Laplacian Estimation Method with Mean-Variance Model and Interaction
From: S.Beal [stuart@c255.ucsf.edu]
Subject: Laplacian Estimation Method with Mean-Variance Model and Interaction
Date: July 16 2002
Pharmacokinetic models and certain other models for a continuous outcome
are generally mean-variance models (see Users Guide VII). With a mean-
variance model, one may use the INTERACTION option - which specifies that
the estmation method is to recognize the existence of an eta/epsilon
interaction - along with the First-Order Conditional Estimation method.
However, with NONMEM Version V, one cannot use the INTERACTION option along
with the Laplacian Estimation method. Some NONMEM users are aware that
with Version VI (in progress), this will be possible. However, doing this
will necessitate the use of the NUMERICAL option, which can compromise pre-
cision and introduce some instability. Here is a way to use the Laplacian
estimation method with Version V in a way that recognizes an eta/epsilon
interaction, and without necessitating the use of the NUMERICAL option.
The technique takes advantage of the LIKELIHOOD option (more precisely, the
-2LL option), and it works as long as L2 records and intraindividual corre-
lations are absent. When the LIKELIHOOD option is used, the model is not
recognized by NONMEM to be a mean-variance model, and so the existence of
epsilons is not recognized. The user will need to model the variance of an
epsilon variable as a theta, with a lower bound of 0 to keep the variance
positive.
Suppose the equation for Y (in epsilons) is given schematically by
PRD = ... ;prediction for the observation
ER = ... ;intraindividual error
Y=PRD+ER
ER could be PRD*EPS(1), or PRD*EPS(1)+EPS(2), or any other error model
involving epsilon variables.
Replace the above with:
PRD = ... ;prediction for the observation
VAR = ... ;variance for intraindividual error
Y=LOG(VAR)+(DV-PRD)**2/VAR
If ER is given as above,
VAR would be PRD**2*THETA(5), or PRD**2*THETA(5)+THETA(6),
where THETA(5) and THETA(6) are the variance parameters of EPS(1) and EPS(2)
(the subscripts 5 and 6 are just taken as examples).
The $ESTIMATION record should include the options MET=COND, LAPLACE, and
-2LL (but not INTER).
There should be no $SIGMA record (epsilons are not recognized when -2LL is
used), and both THETA(1) and THETA(2) (which replace SIGMA11 AND SIGMA22)
should have lower bounds of 0.