From: "HUTMACHER, MATTHEW [Non-Pharmacia/1820]" <matthew.hutmacher@pharmacia.com>
Subject: [NMusers] FOCE objective function
Date: Mon, 12 Nov 2001 15:02:44 -0600
Hello all,
I am sure I am missing something simple here, but could someone explain to
me where the last term in the objective function written on page 4 (Chapter
II) of Manuel 7 comes from? It is the term,
"-(0.5*G+O^(-1)*E)`(O^-1+0.5*H)^(-1)(0.5*G+O^-1*E)" where G=gradient,
O=omega, and H=hessian. I was just working through the derivation for my on
edification and I could only get the first four pieces. Using the notation
from Chapter I and suppressing the subscript i, then l(psi,eta) is the
conditional and arbitrary likelihood and h(eta;O) is the multivariate normal
distribution with mean 0 and covariance matrix O. Let M=l(psi,eta)h(eta;O)
and m()=log{M()}=log{l(psi,eta)}+
log{h(eta:O)}. By the general Laplacian approximation method, the marginal
likelihood is represented by the integral (let INT stand for taking the
integral), INT{exp(m)*d(eta)}=k*|-m"(etahat)|^(-0.5)*exp(m(etahat)), where
etahat maximizes m() and k is a constant. When I take
-2log{INT{exp(m)*d(eta)}}, I get -->
-2log{l(psi,eta)}+log|O|+log|O^(-1)+0.5*H|+etahat`O^(-1)etahat, where
H=l"(psi,eta). Could someone please explain to me my error?
Thanks for your time.
Matt
FOCE objective function
From: "LSEN (Lars Erichsen)" <lsen@novonordisk.com>
Subject: RE: [NMusers] FOCE objective function
Date: Tue, 13 Nov 2001 12:47:41 +0100
In the usual Laplace approximation of the integral,
the first derivative of the exponent is 0 in etahat,
because etahat maximises the exponent. This is not assumed in
the derivation in the NONMEM manual Part VII on page 4 which is why
the last term is not zero.
The integral is L=Int[exp(-0.5*(Phi(eta)+eta'*O*eta))]
where O=inverse of Omega
If etahat minimises Phi(eta)+eta'*O*eta
we have that (Gamma=derivative of Phi):
Gamma(etahat)+2*O*etahat=0, so that
0.5*Gamma(etahat)+O*etahat=0.
I hope this is of use.
Lars Erichsen
Pharmacokinetics, Pharmacometrics
Novo Nordisk
Denmark.