David Müller-Widmann 1, Andreas Noack 1
1 PumasAI (Dover, USA)
Introduction:
Expectation-maximization (EM) algorithms are a well-established alternative to likelihood-approximation-based methods such as FOCE and the Laplace method for parameter estimation in nonlinear mixed-effect models in pharmacometric analysis.
NONMEM’s implementation of Monte Carlo parametric expectation-maximization (MC-PEM) [1,2] uses MAP-centered importance sampling and a numerical optimization M-step [3].
However, inter-individual variability is restricted to normal distributions and transformations of normal distributions such as log-normal and logit-normal distributions, limiting the ability to capture non-standard variability patterns.
Pumas introduces a new experimental MCEM implementation that exploits the exponential family structure of random effect distributions, enabling native support for non-Gaussian random effects including beta, gamma, and inverse-gamma distributions.
Objectives:
– Study the bias and variability of Pumas MCEM estimates on a model with log-normally distributed random effects and compare it to estimation with FOCE and the Laplace method
– Demonstrate the ability of Pumas MCEM to fit models with non-Gaussian random effect distributions by estimating a model with gamma-distributed inter-individual variability
Methods:
Two simulation scenarios were designed.
In the first scenario, a one-compartment IV bolus model with log-normal random effects on volume of distribution and elimination rate, a proportional Gaussian error model, and sparse sampling (3 timepoints) of 1000 subjects, evaluated on 500 simulated datasets.
The design of this scenario closely follows Example 1 of Bauer and Guzy [2].
In the second scenario, the same one-compartment IV bolus model with a gamma-distributed random effect on elimination rate, a log-normal random effect on volume of distribution, a proportional Gaussian error model, and 1000 subjects at the same 3 timepoints, evaluated on 500 simulated datasets.
In both scenarios, parameter estimation was performed using MCEM, FOCE, and the Laplace method in Pumas.
In all simulations, MCEM was run with 50 iterations and 100 initial Monte Carlo samples (increasing polynomially per Fort and Moulines [4]).
Simulations were evaluated based on the mean and standard deviation of parameter estimates, empirical CV of estimates, and absolute and relative bias (%).
Results:
In the first scenario, MCEM showed the smallest relative bias among the three methods, with a maximum absolute relative bias of 0.80% across all parameters, compared to 1.19% for the Laplace method and 3.48% for FOCE.
Empirical CVs of estimates were comparable across all three methods (within 0.2 percentage points for each parameter), indicating that the stochastic nature of MCEM did not introduce appreciable additional variability beyond the sampling variability inherent in the simulation design.
In the second scenario, MCEM again showed the smallest maximum absolute relative bias (1.65%), compared to 2.50% for FOCE and 2.52% for the Laplace method.
Empirical CVs were again comparable across all three methods (within 0.1 percentage points).
Conclusions:
The exponential family sufficient statistics framework underlying Pumas MCEM provides distributional flexibility through native support for non-Gaussian random effects, thereby
extending the range of inter-individual variability models available to pharmacometricians.
The implementation is under active development, with improvements to proposal distributions, convergence diagnostics, and post-estimation inference planned for upcoming releases.
References:
[1] Chan, K. S., & Ledolter, J. (1995). Monte Carlo EM Estimation for Time Series Models Involving Counts. Journal of the American Statistical Association, 90(429), 242–252. https://doi.org/10.1080/01621459.1995.10476508
[2] Bauer, R. J., & Guzy, S. (n.d.). Monte Carlo Parametric Expectation Maximization (MC-PEM) Method for Analyzing Population Pharmacokinetic/Pharmacodynamic Data. In The International Series in Engineering and Computer Science (pp. 135–163). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-48523-0_7
[3] Bauer, R. J. (2019). NONMEM Tutorial Part II: Estimation Methods and Advanced Examples. CPT: Pharmacometrics & Systems Pharmacology, 8(8), 538–556. https://doi.org/10.1002/psp4.12422
[4] Fort, G., & Moulines, E. (2003). Convergence of the Monte Carlo expectation maximization for curved exponential families. The Annals of Statistics, 31(4). https://doi.org/10.1214/aos/1059655912
Reference: PAGE 34 (2026) Abstr 11878 [www.page-meeting.org/?abstract=11878]
Poster: Methodology - Estimation Methods