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Functional forms in joint models for longitudinal and time-to-event data: A practical guide with application and interpretation

This paper provides a practical guide demonstrating that the choice of functional form in joint models for longitudinal and time-to-event data is a critical modeling decision that fundamentally shapes the interpretation of biomarker-risk associations, as illustrated through a comparative analysis of various association structures applied to glioblastoma trial data.

Original authors: Felix Boakye Oppong, Dimitris Rizopoulos, Thierry Gorlia, Nicole Erler

Published 2026-08-20

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CC BY 4.0

Original authors: Felix Boakye Oppong, Dimitris Rizopoulos, Thierry Gorlia, Nicole Erler

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern medicine, doctors often track patients over years, watching how specific numbers in their blood or tissue change day by day. These numbers, known as biomarkers, tell a story about how a disease is behaving. At the same time, researchers are trying to understand when a patient might face a serious health event, such as a relapse or death. For a long time, scientists treated these two stories separately: one story about the changing numbers, and another about the timing of the event. But in reality, these stories are deeply intertwined. A patient's risk of a bad outcome is not just about where their numbers are right now, but how they got there. Did the numbers rise slowly over months, or did they spike suddenly last week? Did they bounce up and down wildly, or did they stay steady? Understanding the precise shape of this journey is crucial for predicting the future, yet figuring out exactly how to translate a complex medical history into a risk prediction has been a difficult puzzle.

A team of researchers set out to solve this puzzle by creating a practical guide for how to connect these two stories. They focused on a specific type of statistical tool called a joint model, which is designed to look at repeated measurements and survival times together. The core of their work was not about inventing new math, but about clarifying how to choose the right "lens" to view the data. They argued that the way a researcher decides to link a biomarker's history to a patient's risk is a critical scientific choice, not just a technical detail. If the wrong lens is chosen, the resulting conclusions about what drives the disease could be misleading. To demonstrate this, they applied their guide to real data from a major clinical trial involving patients with a type of brain cancer called glioblastoma. They used white blood cell counts, a standard measure of immune function, to show how different ways of looking at the same data could lead to different, and sometimes conflicting, understandings of the patient's risk.

The researchers identified several distinct ways to describe a patient's biomarker history, each capturing a different aspect of their medical journey. The most common approach simply looks at the current level of the biomarker at any given moment. It asks, "How high is the white blood cell count right now?" and assumes that this single number determines the risk. However, the team showed that this view misses the bigger picture. Two patients could have the exact same white blood cell count on a specific day, but one might have been rising steadily for weeks while the other was falling. If the speed of that change matters, the simple current-level view fails to distinguish between them.

To address this, the researchers explored more dynamic lenses. One approach looks at the speed of change, or the velocity, asking whether the biomarker is climbing or dropping at this exact moment. Another looks at the acceleration, or how quickly that speed is changing, capturing whether a rise is speeding up or slowing down. They also examined cumulative approaches, which look at the total exposure a patient has had to a biomarker over time. This is like calculating the total area under a curve, representing the burden of the disease over months or years, rather than just a snapshot. They also tested methods that focus on recent changes over a specific window, such as the last thirty days, to see if recent deterioration is more dangerous than long-term trends. Finally, they looked at variability, measuring how much a patient's numbers fluctuate around their average. A patient whose numbers swing wildly might be at higher risk than a patient with the same average but a steady line, because that instability could signal a body struggling to regulate itself.

When the team applied these different lenses to the data from the brain cancer trial, the results highlighted the importance of the chosen method, though the statistical evidence was often uncertain. When they looked only at the current white blood cell count, the link to survival suggested a possible positive association, but with considerable uncertainty as the confidence interval included zero. Similarly, when they looked at the cumulative exposure—the total burden of white blood cells over time—the association was statistically significant, yet the magnitude of the effect required careful interpretation. In other cases, looking at the rate of change or the variability provided different numerical estimates, but the wide confidence intervals for these measures indicated substantial uncertainty and weak evidence for a clear association in this specific dataset. The study found that these different functional forms are not interchangeable; they measure different biological realities. A large number in one model does not mean the same thing as a large number in another, because they are measuring different things, such as a current level versus a total history.

The researchers emphasized that there is no single "best" way to connect these data points. The right choice depends entirely on the biological question being asked. If a disease is driven by a sudden spike, looking at the current level might be best. If it is driven by years of exposure, a cumulative measure is necessary. If the instability of the system is the danger, then measuring variability is key. The paper concludes that researchers must carefully align their statistical choices with their scientific understanding of the disease. By treating the choice of how to link the data as a fundamental part of the scientific inquiry, rather than a hidden technical step, doctors and scientists can avoid drawing misleading conclusions. This clarity ensures that when a model predicts a patient's risk, it is based on the specific feature of their health history that actually matters, leading to more accurate predictions and better care.

Technical Summary: Functional Forms in Joint Models for Longitudinal and Time-to-Event Data

Problem Statement

Joint models for longitudinal and time-to-event data are established tools in clinical research for analyzing the relationship between repeated biomarker measurements and event risks (e.g., death, disease progression). However, a critical yet often under-utilized aspect of these models is the choice of the functional form (or association structure). This choice defines how the biomarker trajectory relates to the event hazard.

Currently, researchers frequently rely on default specifications (typically the "current value" of the biomarker) without considering whether this form captures the clinically relevant characteristics of the biomarker's evolution. Different characteristics—such as the rate of change, cumulative exposure, or variability—may influence prognosis in distinct ways. Ignoring these aspects can lead to incomplete or misleading scientific conclusions regarding the mechanisms linking biomarkers to clinical outcomes. Furthermore, the interpretation of association parameters is heavily dependent on the chosen functional form, making effect sizes non-comparable across different specifications.

