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Longitudinal & Survival Analysis

Joint Modeling of Longitudinal and Survival Data

A practical guide to joint models that connect repeated longitudinal measurements with time-to-event outcomes, including mixed-effects longitudinal models, survival submodels, shared random effects, association structures, dynamic prediction, estimation, interpretation, diagnostics, and implementation in R.

Advanced 20 min read

What You'll Learn

  • Why longitudinal and survival outcomes may need to be modeled jointly
  • How the longitudinal mixed-effects and survival submodels are constructed
  • How shared random effects connect the two processes
  • How current value, slope, and cumulative-history association structures differ
  • How joint models address informative dropout and measurement-error issues
  • How to fit and interpret a joint model in R

Introduction

Many clinical studies collect two fundamentally different types of outcomes from the same patient.

The first is a longitudinal outcome: a biomarker, laboratory measurement, symptom score, tumor measurement, lung function measure, or other variable recorded repeatedly over time.

The second is a time-to-event outcome: death, disease progression, relapse, treatment failure, hospitalization, or another event that may occur at a different time for each patient.

These two processes are often related. A patient's biomarker trajectory may contain information about their subsequent risk of progression or death. Conversely, the occurrence of the event often causes longitudinal measurements to stop being collected.

A joint model explicitly models these processes together.

Key idea: A joint longitudinal-survival model combines a model for repeated measurements with a model for time to event and introduces an association structure that allows the latent longitudinal trajectory to influence the patient's event hazard.

Why Not Analyze the Outcomes Separately?

A natural first approach is to fit a linear mixed model to the longitudinal measurements and a Cox model to the survival outcome.

That can be reasonable when the two processes are effectively independent after conditioning on observed covariates.

However, it becomes problematic when the probability of remaining in the longitudinal study depends on the underlying trajectory.

For example, consider a clinical trial in which a biomarker is measured every four weeks until disease progression. Patients with rapidly worsening disease may progress earlier and therefore contribute fewer biomarker measurements.

The missing longitudinal measurements are then not simply random missingness. They are related to the event process itself.

Important: A joint model is particularly useful when the longitudinal process and the event process are associated and the event process affects whether future longitudinal measurements are observed.

A Clinical Example

Suppose patients with advanced cancer have a biomarker measured at baseline and every four weeks. The primary time-to-event endpoint is overall survival.

Let:

  • \(Y_{ij}\) = biomarker measurement for patient \(i\) at time \(t_{ij}\)
  • \(T_i\) = true event time for patient \(i\)
  • \(C_i\) = censoring time
  • \(D_i\) = event indicator

The observed survival information is:

$$ \tilde T_i=\min(T_i,C_i) $$

and:

$$ D_i=I(T_i\le C_i) $$

The scientific question might be: Does a patient's underlying biomarker trajectory predict their risk of death?

The Two Processes

A joint model consists of two linked components.

1
Longitudinal submodel: describes how the repeated biomarker measurements evolve over time.
2
Survival submodel: describes the patient's hazard of experiencing the event.
3
Association structure: connects the latent longitudinal trajectory to the event hazard.

The Longitudinal Submodel

For a continuous longitudinal biomarker, a linear mixed-effects model is a natural starting point.

A simple random-intercept model is:

$$ Y_i(t) = \beta_0 + \beta_1 t + \boldsymbol{\beta}_X^\top X_i + b_{0i} + \epsilon_i(t) $$

where:

  • \(\beta_0\) is the population-average intercept.
  • \(\beta_1\) is the average longitudinal slope.
  • \(X_i\) represents baseline covariates.
  • \(b_{0i}\) is the patient-specific random intercept.
  • \(\epsilon_i(t)\) is measurement error.

The latent true longitudinal trajectory can therefore be written as:

$$ m_i(t) = \beta_0 + \beta_1t + \boldsymbol{\beta}_X^\top X_i + b_{0i} $$

The distinction between \(Y_i(t)\) and \(m_i(t)\) is fundamental.

\(Y_i(t)\) is the observed biomarker, which contains measurement error. \(m_i(t)\) is the model-based underlying trajectory.

Why this matters: The survival component can be linked to the estimated underlying trajectory rather than directly to a noisy observed biomarker value.

Random Intercepts and Random Slopes

Clinical trajectories frequently differ between patients in both their starting values and their rates of change.

A more flexible model therefore includes a random intercept and random slope:

$$ Y_{ij} = \beta_0 + \beta_1t_{ij} + \boldsymbol{\beta}_X^\top X_i + b_{0i} + b_{1i}t_{ij} + \epsilon_{ij} $$

The corresponding latent trajectory is:

$$ m_i(t) = \beta_0 + \beta_1t + \boldsymbol{\beta}_X^\top X_i + b_{0i} + b_{1i}t $$

The random-effects vector is:

$$ \mathbf b_i= \begin{pmatrix} b_{0i}\\ b_{1i} \end{pmatrix} $$

and is commonly assumed to follow a multivariate normal distribution:

$$ \mathbf b_i \sim N(\mathbf 0,\mathbf D) $$

where \(\mathbf D\) contains the random-intercept variance, random-slope variance, and their covariance.

