Patrick Kofod Mogensen 1, Vijay Ivaturi 1
1 PumasAI (Dover, USA)
Objectives:
Nonlinear mixed-effects models typically assume multivariate normal random effects to describe inter-individual variability. In some applications, more flexible marginal distributions may be desirable to accommodate skewness, heavy-tailed behavior or bounded domains while preserving structured dependence between model parameters. Copulas provide a mechanism to construct joint distributions by combining arbitrary marginal distributions with a specified dependence structure. The objective of this work was to demonstrate simulation and maximum likelihood estimation of correlated non-Gaussian random effects in Pumas.jl [1] using copulas, and to assess the practical feasibility of this approach within a standard nonlinear mixed-effects modeling workflow.
Methods:
A copula-based framework for random effects was implemented in Pumas.jl by integrating functionality from the Copulas.jl [2] package. This extends the use of the Distributions.jl [3] interface to work with correlated continuous random variables that are not Gaussian. Marginal random effects were specified using non-Gaussian distributions from Distributions.jl. Specifically, LogNormal and Gamma distributions for random effects that are positive and unbounded and the Beta distribution for a random effect associated with bioavailability. Dependence between random effects was introduced through the results of Sklar’s Theorem, allowing separate specification of marginal behavior and correlation structure.
Random effects were generated by sampling from the chosen copula directly through the Pumas modeling language. The resulting joint density was incorporated into the nonlinear mixed-effects likelihood formulation in Pumas.jl. Maximum likelihood estimation was performed using first-order approximation methods, enabling simultaneous estimation of marginal distribution parameters and copula dependence parameters. Sampling from copulas is possible in many statistical software systems and has been used in pharmacometrics in the past, for example to simulate Virtual Patients in QSP [4]. However, the direct integration into the pharmacometric modelling as random effects is unique and utilizes the flexibility of Pumas as well as the Julia ecosystem.
The framework was demonstrated using pharmacokinetic models with multiple correlated random effects. Simulation studies show that the approach successfully is able to recover parameters, has no practical numerical stability issues, and convergence behavior under the copula-based specification.
Results:
The copula-based random effects structure was successfully implemented within Pumas.jl, enabling both simulation and likelihood-based estimation of correlated non-Gaussian random effects using first-order methods. Simulation experiments demonstrated consistent recovery of marginal distribution parameters and copula dependence parameters across repeated datasets. Estimation remained numerically stable, with reliable convergence and well-behaved likelihood profiles. The implementation integrates seamlessly with existing Pumas modeling workflows, requiring only modest extensions to standard model specifications.
Conclusions:
Copulas provide a flexible and practical approach for constructing correlated non-Gaussian random effects in nonlinear mixed-effects models. By integrating Copulas.jl with Pumas.jl, simulation and maximum likelihood estimation of such models can be performed within a familiar pharmacometric framework. This approach expands the modeling flexibility available to pharmacometricians while maintaining compatibility with established estimation procedures and workflows.
References:
[1] Rackauckas C. et al., https://doi.org/10.1101/2020.11.28.402297
[2] https://github.com/lrnv/Copulas.jl
[3] https://github.com/JuliaStats/Distributions.jl
[4] PAGE 30 (2022) Abstr 10099 [www.page-meeting.org/?abstract=10099]
Reference: PAGE 34 (2026) Abstr 12122 [www.page-meeting.org/?abstract=12122]
Poster: Methodology - Covariate/Variability Models