From: Bauer, Robert <*Robert.Bauer*>

Date: Wed, 24 Mar 2021 21:19:17 +0000

Ken:

Yes, if $COV MATRIX is not specified, then .cov and .coi contain the respective sandwich versions.

According to Wikipedia ( https://en.wikipedia.org/wiki/Fisher_information ) the proper Fisher information matrix is defined as

“Formally, it is the variance<https://en.wikipedia.org/wiki/Variance> of the score<https://en.wikipedia.org/wiki/Score_(statistics)>, or the expected value<https://en.wikipedia.org/wiki/Expected_value> of the observed information<https://en.wikipedia.org/wiki/Observed_information>”

which is E(S) (at the maximum likelihood positon), and it is equivalent to E(R) (at the maximum likelihood position). So, it seems that neither a finite data R nor finite data S are formally a FIM, but if one is going to use a practically assessed (on finite data) FIM, R is preferred when there are few subjects. With sufficient data available, S approaches R, so that S and R*Sinv*R may be used as well.

Robert J. Bauer, Ph.D.

Senior Director

Pharmacometrics R&D

ICON Early Phase

820 W. Diamond Avenue

Suite 100

Gaithersburg, MD 20878

Office: (215) 616-6428

Mobile: (925) 286-0769

Robert.Bauer

www.iconplc.com<http://www.iconplc.com>

From: Ken Kowalski <kgkowalski58

Sent: Wednesday, March 24, 2021 1:34 PM

To: Bauer, Robert <Robert.Bauer s

Subject: RE: [EXTERNAL] RE: [NMusers] Statistical power computation based on the wald test

Hi Bob,

Just a point of clarification. If the default sandwich estimator is used to estimate the covariance matrix then the .coi file outputs the inverse of this sandwich estimator, ie., R(S^-1)R …correct? If so, and maybe this is just semantics but I don’t think we would refer to R(S^-1)R as the Fisher information matrix. However, both R and S can be considered equivalent FIM under certain regularity conditions. Nevertheless, if one wanted to determine a D-optimal design I suppose maximizing the determinant of R(S^-1)R could be a reasonable thing to do. Your thoughts?

Ken

Kenneth G. Kowalski

Kowalski PMetrics Consulting, LLC

Email: kgkowalski58 ilto:kgkowalski58

Cell: 248-207-5082

From: owner-nmusers s

Sent: Wednesday, March 24, 2021 2:18 PM

To: nmusers

Subject: RE: [EXTERNAL] RE: [NMusers] Statistical power computation based on the wald test

Hello all:

I would just like to add some information to help the discussion along.

In addition to the variance-covariance matrix that is outputted in the .cov file that Ken mentioned, the Fisher information matrix itself (inverse of variance-covariance) is also outputted in the .coi file. Additional files, such as .rmt (R matrix), and .smt (S matrix) are also outputted upon user request ($COV PRINT=RS, for example)

A test related to Wald and log-likelihood ratio tests is the Lagrange Multiplier test. For this purpose, NONMEM outputs the following in the .ext file:

Iteration -1000000008 lists the partial derivative of the log likelihood (-1/2 OFV) with respect to each estimated parameter.

PFIM, POPED, and NONMEM’s $DESIGN calculate the expected FIM with respect to the data, and the expected value R matrix is equivalent to the expected value of the S matrix. That is, Ey(R)= Ey(S).

Several companion/interface software to NONMEM have additional model evaluation facilities, such as stepwise covariate model (scm) building in Perl Speaks NONMEM, and Wald test in PDxPop.

Robert J. Bauer, Ph.D.

Senior Director

Pharmacometrics R&D

ICON Early Phase

820 W. Diamond Avenue

Suite 100

Gaithersburg, MD 20878

Office: (215) 616-6428

Mobile: (925) 286-0769

Robert.Bauer Bauer

www.iconplc.com<http://www.iconplc.com>

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

[Image removed by sender.]<https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient&utm_term=icon>

Virus-free. www.avast.com<https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient&utm_term=link>

Received on Wed Mar 24 2021 - 17:19:17 EDT

Date: Wed, 24 Mar 2021 21:19:17 +0000

Ken:

Yes, if $COV MATRIX is not specified, then .cov and .coi contain the respective sandwich versions.