Why the Random Effects Are So Important

The random effects represent patient-specific deviations from the population trajectory.

For example, two patients may have the same population-average slope but different individual trajectories because their \(b_{0i}\) and \(b_{1i}\) values differ.

These latent patient-specific characteristics can also be associated with survival.

That provides a natural mechanism for connecting the two models.

The Survival Submodel

The survival component is commonly specified using a proportional hazards model.

A general form is:

$$ h_i(t) = h_0(t) \exp \left\{ \boldsymbol{\gamma}^\top X_i + \alpha\,m_i(t) \right\} $$

where:

  • \(h_i(t)\) is the hazard for patient \(i\) at time \(t\).
  • \(h_0(t)\) is the baseline hazard.
  • \(\boldsymbol{\gamma}\) contains baseline covariate effects.
  • \(m_i(t)\) is the latent longitudinal trajectory.
  • \(\alpha\) is the association parameter.

The parameter \(\alpha\) is the central quantity connecting the two processes.

Interpreting the Association Parameter

Suppose the survival model contains:

$$ h_i(t) = h_0(t) \exp\{\gamma^\top X_i+\alpha m_i(t)\} $$

A one-unit increase in the underlying longitudinal value at time \(t\) is associated with a hazard ratio of:

$$ HR=\exp(\alpha) $$

For example, if:

$$ \alpha=0.40 $$

then:

$$ \exp(0.40)\approx1.49 $$

Thus, a one-unit higher underlying biomarker level at the same time is associated with approximately a 49% higher instantaneous event hazard, conditional on the other model components.

Interpret carefully: The association is with the model-defined latent longitudinal quantity, not necessarily with a single observed measurement. The exact interpretation depends on the association structure chosen for the joint model.

Current-Value Association

The simplest and most commonly discussed association structure links the current underlying longitudinal value to the hazard.

$$ h_i(t) = h_0(t) \exp \left\{ \gamma^\top X_i + \alpha m_i(t) \right\} $$

This asks: How does the patient's current underlying biomarker level relate to their current event risk?

For example, a rising tumor marker may be associated with an increasing instantaneous risk of progression.

Current-Slope Association

Sometimes the level of the biomarker is less important than its rate of change.

The survival model can then incorporate the derivative of the longitudinal trajectory:

$$ h_i(t) = h_0(t) \exp \left\{ \gamma^\top X_i + \alpha_1m_i(t) + \alpha_2m_i'(t) \right\} $$

Here \(m_i'(t)\) represents the instantaneous slope.

The scientific question becomes: Does the patient's current rate of biomarker change predict event risk, after accounting for the current biomarker level?

Why Slope Can Tell a Different Story

Consider two patients with the same biomarker value at month six.

Patient A has a stable trajectory. Patient B has been increasing rapidly.

A current-value model considers the patients similar at month six. A current-value-plus-slope model can distinguish them.

Clinical interpretation: The choice between current-value and slope association should be driven by the scientific mechanism. A biomarker may predict outcome because of its absolute level, its rate of change, or both.

Shared Random-Effects Association

Another approach links the survival process directly to the patient's latent random effects.

For example:

$$ h_i(t) = h_0(t) \exp \left\{ \gamma^\top X_i + \boldsymbol{\alpha}^\top\mathbf b_i \right\} $$

This can be useful when the scientific interest is in unobserved patient characteristics represented by the random effects.

However, the interpretation differs from a direct current-value association.

Value, Slope, and Random Effects: Comparison

Association What It Connects Scientific Question
Current value \(m_i(t)\) Does current biomarker level predict event risk?
Current slope \(m_i'(t)\) Does the rate of change predict event risk?
Value + slope \(m_i(t),m_i'(t)\) Do level and trajectory jointly predict risk?
Random effects \(\mathbf b_i\) Do latent patient-specific trajectory characteristics predict risk?
Cumulative exposure \(\int_0^t m_i(s)ds\) Does accumulated biomarker burden predict risk?

Cumulative-Exposure Association

For some biological mechanisms, cumulative exposure is more meaningful than the current value.

The survival model may use:

$$ h_i(t) = h_0(t) \exp \left\{ \gamma^\top X_i + \alpha \int_0^t m_i(s)\,ds \right\} $$

The integral represents accumulated longitudinal exposure up to time \(t\).

This type of association can be useful when prolonged exposure to a biomarker or physiological state is thought to influence event risk.

