MINIMIZATION SUCCESSFUL
> NO. OF FUNCTION EVALUATIONS 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 MATRIX 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 - 08:33:53 EST

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