Dynamic Survival Prediction From Joint Models

Why Predictions Must Change With New Measurements

A survival prediction made at enrollment answers a different question from one made months later, after an individual has remained event-free and new omics measurements have arrived. The later prediction should use both sources of information. A trajectory that appears high-risk may alter the predicted future event probability, while remaining event-free for a period can itself change what is plausible about that individual's latent risk.

A joint model provides a coherent way to update predictions because it represents the longitudinal omics trajectories and event time through shared latent effects. The key is to condition on the individual's measurements and survival up to the prediction time, then average future survival over the resulting distribution of those latent effects. Substituting a single fitted trajectory or a single risk score generally does not perform that averaging.

Model and Conditional Prediction

Let \(T_i\) be the event time for individual \(i\), and let \(Y_i^{(s)}\) denote all omics measurements observed for that individual by time \(s\). Write \(b_i\) for the individual's shared random effects across the omics trajectories and survival process, and \(X_i\) for included baseline covariates. For omics layer \(k\), let \(m_{ik}(t;b_i)\) be its modelled underlying trajectory at time \(t\), conditional on \(b_i\). A common joint-model hazard is

Here \(h_0(t)\) is the baseline hazard, \(\gamma\) is the vector of baseline-covariate coefficients, and \(\alpha_k\) links the current underlying trajectory in layer \(k\) to the event hazard. Conditional on \(b_i\) and the covariates, the corresponding survival function is

The model must specify a joint distribution for the random effects and a longitudinal measurement model for the observed assays. A product of separate assay likelihoods is appropriate only if the model assumes those measurements are conditionally independent given the random effects and included covariates. The survival submodel also specifies how the trajectory enters the hazard; other association structures are possible, but the prediction calculation must match the chosen model.

For fixed model parameters, the latent effects have a conditional posterior distribution proportional to their prior distribution, the likelihood of the observed omics data, and the probability of surviving to \(s\):

The survival factor is essential: the prediction concerns individuals who have survived event-free to \(s\), not a random individual from the enrollment population. With this posterior, the conditional prediction is

The survival ratio is the probability of surviving from \(s\) to \(u\) conditional on \(b_i\). Integrating that ratio over the posterior averages predictions across the latent effects still plausible given the individual's history.

Why the Updating Formula Holds

The formula follows by conditioning on the unobserved random effects. Conditional on \(b_i\), the probability of surviving past \(u\), given survival past \(s\), is \(S_i(u\mid b_i,X_i)/S_i(s\mid b_i,X_i)\). Average this conditional probability over the distribution of \(b_i\) for individuals with the observed history. That distribution is obtained by Bayes' rule.

This argument assumes the fitted joint model correctly describes the relevant longitudinal and event-time distributions, including their dependence through \(b_i\). It also assumes the observed history is used only up to \(s\), and that the individual is known to be event-free at \(s\). In applications, censoring and the observation process must be handled consistently with the model and prediction target; treating an event or assay recorded after \(s\) as known information would invalidate this prediction.

New Measurements Update the Posterior Again

When another assay arrives, its likelihood changes the posterior distribution of the random effects. If the individual also remains event-free between prediction times, that survival contributes further information. The update is not simply a substitution of the new measured value into the hazard: the measurement informs the latent trajectory, and the latent trajectory informs the future hazard.

Baseline and Dynamic Predictions Answer Different Questions

A baseline prediction uses information available at enrollment. A dynamic prediction at \(s\) uses the data available by \(s\) and conditions on survival through \(s\). These should not be described as interchangeable estimates of the same quantity. Even a prediction made at \(s\) using no new omics assay must account for having survived to \(s\); a prediction that also uses the new assay incorporates further individual-specific evidence.

Computation, Diagnostics, and Limitations

The integral over \(b_i\) is usually evaluated numerically, for example by quadrature, Monte Carlo draws from the fitted conditional distribution, or posterior simulation. With simulated draws \(b_i^{(1)},\ldots,b_i^{(M)}\) from \(p(b_i\mid T_i>s,Y_i^{(s)},X_i)\), the conditional survival prediction is approximated by

If model parameters are treated as uncertain, prediction should also average over their uncertainty rather than report only a plug-in estimate. The formula above describes the calculation conditional on fixed parameters; parameter uncertainty can be incorporated by drawing parameters as well as random effects from their fitted uncertainty distribution.

Useful checks include confirming that predicted survival is between 0 and 1 and does not increase as \(u\) increases. Compare predictions with observed outcomes using appropriate survival-prediction assessment, and examine calibration across relevant prediction times and risk ranges. Also inspect whether the fitted longitudinal trajectories and event model adequately represent the observed data. Poor trajectory fit, implausible hazard shapes, weakly estimated association parameters, or highly influential individuals can make individual updates unstable. Estimates may be particularly uncertain when a person has few measurements or assays with substantial measurement error.

A time-dependent Cox model using observed covariate values can be a useful alternative when its covariate representation and measurement process suit the question. However, using a noisy observed assay value directly is not the same as integrating over latent trajectory uncertainty in a joint model. A baseline-only survival model is simpler but cannot update its individual prediction using subsequent omics history. No approach removes the need to define the prediction time, horizon, available data, and target population precisely.