The Full Joint Model

A basic joint model can therefore be represented by two linked equations.

$$ Y_i(t) = m_i(t)+\epsilon_i(t) $$

and:

$$ h_i(t) = h_0(t) \exp \left\{ \gamma^\top X_i+\alpha m_i(t) \right\} $$

The first equation describes the longitudinal observations. The second describes the event process. The association parameter \(\alpha\) connects them.

Why the Longitudinal Process Is Latent

The true biomarker trajectory is not directly observed. Each measurement contains measurement error:

$$ Y_{ij}=m_i(t_{ij})+\epsilon_{ij} $$

Therefore, simply putting the observed biomarker value into a Cox model does not necessarily answer the same question.

A joint model instead estimates the latent trajectory while simultaneously estimating its association with survival.

Key distinction: A time-dependent Cox model using the latest observed biomarker measurement and a joint model are not generally equivalent. The joint model explicitly accounts for measurement error and the stochastic longitudinal trajectory.

Informative Dropout

Suppose a patient experiences an event at time \(T_i\). After the event, their longitudinal biomarker may no longer be measured.

Consequently, the number and timing of longitudinal observations depend partly on the survival process.

This creates a form of informative dropout.

A standard longitudinal mixed model may still be useful under appropriate missingness assumptions, but it does not explicitly model the relationship between the dropout mechanism and the event process.

A joint model provides a framework in which the two processes are modeled simultaneously.

Joint Models and Missing Data

It is important not to interpret a joint model as a universal solution to missing data.

Its validity still depends on assumptions concerning:

  • The longitudinal measurement model
  • The survival model
  • The association structure
  • The censoring mechanism
  • The distribution of random effects
  • The mechanism generating missing longitudinal measurements

A joint model can be more appropriate than a separate analysis when the event process is related to the underlying longitudinal trajectory, but model assumptions still require evaluation.

A Worked Clinical Example

Consider a hypothetical oncology study with 200 patients. A biomarker is measured every four weeks. The primary event is death. Suppose the biomarker is expected to increase as disease worsens.

A random-intercept/random-slope longitudinal model is specified as:

$$ Y_{ij} = \beta_0 + \beta_1t_{ij} + \beta_2\operatorname{Treatment}_i + b_{0i} + b_{1i}t_{ij} + \epsilon_{ij} $$

The survival model is:

$$ h_i(t) = h_0(t) \exp \left\{ \gamma_1\operatorname{Treatment}_i + \alpha m_i(t) \right\} $$

The treatment effect in the longitudinal model describes how treatment changes the biomarker trajectory. The treatment coefficient in the survival model describes its association with event hazard after accounting for the modeled biomarker trajectory.

Suppose the Association Estimate Is Positive

Suppose the fitted association parameter is:

$$ \hat\alpha=0.25 $$

Then:

$$ \exp(0.25)\approx1.28 $$

A one-unit higher latent biomarker value is therefore associated with an approximately 28% higher hazard, conditional on the other variables in the joint model.

Suppose the Association Estimate Is Negative

If instead:

$$ \hat\alpha=-0.50 $$

then:

$$ \exp(-0.50)\approx0.61 $$

A one-unit higher latent longitudinal value would be associated with a hazard approximately 39% lower.

Whether that makes clinical sense depends entirely on the biomarker. A positive association is not inherently desirable or undesirable; it depends on whether higher values represent worsening or improvement.

Treatment Effects in a Joint Model

Joint models can contain treatment effects in both submodels.

For example:

$$ m_i(t) = \beta_0 + \beta_1t + \beta_2A_i + \beta_3A_it + b_{0i} + b_{1i}t $$

where \(A_i\) indicates treatment assignment.

Here:

  • \(\beta_2\) represents a treatment difference in the baseline longitudinal level.
  • \(\beta_3\) represents a treatment difference in longitudinal slope.

The survival component may contain:

$$ h_i(t) = h_0(t) \exp \left\{ \gamma A_i+\alpha m_i(t) \right\} $$

where \(\exp(\gamma)\) is the treatment-associated hazard ratio conditional on the modeled longitudinal process.

Do not automatically interpret \(\exp(\gamma)\) as the total causal effect of treatment. When the longitudinal biomarker lies on the treatment-to-event pathway, conditioning on it changes the interpretation of the treatment coefficient. The scientific estimand must be defined carefully.

Joint Models Are Not Automatically Mediation Analyses

It is tempting to interpret a joint model as a formal mediation analysis:

Treatment → biomarker → survival.

A joint model can certainly help characterize this relationship, but it is not automatically a causal mediation analysis.

Causal mediation requires additional assumptions and a clearly defined causal estimand.

Dynamic Prediction

One of the most powerful applications of joint models is dynamic prediction.

Suppose a patient has survived to time \(t\) and has accumulated biomarker measurements through that time. The model can estimate their future event risk conditional on their observed history.

