Conditional bootstrap for non-linear mixed effects models

2026-05-04

AI Summary

This study addresses the challenge in nonlinear mixed-effects models where conventional methods often fail to adequately account for the hierarchical structure of between-subject and within-subject variability, leading to undercoverage of confidence intervals—particularly for variance components. To overcome this limitation, the authors propose a conditional nonparametric bootstrap (cNP) approach that innovatively integrates the conditional distribution of individual random effects estimated via the SAEM algorithm with residual resampling. This method preserves the original data’s sample size, covariate distribution, and hierarchical structure without requiring explicit stratification. Simulation studies implemented using the saemix package demonstrate that, across various designs and levels of residual variability, cNP substantially improves coverage probabilities compared to classical nonparametric and case bootstrap methods, while excelling in maintaining the integrity of the data’s inherent structure.

Abstract

Background and Objective: Uncertainty in non-linear mixed effect models is often assessed using the Fisher information matrix to derive the standard errors of estimation. The bootstrap is an alternative to the asymptotic method, with different approaches to handle the different levels of individual and population variabilities. The simplest method is the Case bootstrap where the entire vector of individuals is resampled, but this approach does not take into account the hierarchical nature of non-linear mixed effect models (NLMEM). Methods: We propose here a non-parametric bootstrap, cNP, to preserve the structure of the original data. We resample interindividual random effects from the conditional distribution of the individual parameters, obtained as a by-product of the SAEM algorithm, and residuals from their distribution. cNP was implemented in the saemix package for R along with the case, parametric (Par), and non-parametric (NP) residual bootstraps. Coverage rates were compared in a simulation study using sigmoid Emax models, with rich, sparse and unbalanced designs, and 3 levels of residual variability. Results: The asymptotic method tended to produce lower than theoretical coverages for the variance terms. Bootstraps provided more adequate coverage, but none of the approaches maintained coverage when the residual error increased. Overall, the new cNP and the Case provided better coverage than the classical NP. Conclusion: The new conditional non-parametric bootstrap can be used when it is important to preserve the structure of the original dataset, such as the number of observations or the repartition of covariates as it does not require stratification.

Problem

• non-linear mixed effects models
• uncertainty assessment
• bootstrap
• hierarchical structure
• residual variability

Innovation

• conditional non-parametric bootstrap
• non-linear mixed effects models
• SAEM algorithm
• coverage rate
• hierarchical data structure