Joint Longitudinal–Survival Models

Suppose patients with chronic kidney disease have their kidney function tested every 3 months. In some patients kidney function stays stable for a long time, while in others it drops quickly with every test.

What the researcher wants to know isn't simply how much kidney function declined. They want to know how the rate at which kidney function declines is related to the risk of kidney failure.

There's an important problem here. Patients who develop kidney failure may not be able to attend their later scheduled tests. So as time goes on, the remaining test results may come mostly from patients in relatively good condition.

If you analyze the repeated measurements on their own, you may not adequately account for this dropout. On the other hand, if you only run a survival analysis, it can be hard to properly reflect how kidney function changes differently from patient to patient.

The method for dealing with these problems is the joint longitudinal–survival model.

A joint model analyzes values measured repeatedly over time and the time until an event together in a single statistical model. This lets you estimate how the current level or rate of change of a measurement is related to the risk of the event.

In this article we'll look at how joint models are built and how they work, a real study example, interpreting the results, dynamic prediction, how they differ from Cox regression, and the main assumptions and cautions.

The first is longitudinal data: values measured repeatedly at several time points in the same person or unit. The second is survival data: data recording the time until a particular event and whether the event occurred.

For example, a kidney disease study could analyze the following two outcomes together.

A joint model estimates how kidney function changes over time while also analyzing how that change is related to the risk of kidney failure. The important point is that it doesn't just analyze the two outcomes separately and then compare them. A joint model builds the association between the two outcomes directly into the statistical model.

This lets you account for measurement error in the repeated values and for differences in each patient's trajectory, and, under appropriate assumptions, also handle the situation where repeated measurements stop because the event occurred.

For example, you can estimate how much a patient's kidney function declines over time and how the rate of decline differs between patients. The risk of kidney failure, meanwhile, can be analyzed with Cox regression. But running the two analyses separately can cause several problems.

First, repeated measurements contain measurement error.

Kidney function test values can reflect not only true health status but also measurement error and temporary physical conditions. Putting the observed test values straight into a survival model can let this error distort the relationship with risk.

Second, each patient's measurements change in a different way.

Two patients may have the same current kidney function, but one may have stayed stable for a long time while the other declined sharply in recent months. So not only the current value but also the rate of change can be important information.

Third, the event can stop the repeated measurements.

For patients who develop kidney failure, later scheduled test values may not be observed. If this dropout is related to the patient's health, analyzing the repeated-measures data on their own can introduce bias. Joint models are designed to handle these problems in one analytical framework.

Longitudinal submodel

Analyzes how the repeated measurements change over time. A linear mixed-effects model is a typical choice. For example, it can estimate the average change in kidney function along with differences in each patient's starting level and rate of change.

Survival submodel

Analyzes the time until the event and the risk of the event. You can use a Cox-type proportional hazards model, a parametric survival model, and so on.

Linking the two submodels

A joint model links the two submodels through each patient's latent trajectory of the measurement, shared random effects, and so on.

Through this link you can estimate how strongly the level or rate of change of kidney function is related to the risk of kidney failure. But the meaning of the results depends on how the two submodels are linked.

To track their kidney function, the estimated glomerular filtration rate (eGFR) was measured every 3 months. eGFR is a measure of how well the kidneys filter blood; in general, a lower value suggests that kidney function may have declined.

In this example, assume the timing of kidney failure, as defined in advance by the study, was recorded separately.

Study data

The actual number of measurements can differ by patient. Some patients were tested through 36 months, while others stopped earlier because of kidney failure or the end of follow-up. The research question is:

To answer this question, you can fit a joint model that uses both the repeated eGFR measurements and the time to kidney failure.

For example, suppose three patients' kidney function was measured as follows.

Example eGFR changes by patient

Unit: mL/min/1.73m²

All three patients started with an eGFR of 55, but their trajectories differed afterward. Patient A declined gradually from 55 to 52 over 12 months. Patient B fell to 43, and patient C dropped quickly to 30. The longitudinal submodel estimates these differences between patients.

In a mixed-effects model, for example, you can analyze the average downward trend in kidney function across all patients together with differences in each patient's starting level and rate of change. These estimated patient-level trajectories are then linked to the survival submodel.

That said, the five measurements in the table alone aren't enough to conclude that each patient's actual change in kidney function is necessarily linear. In real research, you should consider nonlinear change where needed.

In a joint model, this risk is set up so that it can be related to the estimated trajectory of the repeated measurements. A typical current-value association model can be written as follows.

Here is what each symbol means.

This model analyzes the relationship between the current estimated eGFR level and the risk of kidney failure. For example, a negative α would mean that higher eGFR goes with a lower risk of kidney failure.

When interpreting the size of the coefficient, though, you need to take into account the measurement unit of eGFR and the other variables in the model. Also, this relationship is a statistical association estimated from observational data; it doesn't have the causal meaning that artificially changing eGFR would change the risk by the same ratio.

Typical options include the current value, the rate of change, and the cumulative history.

For example, suppose patients A and B both have a current eGFR of 40. Over the past year, patient A's eGFR fell slightly, from 42 to 40. Patient B's, in contrast, fell quickly from 65 to 40.

In a model that uses only the current value, the two patients have the same current eGFR, so that variable produces no difference in risk between them. But a model that also links the rate of change can reflect the information that the two patients are declining at different rates.

