# RE: OFV higher with FOCEI than FO

From: Bob Leary <bleary>
Date: Wed, 10 Dec 2008 12:10:47 -0500

As shown by X. Wang, FO, FOCE and LAPLACE form a hierarchy of =
approximations.
Both the FO and FOCE methods are based on the same underlying Laplacian =
approximation to the
integral of the joint likelihood function of the random effects (eta's). =

The basic Laplace approximation requires knowledge of
the value of the joint likelihood function at its peak, and the second =
derivatives at the
eta values at which the peak is reached.

Hessian matrix of second derivatives at the peak of the joint likelihood =
function
from first derivatives, but accurately
determines the position of the peak (the empirical Bayes estimates)
in random effects (eta) space
and the function value at the peak (this determination of the EBE's is =
what the 'conditional step'
is all about and is computationally costly.)

Although the underlying Laplacian approximation is based on the local =
behavior of the
joint log likelihood function in the neighborhood of its peak, FO does =
not investigate the behavior
of the joint likelihood function near its peak at all (which is =
basically why FO estimates can be arbitrarily
poor). Instead it guestimates the value at the peak by extrapolating =
from eta=0, using a single Newton step
based on approximate first and second derivatives at eta=0. It also =
simply assigns the FOCE
approximate values of the
second derivatives at eta=0 to the values at the peak in order to =
evaluate the Laplacian approximation.
These additional approximations layered on top of the basic Laplacian =
and FOCE approximations
by FO are quite dubious for significantly nonlinear model functions, and =
often result in very poor quality
parameter estimates compared to FOCE and Laplace.

Strictly speaking. FOCE and FO objective values cannot be compared in =
any consistently meaningful sense.
But loosely speaking, since both FO and FOCE share a common base =
Laplacian approximation, but FO layers
on additional approximations on top of FOCE, the difference in FO vs =
FOCE objective values reflects the
effects of the additional FO approximations. Large differences may =
suggest that the additional FO approximations
have large effects, and make the FO estimates even more suspect relative =
to FOCE.

Robert H. Leary, PhD
Principal Software Engineer
Pharsight Corp.
5520 Dillard Dr., Suite 210
Cary, NC 27511

Phone/Voice Mail: (919) 852-4625, Fax: (919) 859-6871

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-----Original Message-----
From: owner-nmusers
[mailto:owner-nmusers
ayyappa.5.chaturvedula
Sent: Wednesday, December 10, 2008 9:40 AM
To: owner-nmusers
Subject: [NMusers] OFV higher with FOCEI than FO

Dear All,

I am analyzing a data set pooled from 4 clinical studies with rich =
sampling. When I fit a 2 comp oral absorption model with lag time using =
FO, I got successful minimization with COV step, but minimization was =
not successful when I used FO parameter estimates as initial estimates =
for FOCE run. When I used FOCE with INTER minimization was successful =
with COV step but the OFV is much higher (~25000 vs 20000) with FOCEI =
estimation than FO. The parameter estimates make more sense with FOCEI =
than FO. My questions are,

Can we get something like this or I am missing something here?
Can we compare OFV between different estimation methods (my =
understanding is no and OFV in case of FO does not make a lot of sense)? =

Regards,
Ayyappa Chaturvedula
GlaxoSmithKline