Efficient Bayesian inference for non-linear association structures in joint models: A hierarchical approach via INLA

2026-04-20

AI Summary

This study addresses the limitation of conventional joint models that assume a linear relationship between biomarkers and the log-hazard of clinical events, which often fails to capture complex nonlinear associations such as U-shaped patterns. The authors propose a hierarchical Bayesian joint model based on integrated nested Laplace approximation (INLA), employing orthogonal basis functions and a second-order random walk prior to decompose the association structure into a parametric baseline component and a nonparametric smooth deviation term, thereby enabling data-driven flexible modeling. The framework facilitates formal hypothesis testing of linearity and automatically balances model complexity via information criteria. Simulation studies demonstrate accurate recovery of nonlinear trajectories, and application to the Health and Retirement Study reveals a U-shaped association between BMI and mortality risk, along with a nonlinear high-risk effect linked to the rate of weight loss.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Machine Learning: Bayesian Learning

Application Category

• User Modeling, Personalization and Recommendation: User model development and evaluation
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Joint models for longitudinal and time-to-event data are increasingly used in health research to characterize the association between biomarker trajectories and the risk of clinical events. However, these models usually assume a linear relationship between the longitudinal marker and the log-hazard of the event. This assumption is rarely verified and often fails to capture complex biological mechanisms, such as U-shaped risk profiles or plateau effects. In this paper, we propose a fast and stable hierarchical framework for non-linear association structures in joint models using Integrated Nested Laplace Approximations (INLA), implemented in the INLAjoint R package. Our approach builds upon a unified framework where the scaling effect of the marker is decomposed into a parametric baseline (constant and linear components) and a data-driven smooth deviation modeled via an orthogonal basis derived from a second-order random walk. This natural hierarchy allows researchers to adapt model flexibility directly and verify the linearity assumption using standard information criteria. Through simulation studies, we demonstrate that the proposed method accurately recovers complex non-linear trajectories. We illustrate the practical utility of our framework by analyzing the joint association of the current value and current slope of body mass index (BMI) with all-cause mortality in the Health and Retirement Study. This analysis reveals a U-shaped mortality risk for the BMI value, and a non-linear effect for the rate of weight change, where a declining weight trajectory is associated with higher mortality risk.

Problem

• joint models
• non-linear association
• longitudinal and time-to-event data
• linearity assumption
• biomarker trajectories

Innovation

• non-linear association
• joint models
• INLA
• hierarchical modeling
• orthogonal basis