For example: What is this patient's predicted probability of surviving another 12 months, given everything we have observed about their biomarker trajectory through month six?

Conceptually, this is a conditional survival probability:

$$ P(T_i>t+\Delta\mid T_i>t,\mathcal Y_i(t),X_i) $$

where \(\mathcal Y_i(t)\) represents the patient's longitudinal measurement history up to time \(t\).

Why Dynamic Prediction Is Different From a Baseline Prediction

A conventional baseline survival model might predict risk using variables measured at enrollment.

A joint model can update the prediction as new longitudinal measurements become available.

1
At baseline, the model uses baseline covariates and initial biomarker information.
2
At month three, the patient's accumulated biomarker history is incorporated.
3
At month six, the prediction is updated again using the expanded longitudinal history.
4
The resulting prediction is individualized and dynamic rather than fixed at baseline.

The Conditional Prediction Concept

Let:

  • \(t\) = landmark time
  • \(u\) = future prediction horizon
  • \(\mathcal Y_i(t)\) = longitudinal history through \(t\)

The model can estimate:

$$ P(T_i>u\mid T_i>t,\mathcal Y_i(t),X_i) $$

This is particularly valuable when the biomarker is repeatedly updated and clinical decisions need to respond to the patient's evolving trajectory.

Joint Model Likelihood

The joint model is estimated from the combined longitudinal and survival information.

Let:

  • \(Y_i\) = longitudinal observations for patient \(i\)
  • \(\tilde T_i\) = observed survival time
  • \(\delta_i\) = event indicator
  • \(\mathbf b_i\) = patient-specific random effects

Conditional on the random effects, the longitudinal and survival processes are often assumed to be independent.

The contribution for patient \(i\) can be expressed conceptually as:

$$ L_i = \int p(Y_i\mid\mathbf b_i) p(\tilde T_i,\delta_i\mid\mathbf b_i) p(\mathbf b_i) \,d\mathbf b_i $$

The integral over the random effects is essential. The random effects are latent and therefore must be integrated out to obtain the marginal likelihood.

Why estimation can be computationally difficult: The model requires integration over the random-effects distribution, and the survival component may involve a cumulative hazard whose value depends on the latent longitudinal trajectory.

The Survival Likelihood Contribution

For a proportional hazards model, the individual survival contribution can be written as:

$$ p(\tilde T_i,\delta_i\mid\mathbf b_i) = h_i(\tilde T_i)^{\delta_i} S_i(\tilde T_i) $$

where:

$$ S_i(t) = \exp \left[ -\int_0^t h_i(s)\,ds \right] $$

The longitudinal trajectory therefore enters the survival likelihood both through the hazard and through the cumulative hazard.

Why Naive Two-Step Analysis Can Be Problematic

A common shortcut is:

  1. Fit a mixed model to the longitudinal measurements.
  2. Obtain predicted biomarker values.
  3. Insert those predictions into a Cox model.

This can be useful descriptively, but it is not generally equivalent to a fully specified joint model.

The predicted longitudinal values are estimated quantities rather than known covariates. Ignoring their uncertainty can produce biased or overly precise inference in some settings.

General principle: If the scientific objective specifically concerns the association between a latent longitudinal trajectory and event risk, a joint model is preferable to simply treating fitted longitudinal predictions as error-free covariates.

Two-Stage Approaches vs. Full Joint Estimation

Approach Advantages Limitations
Separate models Simple and familiar Does not explicitly model dependence
Two-stage prediction + Cox Easy to implement and interpret Can ignore prediction uncertainty and induce bias
Full joint model Models both processes simultaneously More computationally demanding

Baseline Hazard Specification

The survival component requires a baseline hazard.

Common choices include:

  • Piecewise constant baseline hazards
  • B-spline representations
  • Penalized splines
  • Parametric baseline hazards

The choice can affect model flexibility and computational behavior.

A highly restrictive parametric baseline hazard can be efficient when correctly specified but can be problematic if the shape is inappropriate.

Practical consideration: For clinical applications, flexible baseline-hazard representations are often attractive because the main interest may be the association between the longitudinal process and survival rather than a specific parametric shape for the baseline hazard.

Time Scale Matters

Joint models require careful definition of time.

Possible time scales include:

  • Time since randomization
  • Time since treatment initiation
  • Time since diagnosis
  • Age
  • Time since a landmark event

The longitudinal trajectory and survival process must be aligned on a scientifically meaningful time scale.

Time-Varying Treatment Effects

If the treatment effect changes over time, the survival model may need to incorporate time-varying effects.

For example:

$$ h_i(t) = h_0(t) \exp \left\{ \gamma(t)A_i+\alpha m_i(t) \right\} $$

The exact implementation depends on the software and modeling framework.

