From: Ken Kowalski <*kgkowalski58*>

Date: Wed, 24 Mar 2021 09:35:14 -0400

Hi Ibtihel,

See below for my comments.

Ken

From: owner-nmusers

Behalf Of Hammami, Ibtihel /FR

Sent: Wednesday, March 24, 2021 5:47 AM

To: nmusers

Subject: [NMusers] Statistical power of covariate inclusion in popPK models

Hi,

Thank you all for your answers. I have two follow-up questions:

1. Is it mandatory to use the matrix R as a variance co-variance

matrix to obtain the FIM? In case we have already used other type of

variance co-variance matrix, should we rerun the model with matrix R

setting?

Ken: The Hessian (or R matrix) is the 2nd derivative matrix of -LL which

is equivalent to the FIM. However, you don't have to invert the Hessian to

obtain the covariance matrix. For example, in NONMEM the default estimator

is the sandwich estimator, (R^-1)S(R^-1), where S is the 1st derivative

(gradient) cross-product matrix. Note that R^-1 as well as (R^-1)S(R^-1)

are consistent estimators of the covariance matrix and thus asymptotically,

both should converge to the same result as the sample size increases.

However, most nonlinear regression packages invert the Hessian (R matrix) as

the default estimate of the covariance matrix of the parameter estimates.

2. How critical is the estimation method for the computed SE? In other

words, is it relevant to compare two powers computed based on two standard

errors givens by different estimation algorithms?

For example, if we used FOCE-INTER to minimize our model, could we compare

the power (based on the Wald test) to results given by PFIM which is based

on FO ?

Ken: The R and S matrices involve taking 1st and 2nd derivatives of the

log-likelihood and evaluating them at the final estimates of the parameters.

Thus, if you use different approximations of the likelihood (e.g., FOCEI or

FOCE or FO) and you get different final estimates of the population

parameters you will also get different values for the R and S matrices.

Just for a little bit of background to clarify why we are interested in the

Wald test.

We are working on the comparison of different methods to compute the

statistical power of covariate inclusion in popPK models (SSE (gold

standard), MCMP, PPE , and Wald test). We have also included the Wald test

in our comparison because it is the fastest method and mostly because it

used by optimal designs software. R Therefore, the evaluation of the

accuracy of the power derived by this method could facilitate the bridging

step between model validation and the design of upcoming clinical trials

using optimal design software.

Ken: Most optimal design software (e.g., D-optimality) I imagine would

use the Hessian (R matrix) rather than the sandwich estimator, however, I'm

not familiar with what PFIM uses. You may want to follow up with the PFIM

developers to find out what they use.

Thank you in advance.

Cordially.

Ibtihel HAMMAMI.

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Received on Wed Mar 24 2021 - 09:35:14 EDT

Date: Wed, 24 Mar 2021 09:35:14 -0400

Hi Ibtihel,

See below for my comments.

Ken

From: owner-nmusers

Behalf Of Hammami, Ibtihel /FR

Sent: Wednesday, March 24, 2021 5:47 AM

To: nmusers

Subject: [NMusers] Statistical power of covariate inclusion in popPK models

Hi,

Thank you all for your answers. I have two follow-up questions:

1. Is it mandatory to use the matrix R as a variance co-variance

matrix to obtain the FIM? In case we have already used other type of

variance co-variance matrix, should we rerun the model with matrix R

setting?

Ken: The Hessian (or R matrix) is the 2nd derivative matrix of -LL which

is equivalent to the FIM. However, you don't have to invert the Hessian to

obtain the covariance matrix. For example, in NONMEM the default estimator

is the sandwich estimator, (R^-1)S(R^-1), where S is the 1st derivative

(gradient) cross-product matrix. Note that R^-1 as well as (R^-1)S(R^-1)

are consistent estimators of the covariance matrix and thus asymptotically,

both should converge to the same result as the sample size increases.

However, most nonlinear regression packages invert the Hessian (R matrix) as

the default estimate of the covariance matrix of the parameter estimates.

2. How critical is the estimation method for the computed SE? In other

words, is it relevant to compare two powers computed based on two standard

errors givens by different estimation algorithms?

For example, if we used FOCE-INTER to minimize our model, could we compare

the power (based on the Wald test) to results given by PFIM which is based

on FO ?

Ken: The R and S matrices involve taking 1st and 2nd derivatives of the

log-likelihood and evaluating them at the final estimates of the parameters.

Thus, if you use different approximations of the likelihood (e.g., FOCEI or

FOCE or FO) and you get different final estimates of the population

parameters you will also get different values for the R and S matrices.

Just for a little bit of background to clarify why we are interested in the

Wald test.

We are working on the comparison of different methods to compute the

statistical power of covariate inclusion in popPK models (SSE (gold

standard), MCMP, PPE , and Wald test). We have also included the Wald test

in our comparison because it is the fastest method and mostly because it

used by optimal designs software. R Therefore, the evaluation of the

accuracy of the power derived by this method could facilitate the bridging

step between model validation and the design of upcoming clinical trials

using optimal design software.

Ken: Most optimal design software (e.g., D-optimality) I imagine would

use the Hessian (R matrix) rather than the sandwich estimator, however, I'm

not familiar with what PFIM uses. You may want to follow up with the PFIM

developers to find out what they use.

Thank you in advance.

Cordially.

Ibtihel HAMMAMI.

--

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Received on Wed Mar 24 2021 - 09:35:14 EDT