An End-to-End Prediction Algorithm

From a Fitted Model to a Patient-Specific Forecast

A fitted joint model contains the ingredients for dynamic prediction, but a forecast is not obtained by simply inserting a patient’s latest biomarker value into a hazard equation. The patient’s observed marker history and survival up to the prediction time both inform the latent random effects. The algorithm must account for that updated uncertainty, calculate survival from the landmark to the chosen horizon, and average over the resulting patient-specific distribution.

Let \(s\) be the landmark time, when the patient is known to be alive and event-free, and let \(u>0\) be the prediction horizon measured from that landmark. The target is conditional survival through time \(s+u\), given survival through \(s\) and the observed history available by \(s\). The corresponding event risk over the interval is one minus this conditional survival.

A shared-random-effects model links the longitudinal trajectory to the event process through \(b_i\). For example, a marker model may specify a latent trajectory \(m_i(t)=X_i(t)^\mathsf{T}\beta+Z_i(t)^\mathsf{T}b_i\), while the event hazard may depend on that trajectory or on some of its features. The exact association structure depends on the fitted model; the algorithm below uses that structure as fitted rather than replacing it with a new one.

The Conditioning and Integration Steps

Suppose first that the model parameters are held fixed at \(\theta\), as in a plug-in prediction based on fitted parameter estimates. The patient’s random effects are not known. Their distribution after observing the patient’s history must include the fact that the patient survived event-free to \(s\). If \(p(b_i\mid\theta)\) is the random-effects distribution and \(p(Y_i(s)\mid b_i,\theta)\) is the longitudinal-data likelihood through \(s\), then

where \(S_i(t\mid b_i,\theta)=P(T_i>t\mid b_i,\theta)\) is survival conditional on the random effects and model parameters. This expression assumes the longitudinal and event processes are conditionally independent given the shared random effects and other modeled covariates, and that the specified joint model is appropriate. It also assumes that any censoring information used in fitting or prediction is handled under the model’s censoring assumptions. Survival to \(s\) contributes information: random-effects values associated with lower survival to \(s\) receive less posterior weight among patients known to be alive at \(s\).

For a particular value of \(b_i\), survival over the forecast interval, conditional on survival to the landmark, is \(S_i(s+u\mid b_i,\theta)/S_i(s\mid b_i,\theta)\). Averaging this ratio over the patient-specific distribution of the random effects gives the prediction:

This is a conditional survival probability. It is not the unconditional survival \(S_i(s+u)\), nor is it the survival probability for a hypothetical patient whose random effects are known exactly. Its complement, \(1-\pi_i(s,u;\theta)\), is the predicted probability of experiencing the event between \(s\) and \(s+u\), conditional on the patient’s history and being event-free at \(s\).

An equivalent expression clarifies how the landmark survival information enters. Let \(q_i(b_i)=p(b_i\mid Y_i(s),\theta)\) denote the random-effects distribution updated by the longitudinal history alone. Then

The denominator is the model-based probability of surviving to \(s\), averaged over the marker-informed random-effects distribution. This ratio is equal to averaging the interval survival ratio over the distribution further updated by survival to \(s\). Both forms are useful for checking an implementation.

A Practical End-to-End Algorithm

In practice, the integrals may be evaluated with a numerical method suitable for the fitted model, such as quadrature or Monte Carlo integration. The integration method does not change the target; it approximates the averages in the equations. A complete prediction proceeds as follows.

If parameter uncertainty is part of the requested prediction, repeat the conditional calculation across parameter draws and average the resulting predictions using the appropriate draw weights. If parameters are fixed at estimates, report that the result is a plug-in prediction. These are different uncertainty targets; the calculation should make clear which one is being used.

Why the Landmark Update Matters

The following small calculation isolates the survival update. It is not a separate model: it illustrates how survival to the landmark changes the weights before a future interval is forecast. Suppose the marker history alone gives two random-effects values equal weights, \(0.50\) and \(0.50\). Under the fitted model, their probabilities of surviving from study entry through landmark \(s\) are \(0.90\) and \(0.60\). Given that the patient is known to have survived to \(s\), the normalized weights become

If the two corresponding conditional survival probabilities over the future interval are \(0.80\) and \(0.50\), the forecast is \(0.600(0.80)+0.400(0.50)=0.680\), giving interval risk \(0.320\). All future interval probabilities in this calculation are conditional on survival to \(s\); the survival probabilities \(0.90\) and \(0.60\) are used to update the weights, not multiplied into the reported interval survival a second time.

Checks, Alternatives, and Limitations

Several checks help detect errors before a forecast is reported. The integration weights should be nonnegative and sum to one, and the computed conditional survival should lie between zero and one. For a positive horizon, survival should not increase when the horizon is extended, provided the same conditioning information and model are used. The reported event risk must equal one minus the reported conditional survival. Also verify that the time scale in the baseline hazard, the landmark, and the horizon agree.

A common mistake is to update random-effects weights using the marker history but omit survival to the landmark. That treats the patient as if their event-free status through \(s\) carried no information about their latent trajectory. The correction is to include \(S_i(s\mid b_i,\theta)\) in the posterior weighting, or equivalently to use the numerator-over-denominator survival ratio. A different mistake is to include survival to \(s\) in the updated weights and then multiply the final forecast by survival to \(s\) again. That changes a conditional prediction into a quantity with the wrong conditioning. Once the weights are conditioned on survival to \(s\), average only the interval survival ratios.

The main computational alternatives differ in how they approximate the same integration. A plug-in random-effects prediction evaluates the model at one estimated random-effects value; it is simpler but can understate the impact of uncertainty in the patient’s latent trajectory. Likelihood-based integration, quadrature, and Monte Carlo integration instead average over a distribution of plausible random effects. A plug-in parameter prediction fixes \(\theta\) at its fitted estimate, whereas averaging across parameter draws also reflects parameter uncertainty. Neither computational choice removes the need for a correctly specified joint model or appropriate conditioning.

This algorithm assumes that the fitted model adequately represents the marker and event processes, their dependence through shared random effects, and the relevant covariates. It also requires the future covariate values needed by the event model to be specified or otherwise available under the prediction setup. The calculation does not correct model misspecification, poor extrapolation beyond observed follow-up, or an unsuitable baseline hazard. Sparse histories, measurement error, missing measurements, and uncertain future covariates require additional modeling choices; a forecast should not silently treat those issues as resolved.