Neural ODE enhanced linear mixed effect models for estimating complex association patterns of time-varying covariates with the marker trajectory Supplementary Materials
Abstract
Longitudinal cohort studies produce repeated data that enable the assessment of time-varying association patterns between exposures and health outcomes. Classical linear mixed-effects models (LMMs) can accommodate a large variety of association patterns while accounting for the irregularly spaced, partially observed measurement. But they require the analyst to pre-specify the functional form linking the exposure history to the outcome. We propose the Neural ODE-LMM, which embeds a Neural Or- dinary Differential Equation (Neural ODE) within the linear mixed-effects framework: a learned vector field encodes covariate trajectories into a continuous-time latent state that drives both the fixed- and random-effect design, while preserving the standard LMM observation model. This retains classical likelihood-based inference while learn- ing complex, potentially cumulative, covariate effects flexibly. All parameters are esti- mated by maximising a penalised marginal likelihood. To quantify covariate effects, we introduce contrasts of counterfactual predictions that compare the expected outcome under alternative covariate trajectories with variance estimated via the delta method. In simulations, the model recovers both instantaneous and cumulative-burden effects without prior specification of the functional form. Applied to the Trois-Cit´es (3C) cohort, a population-based study of 7,324 participants, the method reveals trajectory- dependent associations of BMI and fasting glucose with cognitive decline. Neural ordinary differential equations; linear mixed-effects models; longitudinal data; counter- factual predictions; cognitive aging.
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BibTeX
@misc{li2026neural,
title = {{Neural ODE enhanced linear mixed effect models for estimating complex association patterns of time-varying covariates with the marker trajectory Supplementary Materials}},
author = {Zhe Aurore Li and Quentin Clairon and Cécilia Samieri and Rodolphe Thiébaut and Mélanie Prague and Cécile Proust-Lima},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05761770v1},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - Neural ODE enhanced linear mixed effect models for estimating complex association patterns of time-varying covariates with the marker trajectory Supplementary Materials
AU - Zhe Aurore Li
AU - Quentin Clairon
AU - Cécilia Samieri
AU - Rodolphe Thiébaut
AU - Mélanie Prague
AU - Cécile Proust-Lima
PY - 2026
DA - 2026/09/23
PB - HAL
UR - https://hal.science/hal-05761770v1
AB - Longitudinal cohort studies produce repeated data that enable the assessment of time-varying association patterns between exposures and health outcomes. Classical linear mixed-effects models (LMMs) can accommodate a large variety of association patterns while accounting for the irregularly spaced, partially observed measurement. But they require the analyst to pre-specify the functional form linking the exposure history to the outcome. We propose the Neural ODE-LMM, which embeds a Neural Or- dinary Differential Equation (Neural ODE) within the linear mixed-effects framework: a learned vector field encodes covariate trajectories into a continuous-time latent state that drives both the fixed- and random-effect design, while preserving the standard LMM observation model. This retains classical likelihood-based inference while learn- ing complex, potentially cumulative, covariate effects flexibly. All parameters are esti- mated by maximising a penalised marginal likelihood. To quantify covariate effects, we introduce contrasts of counterfactual predictions that compare the expected outcome under alternative covariate trajectories with variance estimated via the delta method. In simulations, the model recovers both instantaneous and cumulative-burden effects without prior specification of the functional form. Applied to the Trois-Cit´es (3C) cohort, a population-based study of 7,324 participants, the method reveals trajectory- dependent associations of BMI and fasting glucose with cognitive decline. Neural ordinary differential equations; linear mixed-effects models; longitudinal data; counter- factual predictions; cognitive aging.
ER -