Nonlinear Longitudinal Trajectories

A linear trajectory is not always appropriate.

Clinical biomarkers may rise rapidly and then plateau, decline initially and then stabilize, or follow other nonlinear patterns.

The longitudinal model can incorporate:

  • Polynomial terms
  • Restricted cubic splines
  • B-splines
  • Piecewise linear terms
  • Nonlinear mixed-effects functions

For example:

$$ m_i(t) = \beta_0 + \beta_1t + \beta_2t^2 + b_{0i} + b_{1i}t $$

The survival model can then use the resulting nonlinear trajectory.

Important: If the longitudinal trajectory is nonlinear, the meaning of a slope association becomes time-dependent. A derivative \(m_i'(t)\) can change substantially over the follow-up period.

Non-Gaussian Longitudinal Outcomes

Joint modeling is not restricted to continuous Gaussian biomarkers. Depending on the framework, the longitudinal outcome may be:

  • Binary
  • Ordinal
  • Count
  • Continuous
  • Bounded or transformed measurements

For example, a binary longitudinal outcome may use a generalized linear mixed model as its longitudinal component.

The basic principle remains the same: model the longitudinal process and connect its latent trajectory or random effects to the survival process.

Competing Risks

Joint modeling becomes more complex when patients can experience different types of events.

For example:

  • Death from disease
  • Death from another cause
  • Treatment discontinuation
  • Disease progression

The survival component may need to contain cause-specific hazards or another competing-risk framework.

The association between the longitudinal process and each event type may also differ.

Recurrent Events

Some clinical outcomes occur repeatedly rather than only once. Examples include:

  • Hospitalizations
  • Asthma exacerbations
  • Seizures
  • Infections
  • Bleeding episodes

Joint modeling can be extended to recurrent-event settings, although the event submodel becomes more complex.

Terminal Events and Informative Censoring

In some studies, death is a terminal event that prevents subsequent recurrent or longitudinal observations.

This is another setting in which joint modeling can be particularly useful. The terminal event and longitudinal/recurrent process may be strongly dependent.

Model Diagnostics

A joint model should not be accepted simply because the optimizer converged.

Diagnostics should examine both components.

Longitudinal Diagnostics

  • Residual distributions
  • Observed versus fitted trajectories
  • Random-effects distributions
  • Covariance structure
  • Time functional form
  • Heteroscedasticity

Survival Diagnostics

  • Proportional hazards assumptions
  • Predicted versus observed survival
  • Baseline hazard behavior
  • Calibration
  • Influential observations

Joint-Model Diagnostics

  • Sensitivity to association structure
  • Sensitivity to longitudinal time specification
  • Sensitivity to random-effects structure
  • Predictive calibration
  • Dynamic prediction performance

Model Comparison

Candidate joint models can differ in:

  • Random-effects structure
  • Longitudinal time function
  • Association structure
  • Baseline hazard representation
  • Covariate specification

Information criteria such as AIC can sometimes help compare candidate models, but model selection should not be based solely on a single numerical criterion.

Clinical interpretability and predictive performance are also important.

Checking the Association Structure

Suppose the primary scientific question is whether a biomarker's current level predicts survival. A current-value association may be appropriate.

But if biology suggests that rapid deterioration is more important than the absolute level, a slope association may be more appropriate.

A useful sensitivity analysis can therefore compare:

Model Association
Model 1 Current value \(m_i(t)\)
Model 2 Current slope \(m_i'(t)\)
Model 3 Current value + slope
Model 4 Cumulative exposure

If the substantive conclusion changes dramatically across plausible association structures, that uncertainty should be acknowledged.

R Implementation

Several R packages and frameworks can be used for joint modeling. A commonly used approach is the JM / JMbayes2 family of methods, while other packages provide specialized joint-modeling implementations.

A general workflow begins by fitting the longitudinal and survival models. 

library(nlme)
library(survival)
library(JMbayes2)

long_fit <- lme(
  biomarker ~ time + treatment,
  random = ~ time | id,
  data = long_data
)

surv_fit <- coxph(
  Surv(surv_time, event) ~ treatment,
  data = surv_data,
  x = TRUE
)

The exact joint-model fitting syntax depends on the package and the desired association structure.

For a package supporting a joint longitudinal-survival formulation, the model can then be constructed from the fitted longitudinal and survival components. 

joint_fit <- jm(
  surv_fit,
  long_fit,
  time_var = "time"
)

summary(joint_fit)

The precise arguments should be checked against the version of the package being used because joint-modeling software has evolved substantially.

Inspecting the Longitudinal Model

summary(long_fit)

ranef(long_fit)

intervals(long_fit)

Before fitting the joint model, verify that the longitudinal model is scientifically and statistically reasonable.

For example, determine whether a random slope is needed and whether the relationship with time is adequately represented.

