# RE: OFV higher with FOCEI than FO

From: Matt Hutmacher <matt.hutmacher>
Date: Wed, 10 Dec 2008 14:04:27 -0500

Hi Bob,

I would just add one point of clarification. My understanding is that the
FOCE approximate is a Laplace-based approximation (related to it) only if
the within subject residual error model does not contain any
subject-specific random effects.

Wolfinger R (1993). Laplace's approximation for nonlinear mixed models.
Biometrika 80, 791-795.

Vonesh ER, Chinchilli VM (1997). Linear and nonlinear models for the
analysis of repeated measurements. Marcel Dekker.

Matt

From: owner-nmusers
Behalf Of Bob Leary
Sent: Wednesday, December 10, 2008 12:11 PM
To: ayyappa.5.chaturvedula
nmusers
Subject: RE: [NMusers] OFV higher with FOCEI than FO

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

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

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-----Original Message-----
From: owner-nmusers
Behalf Of 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