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

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>

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