# RE: OFV higher with FOCEI than FO

From: Matt Hutmacher <matt.hutmacher>
Date: Thu, 11 Dec 2008 10:45:23 -0500

Yaning,

Perhaps I was not clear in my email. I should have stated it more
explicitly in the following;

For the normal density case then application of the Laplace approximation
yields

-2LL = (y-f(eta))'SIG^-1(y-f(eta)+eta'*OM^-1*eta+log|SIG|

Where y are the data, f is the mean function, eta is the subject specific
random variable, SIG is the intrasubject residual variance, OM is the
between subject variance of the etas. If SIG depends on eta, then the
extended least squares form, ie

-2LL =( y-f(etahat)-G*etahat)'MSIG^-1(y-f(etahat)-Getahat)+log(MSIG)

Where MSIG=G*OM^-1*G+SIG no longer represents a Laplace based approximation
to the marginal distribution of y. Now it can be approximately Laplacian
based by various procedures, but it is not Laplacian based anymore.

See Page 345 of Vonesh. Note that Wolfinger shows this derivation.

Matt

From: owner-nmusers
Behalf Of Wang, Yaning
Sent: Wednesday, December 10, 2008 8:45 PM
To: Matt Hutmacher; Bob Leary; ayyappa.5.chaturvedula
owner-nmusers
Subject: RE: [NMusers] OFV higher with FOCEI than FO

Matt:

That's not true. Those two references are discussing when the linearized
structure model can also be derived from direct Laplacian approximation of
the marginal likelihood. When there is an interaction between residual and
between subject variability (or residual error model contain
subject-specific random effect), linearizing the structure model around
eta_hat cannot be derived from the Laplacian approximation any more. But in
NONMEM, FOCE with interaction (when residual error model contain
subject-specific random effect) is still derived from Laplacian
approximation. In other words, NONMEM does not linearize the structure model
for FOCE with interaction case. I discussed this in details in my paper (1).
Adding the following splus code to the splus code in my paper and using the
simple numerical example, you can see how NONMEM is calculating the
objective function for FOCE with interaction. These things are further
visualized in my talk recently put on ACCP webpage
(http://www.accp1.org/pharmacometrics/PopPKCourse.html).

Yaning

#reproduce NONMEM result using my equation 28 which is further approximation
of Laplacian method
sum<-0
for (i in 1:10) {
data1<-data[data\$ID==i,]
cov<-data1\$fp%*%t(data1\$fp)*omega+diag(data1\$f**2)*eps+2*data1\$fp%*%t(data1\$
fp)*omega*eps
cov1<-diag(data1\$f**2)*eps
ginv<-solve(cov1)
sec<-t(data1\$DV-data1\$IPRE)%*%ginv%*%(data1\$DV-data1\$IPRE)+data1\$ETA1[1]**2/
omega
frs<-determinant(cov, logarithm=T)\$modulus[[1]]
sum1<-sec+frs
sum<-sum+sum1
}
sum#39.45756 same as NONMEM OFV 39.458

1. Yaning Wang. Derivation of various NONMEM estimation methods. Journal of
Pharmacokinetics and pharmacodynamics. 34:575-93 (2007)

Yaning Wang, Ph.D.
Office of Clinical Pharmacology
Office of Translational Science
Center for Drug Evaluation and Research
Phone: 301-796-1624
Email: yaning.wang

"The contents of this message are mine personally and do not necessarily
reflect any position of the Government or the Food and Drug Administration."

_____

From: owner-nmusers
Behalf Of Matt Hutmacher
Sent: Wednesday, December 10, 2008 2:04 PM
To: 'Bob Leary'; ayyappa.5.chaturvedula
owner-nmusers
Subject: RE: [NMusers] OFV higher with FOCEI than FO

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

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

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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