Methodology

The paper provides a structured overview and practical guide to various functional forms used to link longitudinal and survival processes. The methodology involves:

Theoretical Classification: The authors categorize association structures into:
- Instantaneous forms: Linking hazard to features evaluated at time $t$ (Current Value, Time-dependent Slope/Velocity, Acceleration/Curvature).
- Cumulative forms: Summarizing historical exposure (Area Under the Curve, Restricted Cumulative, Weighted Cumulative).
- Change-based forms: Quantifying deviations from baseline or recent windows (Delta, Restricted Delta).
- Shared Random Effects: Linking hazard directly to latent subject-specific deviations.
- Variability-based forms: Linking hazard to the instability or variance of the biomarker trajectory.
Implementation and Software: The guide focuses on the implementation of these forms using the R package JMbayes2 (version 0.6-0). The paper details the syntax required to specify these structures, noting that while some forms (like slope and acceleration) require numerical derivatives (finite differences) or specific design matrices, JMbayes2 handles these computations internally.
Empirical Illustration: The authors apply these functional forms to data from the MIRAGE glioblastoma trial (EORTC 1709 / Canadian Cancer Trials Group CE.8).
- Data: 680 patients with newly diagnosed glioblastoma.
- Longitudinal Outcome: White Blood Cell (WBC) count, measured weekly during radiotherapy.
- Time-to-Event Outcome: Overall Survival (OS).
- Modeling: A linear mixed-effects model with natural cubic splines was used for the longitudinal submodel to capture non-linear WBC trends. A Cox proportional hazards model was used for the survival submodel, adjusted for baseline covariates (age, treatment, surgery extent, KPS, MGMT status).
- Variability Approach: For the variability-based form, a two-step approach was employed: first, fitting a mixed model to extract absolute residuals; second, modeling these residuals as a new longitudinal outcome linked to the hazard.

Key Contributions

Comprehensive Taxonomy: The paper systematically classifies and compares a wide range of association structures, including less commonly discussed forms like acceleration (curvature) and variability-based associations.
Practical Implementation Guide: It provides explicit R syntax and workflow instructions for JMbayes2, demonstrating how to move beyond default settings to test specific biological hypotheses.
Interpretation Framework: The authors clarify that the scale and clinical meaning of the association parameter ( $\alpha$ ) are intrinsic to the chosen functional form. They emphasize that parameters from different forms (e.g., current value vs. slope) are not directly comparable in magnitude.
Variability Modeling: The paper details a pragmatic two-step approach to incorporate biomarker variability (instability) into standard joint modeling software, addressing the lack of native support for heteroscedastic joint models in many packages.
Illustrative Application: By applying all forms to the same MIRAGE dataset, the paper demonstrates how different functional forms can yield different effect interpretations and, in some cases, different substantive conclusions regarding the biomarker-risk relationship.

Results

The application to the MIRAGE trial data yielded the following observations:

Current Value: A 1 unit increase in WBC was associated with a 1.1% increase in the instantaneous log-hazard of death ( $\alpha = 0.011$ ), though the 95% credible interval included zero, indicating uncertainty.
Time-dependent Slope: Subjects with more rapid increases in WBC showed a higher risk of death, but the credible interval was wide ( $[-1.821, 4.751]$ ), suggesting weak evidence for a clear association in this dataset.
Acceleration: A positive association was found between the curvature of the WBC trajectory and survival risk ( $\alpha = 1464.33$ ). The large magnitude reflects the scale of the second derivative rather than an extreme biological effect.
Cumulative Exposure: The average historical WBC exposure showed a statistically significant positive association with death risk ( $\alpha = 0.051$ , 95% CI $[0.015, 0.090]$ ). Restricted cumulative effects over 30 days showed a weaker, non-significant association.
Change (Delta): The change from baseline showed a negative point estimate but wide uncertainty. The change over a 30-day window suggested a potential positive association with risk, though again, the credible interval included zero.
Variability: Greater within-subject variability in WBC was positively associated with the hazard of death ( $\alpha = 0.236$ ), though the interval $[-0.045, 0.549]$ indicated substantial uncertainty.

The results highlight that different functional forms target distinct features of the biomarker trajectory. For instance, cumulative forms captured longer-term exposure effects that instantaneous forms missed, while variability forms offered an alternative prognostic signal based on instability.

Significance and Claims

The paper asserts that the choice of functional form is a key modeling decision that directly determines the interpretation of the biomarker-risk association. It is not merely a technical detail but a scientific one that shapes the underlying assumptions about how the longitudinal process influences risk.

The authors claim that:

Alignment is Essential: There must be a careful alignment between the scientific question (e.g., "Does current level matter?" vs. "Does cumulative burden matter?") and the chosen functional form to ensure valid interpretation and transparent reporting.
No Universal Form: No single association structure is universally appropriate. The selection should be guided by clinical context, biological plausibility, and the specific aspect of the biomarker trajectory believed to be relevant.
Interpretability: While flexible models (like those using splines or random effects) are powerful, trajectory-based formulations (linking hazard to current value, slope, etc.) are often preferable for understanding biological mechanisms compared to shared random effects, which can obscure interpretation.
Practical Utility: The variety of functional forms available allows researchers to capture complex relationships present in clinical data that default specifications might miss, potentially avoiding misleading conclusions.

The paper concludes by acknowledging limitations, such as the reliance on a single dataset for illustration and the approximate nature of the two-step variability approach, but maintains that providing a practical guide to these diverse structures is a necessary step for improving the rigor of joint modeling in applied research.