From: Bob Leary <*bleary*>

Date: Wed, 25 Feb 2009 14:08:37 -0500

Hi Ethan, that's possible if the S matrix is 'algorithmically' singular =

but not exactly singular.

In this case the pseudoinverse is not a continuous function of the =

elements of the matrix.

For example, consider a 2 by 2 matrix with

a11=1, a22=eps, and a12=a21=0, where eps is equal to a very =

small number.

The true inverse (and pseudoinverse) is ainv11=1, ainv22=1/eps, =

ainv12,ainv21=0. At eps=0,

the pseudoinverse of A is ainv11=1, ainv22=0, a very different =

result than for any arbitrarily small but still positive eps.

For example, in MATLAB (which uses state of the art routines),

for eps=1.d-15 , pinv(A), the pseudoinverse of A, is exactly and =

correctly computed as

a11=1., a22 = 1.d15. But at eps=1.d-20, pinv(A) is computed as =

ainv11=1, ainv22=0,

the same result as if eps=0. So somewhere between eps=1.d-15 and =

1.d-20 (which is right around

the relative precision of double precision computations), the result =

returned by the pseudoinverse changes from

positive definite to not positive definite (interstingly, inv(A), the =

matlab matrix inverse routine, still

returns the exactly correct results for eps=1.d-20). If the criterion =

that NM uses to determing 'algorithmic singularity'is a high condition =

number estimate that is set to something like 1.d10, and the true =

conditon number of S is around 1.d15, then a good pseudoinverse routine =

returns a positive definite matrix but NM decides

(probably quite justifiably) that the original S matrix is =

'algorithmically' singular.

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

This email message (including any attachments) is 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. If you are not =

the intended recipient, please contact the sender by reply email and =

destroy all copies of the original message.

-----Original Message-----

From: Ethan Wu [mailto:ethan.wu75

Sent: Wednesday, February 25, 2009 12:49 PM

To: Bob Leary; nmusers

Subject: Re: [NMusers] var-cov matrix issue?

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 <bleary

To: nmusers

Sent: Wednesday, February 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 =

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

*> This email message (including any attachments) is 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. If you are not =

the intended recipient, please contact the sender by reply email and =

destroy all copies of the original message.

-----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 <http://www.quantpharm.com/>

e-mail: LGibiansky at quantpharm.com <http://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 *

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
*

*> "
*

*>
*

*> 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
*

*>
*

*>
*

Received on Wed Feb 25 2009 - 14:08:37 EST

Date: Wed, 25 Feb 2009 14:08:37 -0500

Hi Ethan, that's possible if the S matrix is 'algorithmically' singular =

but not exactly singular.

In this case the pseudoinverse is not a continuous function of the =

elements of the matrix.

For example, consider a 2 by 2 matrix with

a11=1, a22=eps, and a12=a21=0, where eps is equal to a very =

small number.

The true inverse (and pseudoinverse) is ainv11=1, ainv22=1/eps, =

ainv12,ainv21=0. At eps=0,

the pseudoinverse of A is ainv11=1, ainv22=0, a very different =

result than for any arbitrarily small but still positive eps.

For example, in MATLAB (which uses state of the art routines),

for eps=1.d-15 , pinv(A), the pseudoinverse of A, is exactly and =

correctly computed as

a11=1., a22 = 1.d15. But at eps=1.d-20, pinv(A) is computed as =

ainv11=1, ainv22=0,

the same result as if eps=0. So somewhere between eps=1.d-15 and =

1.d-20 (which is right around

the relative precision of double precision computations), the result =

returned by the pseudoinverse changes from

positive definite to not positive definite (interstingly, inv(A), the =

matlab matrix inverse routine, still

returns the exactly correct results for eps=1.d-20). If the criterion =

that NM uses to determing 'algorithmic singularity'is a high condition =

number estimate that is set to something like 1.d10, and the true =

conditon number of S is around 1.d15, then a good pseudoinverse routine =

returns a positive definite matrix but NM decides

(probably quite justifiably) that the original S matrix is =

'algorithmically' singular.

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

This email message (including any attachments) is 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. If you are not =

the intended recipient, please contact the sender by reply email and =

destroy all copies of the original message.

-----Original Message-----

From: Ethan Wu [mailto:ethan.wu75

Sent: Wednesday, February 25, 2009 12:49 PM

To: Bob Leary; nmusers

Subject: Re: [NMusers] var-cov matrix issue?

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 <bleary

To: nmusers

Sent: Wednesday, February 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 =

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

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. If you are not =

the intended recipient, please contact the sender by reply email and =

destroy all copies of the original message.

-----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 <http://www.quantpharm.com/>

e-mail: LGibiansky at quantpharm.com <http://quantpharm.com/>

tel: (301) 767 5566

Bachman, William wrote:

You

bootstrap,

------------------------------------------------------------------------

if

------------------------------------------------------------------------

model

overparamaterization.

justin.wilkins

-----

to

0.

OF S

-------------------------------------------------------------------------=

-----

privileged information

address. If you

disclosure, copying,

prohibited. If

the sender, so that

message.

Received on Wed Feb 25 2009 - 14:08:37 EST