Inspecting the Survival Model

summary(surv_fit)

Check:

  • Covariate effects
  • Event counts
  • Proportional-hazards assumptions
  • Potentially influential observations
  • Choice of time scale

Interpreting Joint-Model Output

A joint-model summary may contain estimates from both submodels.

A useful reporting structure is:

Component Parameter Interpretation
Longitudinal Time effect Average biomarker trajectory
Longitudinal Treatment × time Difference in trajectory between treatments
Survival Treatment coefficient Conditional treatment-associated hazard ratio
Association \(\alpha\) Relationship between latent longitudinal process and event hazard

Hazard-Ratio Interpretation of the Association

Suppose:

$$ \hat\alpha=0.18 $$

Then:

$$ HR=\exp(0.18)\approx1.20 $$

Thus, a one-unit increase in the latent current biomarker value is associated with approximately a 20% increase in instantaneous event hazard.

If the biomarker is measured in clinically unfamiliar units, rescaling can make the result easier to interpret.

Rescaling the Association

Suppose the biomarker is measured in units where a clinically meaningful change is 10 units. If the one-unit association parameter is \(\alpha\), the hazard ratio for a 10-unit increase is:

$$ HR_{10}=\exp(10\alpha) $$

This is often more interpretable than reporting the hazard ratio for a tiny one-unit change.

Dynamic Prediction in Practice

Suppose a patient has survived to month six. Their biomarker history shows persistent deterioration. The joint model can use the complete observed trajectory through month six to update the patient's future survival prediction.

Conceptually:

$$ \text{Posterior trajectory} \rightarrow \text{future hazard} \rightarrow \text{conditional survival probability} $$

This makes joint models particularly attractive for personalized monitoring and longitudinal biomarker applications.

Prediction vs. Explanation

Joint models can be used for two different broad goals.

Association and inference

The goal is to estimate how the longitudinal trajectory is associated with event risk.

Prediction

The goal is to predict an individual patient's future event risk using their longitudinal history.

These goals overlap, but they are not identical. A model optimized for prediction may contain variables and flexible structures that are less attractive for a simple explanatory analysis.

Calibration of Dynamic Predictions

A prediction model should be evaluated using more than discrimination. Useful measures include:

  • Calibration
  • Prediction error
  • Time-dependent discrimination
  • Observed versus predicted survival
  • Internal validation
  • External validation when possible

Bootstrap validation or cross-validation can be particularly useful when dynamic prediction is a major objective.

Landmark Analysis vs. Joint Modeling

Landmark analysis is another approach for incorporating longitudinal information into survival prediction.

At a chosen landmark time, the most recent biomarker value or summary of longitudinal history can be entered into a survival model.

Joint models differ because they explicitly model the underlying longitudinal trajectory and its relationship with the event process.

Feature Landmark Analysis Joint Model
Longitudinal trajectory model Usually no Yes
Measurement error modeling Usually limited Yes
Dynamic prediction Yes Yes
Informative dropout modeling Limited Explicitly addressed through joint structure
Computational complexity Lower Higher

Time-Dependent Cox Models vs. Joint Models

A time-dependent Cox model can incorporate an observed biomarker measurement that changes over time.

For example:

$$ h_i(t) = h_0(t) \exp\{\gamma^\top X_i+\eta Y_i(t)\} $$

This may be useful, but \(Y_i(t)\) is observed with measurement error and may only be available at discrete visits.

A joint model instead uses the latent trajectory \(m_i(t)\).

$$ h_i(t) = h_0(t) \exp\{\gamma^\top X_i+\alpha m_i(t)\} $$
Practical distinction: A time-dependent Cox model treats the time-varying covariate as an observed predictor. A joint model treats the longitudinal biomarker as a stochastic measurement process and explicitly models the latent trajectory.

Common Mistakes

  1. Automatically using a joint model. A joint model is useful when the two processes are meaningfully associated, not simply because both longitudinal and survival data exist.
  2. Ignoring the longitudinal model. A poorly specified trajectory can produce misleading survival associations.
  3. Assuming linearity without checking it. Many biomarkers have nonlinear trajectories.
  4. Using the wrong association structure. Current value, slope, cumulative exposure, and random effects answer different scientific questions.
  5. Treating predicted longitudinal values as error-free. This can understate uncertainty in a two-stage analysis.
  6. Ignoring informative dropout. Patients who experience the event may systematically contribute fewer longitudinal observations.
  7. Overinterpreting the treatment coefficient. The treatment effect conditional on a longitudinal biomarker is not necessarily the total treatment effect.
  8. Ignoring the time scale. Both submodels must use a coherent and scientifically meaningful time scale.
  9. Failing to assess model sensitivity. Results may depend on the random-effects structure or association specification.
  10. Reporting only the association hazard ratio. The longitudinal trajectory and survival covariate effects are also important for interpreting the complete model.

