What does "Fisher Score" mean?

[0521]

Hi supporter

After activating the “NLME” license, there is an option for “Stderr Method.” under the “Run Mode” tab of the Phoenix Model object.

This is the method of computing the standard errors.
I have some questions about the options:

1. The “Hessian” option is described in great detail in the help, which uses R-1 to calculate standard errors.

2. The “Sandwich” option may calculate standard errors using the R-1SR-1 approach.

3. What does the “Fisher Score” option mean?
   3.1 Use the 4S-1 method to calculate standard errors?
   3.2 Use the inverse of Fisher’s information matrix to calculate standard errors?

Cheers,

0521

[SimonDavis]

I don’t know if you have got the hang of searching the new online help in version 8.2 but searching for fisher score will quickly take you to the page on Run Options;

https://onlinehelp.certara.com/phoenix/8.2/index.html?#t=topics%2Fnlmeuirunoptns.htm

It’s quite brief;

Fisher Score: The Fisher Score method is fast and robust, but less precise than the Sand ­wich and Hessian methods.

does that help any? what specifically would you like to know?

Simon.

[0521]

Dear Simon,

I read the paragraph on “Fisher Score” in the new online help.

But I still don’t understand how “Fisher Score” is. Can you give some reference information to help me understand how Fisher Score is?

I also searched and found the following page. Does the meaning of the “Fisher Score” on this page have the same meaning as the “Fisher Score” in Phoenix?
https://en.wikipedia.org/wiki/Scoring_algorithm

Thank you in advance,

Cheers,

0521

[SimonDavis]

Thanks for your comments, I’ve noted some iumporvements to documentation under SUP-789 and whilst speaking with one of the Developers he pointed me to this reference.

https://cran.r-project.org/web/packages/nmw/nmw.pdf

(Understanding Nonlinear Mixed Effects Modeling for Population Pharmacokinetics)

R Matrix R matrix of NONMEM, the second derivative of log likelihood function with respect to estimation parameters

The R matrix is the Fisher information matrix constructed from the second derivative of the objective function with respect to the various parameters estimated.

R matrix is the same as Hessian in NLME

S Matrix S matrix of NONMEM, sum of individual cross-product of the first derivative of log likelihood function with respect to estimation parameters

The more approximate S matrix (score matrix) is constructed from the first derivatives of each individual’s objective function contribution with respect to the various parameters estimated

S matrix is the same as Fisher score in NLME

The variance–covariance of the estimates is evaluated as R−1 or S−1. By default the variance–covariance is evaluated as R-1SR-1 (Sandwich in NLME**)**

Note that in 8.2 new option ‘AutoDetect’ is added.

When selected, NLME automatically chooses the standard error calculation method. Specifically, if both Hessian and Fisher score methods are successful, then it uses the Sandwich method. Otherwise, it uses either the Hessian method or the Fisher score method, depending on which method is successful. The user can check the Core Status outputs to see which method is used.

Simon.

PS for the ‘nerds’ ;0), you can review internals of calculations using debug CovStep() function of that package.

[0521]

@sdavis

Dear Simon,

Thank you for your clarification.

If you are correct, I think it is not appropriate to use the “Fisher Score” naming method in Phoenix.

the reason:

1. “Fisher Score” is easy to reminiscent or search for ‘Fisher Information matrix’ instead of “S matrix”, or “outer-product-of-the-gradient”, such as I searched for “Fisher Score” result:
   https://en.wikipedia.org/wiki/Scoring_algorithm
   In this entry, “Fisher Score” means ‘Fisher Information’
   [attachment=3433:0521-01.png]

2. In the book《Econometric Theory and Methods》-Oxford University Press, USA (2003)Russell Davidson, James G. MacKinnon, Section 10.4 points out four methods for estimating the covariance matrix based on the maximum likelihood method:
   [attachment=3434:0521-02.png]
   [attachment=3435:0521-03.png]

I compared this with the NONMEM method, obviously
NONMEM uses the first method (R-1), the third method (S), and the fourth method (R-1SR-1).

For Phoenix, it is clear that
the “Hessian” method corresponds to the first method,
and the “Sandwich” method corresponds to the fourth method.
From the name, the “Fisher Score” method seems to correspond to the second method , but NONMEM uses the third method.
So I propose My question.
“3. What does the “Fisher Score” option mean?”
3.1 Is “third method”
3.2 or Is “second method”

Cheers,
0521