# 2004

Uppsala, Sweden

**Accurate Maximum Likelihood Estimation for Parametric Population Analysis **

R. H. Leary (1), R. Jelliffe (2), A. Schumitzky (2), R.E. Port (3)

(1) San Diego Supercomputer Center, University of California, San Diego; (2) Laboratory of Applied Pharmacokinetics, University of Southern California School of Medicine, Los Angeles; (3) German Cancer Research Center, Heidelberg

**Objectives:** The statistical performance of estimators of fixed and random effects in parametric population PK analyses can be strongly degraded by the use of likelihood approximations such as FO and FOCE. Here we investigate the computational viability and statistical performance of a new implementation of a parametric expectation-maximization (PEM) method originally formulated by Schumitzky in 1994 [1] that maximizes an accurate parametric likelihood function.

**Methods:** The PEM algorithm alternates between computing various integrals representing expectations over the random effects distribution (the expectation step) and updating the random effects distribution (the maximization step). Remarkably, even though the accurate likelihood function is being maximized, no explicit evaluations of that function are necessary, although the expectation integrals are similar in form and computational difficulty to the integrals defining the likelihood function. The true likelihood can be shown to increase monotonically during each EM step provided the expectation integrals are evaluated accurately. The required numerical integrals were computed with high accuracy numerical techniques using quadrature points defined by low discrepancy (quasirandom) sequences. The algorithm was also instrumented with an explicit accurate likelihood evaluation at each iteration to verify the theoretical monotonic improvement of the likelihood function and convergence to a maximum. A variety of simulated one compartment IV bolus test cases spanning both sparse and rich data cases as well as different levels of inter-individual variability were analyzed with PEM as well as methods (IT2B, NONMEM) employing FO and FOCE integral approximations.

**Results:** Under all conditions tested the PEM algorithm converged rapidly and monotonically to a maximum likelihood solution, typically in 20-30 iterations. This was somewhat surprising, given the known linear (and often slow) convergence rate for EM algorithms in other contexts such as mixing distribution estimation and nonparametric population PK analysis. Analyses of individual cases generally took on the order of 1 to 10 minutes for several hundred simulated subjects on a current generation PC. The algorithms that use approximations converged to solutions that were suboptimal to varying degrees with respect to the true likelihood, with the FO method typically showing the largest likelihood deviation from the maximum likelihood solution and the poorest statistical performance. Overall, the statistical performance of the PEM estimators was markedly superior with respect to bias, consistency, and in particular statistical efficiency, the latter by factors up to 3 relative to FOCE and 100 relative to FO for cases with sparse data and/or high inter-individual variability.

**Reference: **[1] A. Schumitzky, A, EM Algorithms and Two Stage Methods in Pharmacokinetic Population Analysis, LAPK Technical Report 94-3; also in D. Z. D'Argenio, ed., Advanced Methods of Pharmacokinetic and Pharmacodynamic Systems Analysis, Plenum Press, New York, 1995, pp. 145-160.