A Practical Joint-Modeling Workflow

1
Define the clinical scientific question linking the longitudinal and survival processes.
2
Define the time scale and align longitudinal and survival records.
3
Explore individual longitudinal trajectories.
4
Specify the longitudinal mixed-effects model.
5
Assess fixed effects, random effects, covariance structure, and time functional form.
6
Specify the survival model and baseline hazard.
7
Choose a scientifically justified association structure.
8
Fit the joint model using appropriate estimation methods.
9
Evaluate convergence and numerical stability.
10
Assess longitudinal and survival model diagnostics.
11
Perform sensitivity analyses for association and trajectory specifications.
12
If prediction is a goal, evaluate dynamic prediction and calibration.

What Should Be Prespecified?

For a clinical trial or prospective clinical study, the statistical analysis plan should describe the joint-modeling strategy before examining the final outcome data.

At minimum, specify:

  • Longitudinal endpoint
  • Time-to-event endpoint
  • Time scale
  • Longitudinal time function
  • Fixed-effect covariates
  • Random-effects structure
  • Covariance structure
  • Survival covariates
  • Baseline hazard representation
  • Primary association structure
  • Handling of missing longitudinal measurements
  • Censoring assumptions
  • Model diagnostics
  • Sensitivity analyses
  • Prediction methodology, if applicable

Interpretation in a Clinical Trial

Suppose a trial finds that treatment significantly slows biomarker progression and that the biomarker trajectory is strongly associated with survival.

The results can support a coherent clinical story:

A
Treatment changes the longitudinal disease-related biomarker trajectory.
B
Patients with less favorable latent trajectories have higher event hazards.
C
Repeated biomarker information can therefore improve understanding or prediction of survival.

However, this does not by itself establish that the biomarker is a causal mediator of treatment benefit.

When Joint Modeling Is Particularly Attractive

Joint modeling is especially useful when:

  • A biomarker is repeatedly measured over time.
  • A clinically important time-to-event endpoint is also collected.
  • Longitudinal measurements stop after an event.
  • Patients have heterogeneous trajectories.
  • Measurement error is important.
  • The underlying trajectory is expected to predict event risk.
  • Dynamic prediction is a major objective.
  • Informative dropout is a concern.

When a Joint Model May Not Be Necessary

A joint model may be unnecessary when:

  • The longitudinal and survival outcomes are scientifically unrelated.
  • The longitudinal measurements are merely descriptive.
  • The survival analysis does not depend on longitudinal information.
  • The longitudinal process is adequately summarized by a prespecified baseline variable.
  • The sample size is too small to support the complexity of the proposed model.
  • The additional assumptions cannot be justified.
More sophisticated does not automatically mean better. The joint model should be chosen because its assumptions and estimand match the scientific question—not simply because it is more advanced than separate models.

Joint Modeling and Sample Size

Joint models can require substantial information because several components are estimated simultaneously.

Adequate information may be needed for:

  • Longitudinal fixed effects
  • Random-effects variances
  • Covariance parameters
  • Survival effects
  • Association parameters
  • Potential nonlinear trajectory terms

Simulation is often preferable to relying on simple closed-form sample-size calculations when the joint-model parameter is a primary target.

Simulation-Based Planning

A simulation study can generate data under assumed:

  • Longitudinal trajectory parameters
  • Random-effects distributions
  • Measurement-error variance
  • Event hazard parameters
  • Association parameters
  • Censoring mechanisms
  • Visit schedules
  • Dropout mechanisms

The simulated datasets are then analyzed using the proposed joint-model procedure.

Important operating characteristics include:

  • Bias
  • Empirical standard error
  • Coverage probability
  • Power
  • Convergence rate
  • Prediction accuracy

Convergence Problems

Joint models can encounter numerical difficulties. Possible warning signs include:

  • Non-convergence
  • Very large standard errors
  • Nearly singular covariance matrices
  • Extreme random-effect estimates
  • Unstable association estimates
  • Different estimates from different starting values

Potential causes include:

  • Too much model complexity
  • Sparse longitudinal measurements
  • Few observed events
  • Weak identification of the association parameter
  • Overly flexible trajectories
  • Strongly correlated random effects
Convergence is not merely a software issue. Poor convergence can indicate that the data do not contain enough information to reliably estimate the proposed model.

Model Simplification

If a complex model is unstable, consider simplifying it systematically.

For example:

  1. Start with a clinically plausible longitudinal time function.
  2. Use a random intercept before adding a random slope.
  3. Use the simplest scientifically defensible association structure.
  4. Increase complexity only when supported by the data and study objectives.

The goal is not to fit the most complicated model possible. The goal is to fit a model that is sufficiently rich to answer the scientific question while remaining identifiable and interpretable.