According to Wikipedia ( https://en.wikipedia.org/wiki/Fisher_information ) the proper Fisher information matrix is defined as

“Formally, it is the variance<https://en.wikipedia.org/wiki/Variance> of the score<https://en.wikipedia.org/wiki/Score_(statistics)>, or the expected value<https://en.wikipedia.org/wiki/Expected_value> of the observed information<https://en.wikipedia.org/wiki/Observed_information>”

which is E(S) (at the maximum likelihood positon), and it is equivalent to E(R) (at the maximum likelihood position). So, it seems that neither a finite data R nor finite data S are formally a FIM, but if one is going to use a practically assessed (on finite data) FIM, R is preferred when there are few subjects. With sufficient data available, S approaches R, so that S and R*Sinv*R may be used as well.

Robert J. Bauer, Ph.D.

Senior Director

Pharmacometrics R&D

ICON Early Phase

820 W. Diamond Avenue

Suite 100

Gaithersburg, MD 20878

Office: (215) 616-6428

Mobile: (925) 286-0769

Robert.Bauer

www.iconplc.com<http://www.iconplc.com>

From: Ken Kowalski <kgkowalski58

Sent: Wednesday, March 24, 2021 1:34 PM

To: Bauer, Robert <Robert.Bauer s

Subject: RE: [EXTERNAL] RE: [NMusers] Statistical power computation based on the wald test

Hi Bob,

Just a point of clarification. If the default sandwich estimator is used to estimate the covariance matrix then the .coi file outputs the inverse of this sandwich estimator, ie., R(S^-1)R …correct? If so, and maybe this is just semantics but I don’t think we would refer to R(S^-1)R as the Fisher information matrix. However, both R and S can be considered equivalent FIM under certain regularity conditions. Nevertheless, if one wanted to determine a D-optimal design I suppose maximizing the determinant of R(S^-1)R could be a reasonable thing to do. Your thoughts?

Ken

Kenneth G. Kowalski

Kowalski PMetrics Consulting, LLC

Email: kgkowalski58 ilto:kgkowalski58

Cell: 248-207-5082

From: owner-nmusers s

Sent: Wednesday, March 24, 2021 2:18 PM

To: nmusers

Subject: RE: [EXTERNAL] RE: [NMusers] Statistical power computation based on the wald test

Hello all:

I would just like to add some information to help the discussion along.

In addition to the variance-covariance matrix that is outputted in the .cov file that Ken mentioned, the Fisher information matrix itself (inverse of variance-covariance) is also outputted in the .coi file. Additional files, such as .rmt (R matrix), and .smt (S matrix) are also outputted upon user request ($COV PRINT=RS, for example)

A test related to Wald and log-likelihood ratio tests is the Lagrange Multiplier test. For this purpose, NONMEM outputs the following in the .ext file:

Iteration -1000000008 lists the partial derivative of the log likelihood (-1/2 OFV) with respect to each estimated parameter.

PFIM, POPED, and NONMEM’s $DESIGN calculate the expected FIM with respect to the data, and the expected value R matrix is equivalent to the expected value of the S matrix. That is, Ey(R)= Ey(S).

Several companion/interface software to NONMEM have additional model evaluation facilities, such as stepwise covariate model (scm) building in Perl Speaks NONMEM, and Wald test in PDxPop.

Robert J. Bauer, Ph.D.

Senior Director

Pharmacometrics R&D

ICON Early Phase

820 W. Diamond Avenue

Suite 100

Gaithersburg, MD 20878

Office: (215) 616-6428

Mobile: (925) 286-0769

Robert.Bauer Bauer

www.iconplc.com<http://www.iconplc.com>

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

[Image removed by sender.]<https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient&utm_term=icon>

Virus-free. www.avast.com<https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient&utm_term=link>

(image/jpeg attachment: image001.jpg)