Concept Architecture
Concept
Theoretically, a Joint Model is a statistical model that simultaneously analyses longitudinal measurements and time-to-event outcomes within a unified modelling framework. It was developed to account for the dependence between repeatedly measured biomarkers or clinical outcomes and the risk of an event, thereby reducing bias arising from informative dropout or measurement error. In health economics, joint models are used to improve estimates of disease progression, survival and treatment effectiveness when longitudinal patient data influence subsequent clinical outcomes.
Mathematically, a joint model combines a longitudinal submodel, typically a linear mixed-effects model, with a survival submodel linked through shared random effects or an association structure. The longitudinal process describes changes in repeated measurements over time, while the survival component models the hazard of an event conditional on the evolving longitudinal trajectory. Model parameters are estimated simultaneously using maximum likelihood or Bayesian methods.
In practice, joint models are fitted to individual patient datasets containing repeated biomarker measurements and survival outcomes. They are widely applied in oncology, cardiovascular disease and chronic disease research to estimate dynamic prognosis and improve long-term survival predictions. Health economic models use joint model outputs to generate more accurate estimates of life expectancy, quality-adjusted life-years and healthcare costs.
Purpose
Used to jointly model longitudinal outcomes and survival data, improve estimation of treatment effects and disease progression, and generate more accurate long-term predictions for health economic evaluation.
Mathematical Formulae
Primary Formula
Longitudinal submodel:
y?(t) = x?(t)?? + z?(t)?b? + �?(t)
Survival submodel:
h?(t) = h?(t)exp(x??? + �m?(t))
where:
- y?(t) = longitudinal outcome
- b? = subject-specific random effects
- �?(t) = random error
- h?(t) = baseline hazard
- m?(t) = predicted longitudinal trajectory
- � = association parameter
Supporting Formulae
Joint likelihood:
L = L_longitudinal ? L_survival
Model parameters are estimated using:
?? = arg max L(?)
Related Mathematical Methods
- Linear mixed-effects models
- Cox proportional hazards model
- Maximum likelihood estimation
- Bayesian estimation
- Shared random-effects models
- Survival analysis
- Longitudinal data analysis
Example
An oncology study records tumour biomarker levels every three months while monitoring overall survival. Because biomarker progression influences mortality risk, a joint model simultaneously estimates the biomarker trajectory and survival process. The resulting survival predictions are used to estimate lifetime quality-adjusted life-years and incremental cost-effectiveness within the economic evaluation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| TREND | =TREND(B2:B20,A2:A20,A21) | Produce exploratory longitudinal predictions prior to formal joint modelling. |
| LINEST | =LINEST(B2:B20,A2:A20,TRUE,TRUE) | Estimate preliminary longitudinal regression coefficients. |
| EXP | =EXP(B2) | Convert the linear predictor to the hazard scale. |
| SUMPRODUCT | =SUMPRODUCT(CoefficientRange,CovariateRange) | Calculate the linear predictor for the survival component. |
VBA (Optional)
Automate preparation of linked longitudinal and survival datasets and generate summary outputs from jointly fitted models for health economic analyses.
Sources
- Rizopoulos D. Joint Models for Longitudinal and Time-to-Event Data: With Applications in R.
- Tsiatis AA, Davidian M. Joint modelling of longitudinal and time-to-event data: an overview.
- Ibrahim JG, Chu H, Chen LM. Basic concepts and methods for joint models of longitudinal and survival data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 15: Cost-effectiveness modelling using patient-level simulation — Davis, Stevenson, Tappenden & Wailoo, TSD 15 ed., 2014 (NICE Decision Support Unit (University of Sheffield))
Guidance on individual patient-level (microsimulation) cost-effectiveness modelling — when to use it in preference to cohort models, how to structure it, and how to handle the associated computational and uncertainty challenges.
Frequently Asked Questions (6)
What is a joint model?
A statistical approach analysing a longitudinal outcome, such as a repeated biomarker, alongside a time-to-event outcome, accounting for the link between them.
Source: Rizopoulos 2012
Why analyse a biomarker and survival together in a joint model?
A repeatedly measured marker, such as a blood test tracked over time, and the timing of an event, such as death, are often related, with the marker's trajectory carrying information about impending risk. Analysing them separately can mislead, because a patient's measurements stop when they die, and the reason for the missing data is tied to the outcome. A joint model analyses both together, using the marker's path to inform the risk of the event and handling the linked missingness. This shared analysis avoids the bias. Rizopoulos (2012) describes joint models.
Source: Rizopoulos 2012
How does a joint model work?
A joint model combines a longitudinal sub-model, describing how the repeated measure changes over time, typically a mixed-effects model, with a survival sub-model, describing the risk of the event, linked through shared parameters or the longitudinal outcome's trajectory. The link allows the current or past value of the biomarker to influence the hazard. Estimating the two sub-models jointly accounts for their association, so that the effect of the evolving measure on event risk is captured, and each outcome informs the other.
Source: Rizopoulos 2012
Why are joint models used?
Joint models are used because analysing a longitudinal biomarker and a time-to-event outcome separately can give biased or incomplete results when the two are related: the biomarker may be measured with error and may be associated with dropout or the event, and its trajectory may predict risk. A joint model accounts for these links, giving valid inference on how the evolving measure relates to event risk and correctly handling measurement error and informative dropout. This makes joint models valuable where a changing marker and survival are interdependent.
Source: Rizopoulos 2012
What problems do joint models address?
Joint models address problems arising when a longitudinal outcome and an event outcome are related: measurement error in the biomarker, which biases naive analyses; informative dropout, where the event or missingness depends on the biomarker, biasing longitudinal estimates; and the wish to use the evolving marker as a time-dependent predictor of risk without bias. By modelling both outcomes together with their association, joint models handle these issues, providing correct inference that separate analyses of each outcome cannot, especially when the marker and event are interdependent.
Source: Rizopoulos 2012
Where are joint models applied?
Joint models are applied where a repeatedly measured biomarker and a time-to-event outcome are analysed together, such as tracking a biomarker like a laboratory value or disease measure alongside survival or progression, in clinical and epidemiological studies. They are used to study how the biomarker's trajectory relates to event risk, to make dynamic predictions of risk from the evolving marker, and to handle measurement error and informative dropout. In health research, joint models support the analysis of longitudinal and survival data whose interdependence matters.
Source: Rizopoulos 2012
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 21 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/joint-model
- Term code
- HE-EM-SM-036
Stable URI · Machine-readable · Resolvable · CC BY 4.0