So the association structure should be chosen according to what the researcher considers important: the current health status or the speed at which it is worsening. A different association structure can also change the units and the interpretation of the coefficients.

To make interpretation easier, eGFR is expressed per 5-unit increase.

Survival submodel results

These values were not calculated from real patient data.

The hazard ratio for eGFR was 0.78. This means that, with age and diabetes status the same and other model conditions equal, an estimated eGFR 5 units higher went with an instantaneous risk of kidney failure about 22% lower.

For example, for a 10-unit difference in eGFR, assuming the same log-linear relationship, the hazard ratio is:

So an eGFR 10 units higher can be interpreted as being associated with an instantaneous risk about 39% lower. The hazard ratio for patients with diabetes was 1.46: with other model conditions equal, their instantaneous risk of kidney failure was about 46% higher than that of patients without diabetes.

The 95% confidence interval for age, on the other hand, includes 1, so no statistically significant association was found at the 0.05 significance level.

Because this example used a current-value association model, it doesn't include a hazard ratio for the rate of eGFR decline itself. If you need a separate interpretation of the rate of decline, you have to fit a model that adds that association term.

For example, suppose patients whose kidney function worsens quickly tend to develop kidney failure earlier. Once kidney failure occurs, the study's planned eGFR follow-up may stop.

In that case, the patients who still have test records at 24 or 36 months are likely to be those with relatively good kidney function. So if you calculate the mean eGFR of the remaining patients without accounting for the event, the change in kidney function across all patients can look more gradual than it really is.

This is called the informative dropout problem. A joint longitudinal–survival model can address it by modeling the repeated-measurement process and the event process together.

This is called dynamic prediction. For example, suppose a patient has not had kidney failure in the 6 months since starting treatment and has had eGFR measured three times during that period. Using this patient's test history, you can estimate the conditional probability of not developing kidney failure over the next 12 months.

When new test results are added at 9 and 12 months, the prediction can be recalculated to reflect them.

Example of dynamic prediction for one patient

Assume these are predicted probabilities calculated on the condition that kidney failure has not occurred by each time point.

You can see that as this patient's eGFR declines, the predicted probability of kidney failure over the next 12 months rises. Note, though, that this is not a repeated prediction for the same 12-month window. Each is a prediction made at that time point for a new period, the following 12 months.

Whether individual predicted probabilities are suitable for real clinical decisions also needs separate validation. Even if a prediction model performs well on the data it was built from, there's no guarantee it will perform as well when applied to other hospitals or patient populations.

A mixed-effects model is suitable when you want to know how kidney function changes over time. Cox regression is useful when you're interested in the relationship between explanatory variables and the risk of an event. With time-varying covariates, you can also bring changes in a measurement into a survival model.

But when there is measurement error, or when the end of repeated measurements is related to the event, an ordinary time-varying Cox regression may not be enough. A joint model is different in that it analyzes the change in the repeated measurements themselves and the occurrence of the event at the same time, and models the association between the two processes.

Because computation and model specification are more complex, however, not every survival analysis needs a joint model.

Shape of change in the repeated measurements

We assumed eGFR declines linearly over time, but in reality it may start falling sharply at some point. Where needed, consider splines or nonlinear change.

Structure of between-patient variation

Check whether patients differ only in their starting level or also in their rate of decline. An inappropriate random-effects structure can lead to a wrongly estimated trajectory.

Assumptions of the survival model

If the survival submodel uses a proportional hazards structure, check that assumption and whether the effects change over time.

Choice of association type

Results can differ depending on whether you link the risk to the current eGFR level or also consider the rate of decline or the cumulative history.

Order of measurement times and events

Test values from after kidney failure must not be used to explain the risk at earlier times. Use only the information that would actually be available at the prediction time.

Model convergence and stability

Joint models can have many parameters to estimate and involve complex computation. So check convergence, the random-effects estimates, the uncertainty in the parameters, and so on.

Validating predictive performance

If dynamic prediction is the goal, check not only discrimination but also the calibration of the predicted probabilities and how performance changes over time. If possible, it's best to carry out external validation on separate patient data.

In particular, the association type isn't just a technical setting; it's a choice that directly affects the research question. The current level and the rate of decline have different clinical meanings, so they must be interpreted separately.

You can also visualize each patient's trajectory and check how well the model fits the observed values. Using the study we've been looking at, you could write it up as follows.

A real report should also present the number of repeated measurements, the actual number of events, the number censored, the estimation method, the coefficients of the longitudinal submodel, the association coefficients linking the two models, and the model diagnostics. If you performed dynamic prediction, clearly state the prediction times and prediction horizon, and it's best to also report the results of validating the prediction model's performance.

Where a standard mixed-effects model focuses on changes in repeated measurements and Cox regression focuses on the risk of an event, a joint model estimates the association between the two outcomes in a single statistical framework.

For example, with patient data on repeated kidney function measurements, you can analyze how the current level or rate of decline of eGFR is related to the risk of kidney failure.

It can also handle the problem of repeated measurements stopping because of the event, under appropriate model assumptions, and it can be used for dynamic prediction, updating each patient's event risk as new test results come in.

A joint model does, however, require both submodels and the association structure to be specified appropriately, and estimation and validation can be more complex than in ordinary regression.