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. ------------------