RE: 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. The FOCE method adds 1 additional approximation to get the 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 that the additional FO approximations 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 This email message (including any attachments) is for the sole use of the intended recipient and may contain confidential and proprietary information. Any disclosure or distribution to third parties that is not specifically authorized by the sender is prohibited. If you are not the intended recipient, please contact the sender by reply email and destroy all copies of the original message.