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.
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