Robert H. Leary and Michael R. Dunlavey
Pharsight Corporation
Context: For EM-based NLME algorithms such as QRPEM in Phoenix NLME and IMP in NONMEM, a Monte Carlo or quasi-Monte Carlo integration to estimate moments of a target posterior is performed using (usually) a Gaussian importance sampling distribution (ISD). The ISD is tuned to mimic the target distribution fairly closely in the modal region, while maintaining somewhat wider tails. Adequately satisfying both objectives can be difficult, but is critically important to the success of the overall NLME algorithm. The ISD mean and covariance are typically based on the estimated mean and covariance of the posterior from the previous iteration, but with a scale factor>1 applied to the covariance to widen the tails. The window of workable scale factors may be narrow, not knowable in advance, and the usual default value can easily fail.
Methods: The work of Hesterberg [1] suggests that the use of an ISD that is a ‘defensive mixture’ of two or more Gaussian distributions with differing tail coverage characteristics can be much more robust than a single Gaussian. Owen and Zhou [2] suggest use of Hesterberg’s defensive mixtures in conjunction with control variates to improve accuracy and avoid possible pathologies in the mixture sampling case. We introduced both techniques into the QRPEM algorithm and compared the results with the traditional single Gaussian ISD on variety of models.
Results: Greatly improved robustness with respect to user mis-specification of the covariance scaling parameter was demonstrated with a defensive mixture of two Gaussian components relative to a single Gaussian. The width of the window of workable scaling factors increased dramatically, typically by more than an order of magnitude, thereby greatly reducing the possibility of NLME failure due to a poorly scaled Gaussian ISD. With respect to control variates, we could not detect any examples of the types of pathologies suggested in [2], but did find that the use of control variates noticeably improved overall accuracy and precision for a given sample size.
Conclusions: Defensive mixture importance sampling greatly improves the robustness of importance sampling-based EM algorithms for PK/PD NLME estimation.
References:
[1] Hesterberg, T. “Weighted Average Importance Sampling and Defensive Mixture Distributions”, Technical Report No. 148, Division of Biostatistics, Stanford University, Stanford, California, 1991.
[2] Owen, A. and Y. Zhou, “Safe and Effective Importance Sampling”, JASA 95, 2000, pp. 135-143.
Reference: PAGE 23 (2014) Abstr 3120 [www.page-meeting.org/?abstract=3120]
Poster: Methodology - Estimation Methods