Reporting Results

A publication should generally report enough information for readers to understand both components of the joint model.

A useful results table might include:

Parameter Estimate SE 95% CI Interpretation
Longitudinal time effect \(\beta_1\) ... ... Average trajectory
Treatment × time \(\beta_3\) ... ... Treatment difference in slope
Survival treatment effect \(\gamma\) ... ... \(\exp(\gamma)\) hazard ratio
Longitudinal-survival association \(\alpha\) ... ... \(\exp(\alpha)\) hazard ratio

What the Association Parameter Does Not Tell You

The association parameter does not automatically establish:

  • Causality
  • Mediation
  • Surrogate validity
  • Treatment efficacy
  • Clinical utility

It describes the association between the modeled longitudinal process and event risk under the assumptions of the joint model.

Joint Models and Surrogate Endpoints

Longitudinal biomarkers are sometimes investigated as potential surrogate endpoints.

A strong association between biomarker trajectory and survival is encouraging, but it does not by itself demonstrate that treatment effects on the biomarker reliably predict treatment effects on survival.

Formal surrogate validation generally requires stronger evidence across treatments and trials.

Common Questions

Does every longitudinal-survival analysis require a joint model?

No. The decision depends on the scientific relationship between the processes, the missingness mechanism, the measurement-error structure, and the analysis objective.

Is a joint model just a mixed model plus a Cox model?

Conceptually, it contains those two components, but the key feature is that they are linked and estimated jointly through a specified association structure.

Can the longitudinal outcome be binary?

Yes. Generalized longitudinal models can be used in joint-modeling frameworks, although the details of the latent trajectory and association structure become more complicated.

Can the survival outcome be recurrent?

Yes. Extensions exist for recurrent-event processes, competing risks, and terminal events.

Can joint models be used for prediction?

Yes. Dynamic prediction is one of their most important applications.

Does a significant association prove the biomarker causes survival?

No. It establishes statistical association under the model, not causality.

Final Practical Checklist

1
Confirm that the longitudinal and survival processes are scientifically related.
2
Define the common time scale.
3
Explore individual longitudinal trajectories.
4
Choose an appropriate longitudinal mixed-effects structure.
5
Choose a survival model and baseline hazard.
6
Choose the association structure based on the scientific mechanism.
7
Fit the full joint model.
8
Check convergence and model diagnostics.
9
Perform sensitivity analyses for trajectory and association assumptions.
10
Evaluate dynamic prediction if individual risk prediction is an objective.

The Most Important Concept

The most important idea in joint modeling is that the longitudinal and survival outcomes are not treated as unrelated pieces of data.

Instead, the analysis recognizes that each patient has an underlying, partially observed longitudinal trajectory and that this trajectory may be related to their instantaneous event risk.

The longitudinal model estimates:

$$ m_i(t) $$

while the survival model uses that trajectory through an association structure such as:

$$ h_i(t) = h_0(t) \exp \left\{ \gamma^\top X_i+\alpha m_i(t) \right\} $$

The association parameter \(\alpha\) quantifies how the modeled longitudinal process relates to event risk.

This framework provides a principled way to handle repeated measurements, measurement error, informative dropout, patient-specific trajectories, and dynamic prediction within one coherent statistical model.

Bottom line: Joint modeling of longitudinal and survival data combines a longitudinal mixed-effects model with a survival model and explicitly links the two through a prespecified association structure. It is especially valuable when repeated biomarker measurements are related to a time-to-event outcome and when the occurrence of the event affects the availability of subsequent longitudinal measurements. The choice of current-value, slope, cumulative-exposure, or random-effects association should be driven by the scientific question. Joint models can also provide dynamic, patient-specific survival predictions that are updated as new longitudinal information becomes available.

References

Wulfsohn, M.S. & Tsiatis, A.A. (1997). A joint model for survival and longitudinal data measured with error. Biometrics, 53(1), 330–339.

Tsiatis, A.A. & Davidian, M. (2004). Joint modeling of longitudinal and time-to-event data: an overview. Statistica Sinica, 14, 809–834.

Rizopoulos, D. (2012). Joint Models for Longitudinal and Time-to-Event Data: With Applications in R. Chapman & Hall/CRC.

Rizopoulos, D. (2011). Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data. Biometrics, 67(3), 819–829.

Rizopoulos, D., Hatfield, L.A., Carlin, B.P. & Takkenberg, J.J.M. (2014). Combining dynamic predictions from joint models for longitudinal and time-to-event data. Biometrics, 70(1), 10–19.

Hickey, G.L., Philipson, P., Jorgensen, A. & Kolamunnage-Dona, R. (2018). Joint modelling of time-to-event and multivariate longitudinal outcomes: recent developments and issues. BMC Medical Research Methodology, 18, 117.

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