**QRPEM, A Quasi-Random Parametric EM Method **

Robert H. Leary* and Michael Dunlavey

Pharsight Corporation, Cary, NC, USA

**Context:** Monte Carlo (MC) parametric EM algorithms such as MCPEM and SAEM are attractive alternatives to traditional FO, FOCE and LAPLACE parametric algorithms. Recently QRPEM (Quasi-Random Parametric EM), a new implementation of an importance sampling based parametric EM algorithm, has been added to the Pharsight Phoenix NLME software. QRPEM differs fundamentally from MCPEM algorithms in that sampling of the posterior distributions is based on "quasi-random" sequences rather than the more usual pseudo-random sequences. Additionally, QRPEM implements the SIR (Sampling-Importance-Resampling) algorithm (Ref. [1, 2]) as an accelerant for cases where mu-modeling is not or cannot be used.

**Objectives:** Implement the QRPEM algorithm, including the SIR acceleration technique for non-mu-modeled cases, and verify

- a) QRPEM parameter estimates usually closely approximate true maximum likelihood estimates
- b) The theoretical advantages of QR relative to MC sampling are usually achieved in practice
- c) The SIR algorithm is an effective accelerant for the non-mu-modeled case.

**Methods:** The QRPEM algorithm was implemented with Sobol low discrepancy sequences and a SIR algorithm for accelerating non-mu-modeled cases. Test cases were selected from the literature, including the Monolix test cases from Ref. [3], the PK-PD models from Ref. [4], as well as locally generated. Comparative evaluations were performed among QRPEM with and without SIR acceleration, Adaptive Gaussian Quadrature (AGQ), MCPEM (QRPEM with QR sampling replaced by MC sampling), and ELS FOCE (with interaction). Additionally, individual posterior integrals were extracted from several test cases and evaluated using QR and MC sampling to compare the error decay rates with increasing sample size.

**Results:**

- a) Evaluation of individual integrals consistently confirmed the theoretically superior O(
*N*) vs. O(^{-1}*N*) asymptotic error decay behaviour of QR sampling.^{-1/2} - b) QRPEM and AGQ typically converged to nearly the same parameter and likelihood values, which were usually closer (and in some cases much closer) to the true ML estimates than MCPEM at the same sample size. .
- c) RMSE of parameter estimates for replicates of simulated data sets typically are smaller with QRPEM than ELS FOCE.
- d) The SIR technique is remarkably effective, generally providing speed-ups of at least 2X and in some cases up to 10X without significant degradation of parameter estimates.

**References:**[1] Rubin, D.B. (1987)

*J. Am. Stat. Assoc.*, 82, 543-546 (1987).

[2] J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer, 2008.

[3] Exprimo, "Software Evaluation: Simulation of PK Datasets for the evaluation of the Monolix PK Library", June 6, 2007, Monolix Project.

[4] J. C. Pinheiro and D. M. Bates, Mixed Effects Models in S and S-PLUS, Springer,2000.