First-Visit and Follow-Up Predictions

Why Compare First-Visit and Follow-Up Predictions?

A prediction made when a patient first has a biomarker measurement is based on limited longitudinal information. At a later visit, the clinician may have additional measurements, and the patient is known to have remained event-free up to that time. Both facts can change the prediction. The later risk need not be higher or lower: the direction depends on the measurements, the event model, and how survival up to follow-up changes the information about the patient’s latent trajectory.

A fair comparison also requires care about the prediction target. A first-visit prediction for the next year and a follow-up prediction for the next year refer to different calendar intervals. Each is conditional on the patient being event-free at its own prediction time. Neither is simply the original prediction recalculated without changing its conditioning information.

Information and Assumptions at Each Prediction Time

Let \(T_i\) denote patient \(i\)’s event time, \(Y_i(t_{ij})\) a biomarker measurement at time \(t_{ij}\), and \(b_i\) the patient’s latent random effects. Write \(\mathcal H_i(s)\) for the observed biomarker history through time \(s\), together with baseline covariates. A prediction at time \(s\) is made only for a patient known to satisfy \(T_i>s\). At a first biomarker visit \(s_0\), the history may contain only that visit’s measurement; at a later time \(s_1>s_0\), it can include the first measurement and all subsequent observations through \(s_1\).

For clarity, consider the usual shared-random-effects joint-model structure. Conditional on \(b_i\), the longitudinal measurements have a specified model, for example \(Y_i(t_{ij})=m_i(t_{ij})+\epsilon_{ij}\), where \(m_i(t)\) is the latent marker trajectory and \(\epsilon_{ij}\) is measurement error. The event hazard conditional on \(b_i\) is \(h_i(t\mid b_i)\), and the corresponding survival function is \(S_i(t\mid b_i)=\exp\{-\int_0^t h_i(u\mid b_i)\,du\}\). The hazard may depend on the latent current value, slope, or another specified feature of the trajectory.

The formula below relies on a correctly specified longitudinal and event model, a suitable model for the random effects, and valid measurement and event-time data. In particular, censoring must be independent of the event process conditional on the modeled information, or otherwise handled appropriately. If visit timing is informative after conditioning on the model, it too may need to be modeled. These assumptions do not make the prediction causal: it describes risk under the fitted model, not the effect of changing a biomarker.

How the Follow-Up Information Enters

The key object is the distribution of the latent random effects after observing the patient’s history and survival to the prediction time. By Bayes’ rule, its density is proportional to the prior random-effects density times the longitudinal-data likelihood times the event-free likelihood:

Here \(f(\mathbf y_i(s)\mid b)\) is the likelihood of the biomarker observations through \(s\), and \(f_b(b)\) is the random-effects density. The survival factor is essential: remaining event-free is more likely under latent trajectories associated with lower hazard. At follow-up, the measurement likelihood includes more data and the survival factor extends to the later time. Thus the posterior distribution of \(b_i\), and consequently the predicted risk, can differ from the first-visit prediction.

Conditional on a particular \(b\), the probability of an event in \((s,s+w]\), given survival through \(s\), is \(1-S_i(s+w\mid b)/S_i(s\mid b)\). Averaging this conditional risk over the updated distribution of \(b\) gives:

This follows from the law of total probability: condition first on \(b\), calculate the conditional interval risk, and average over the distribution of \(b\) given everything observed by \(s\). The survival information appears in two related places: it updates the distribution \(\pi_i\), and the survival ratio defines the conditional interval risk for each \(b\). Do not replace the ratio by \(1-S_i(s+w\mid b)\), which is risk from time zero rather than risk conditional on survival to \(s\).

Worked Comparisons

Interpretation, Diagnostics, and Alternatives

A follow-up prediction is useful because it answers a new clinical question with a richer history: given what has been measured so far and the fact that the patient has reached this visit event-free, what is the risk over the next specified interval? It should be reported with its landmark and horizon, such as “estimated risk from year 1 to year 2, conditional on event-free status at year 1,” rather than simply “risk is 10%.” The prediction is conditional on the fitted model and observed information; it is not a guarantee about the individual outcome.

Useful checks include examining calibration separately at first-visit and follow-up landmarks, comparing predicted and observed event frequencies over the stated horizons while accounting for censoring, and checking whether predictions behave plausibly as histories accumulate. A marked change after a small measurement update may indicate a very influential marker, sparse support in the fitted model, or a coding problem. Confirm that the follow-up history contains only measurements available by that landmark, and that the event-free condition is included exactly once in the calculation. Validation must reproduce the same landmark and horizon definitions used for prediction.

A baseline-only prediction uses fewer patient-specific longitudinal details and cannot make the same history-based update. A landmark survival model can instead be fitted separately among patients event-free at each chosen landmark, using covariates available there; it need not represent the shared latent trajectory in the same way. The joint-model approach links the marker and event processes through latent effects, but its predictions depend on that model’s assumptions. Neither approach is automatically better calibrated in every setting.

Common Pitfalls and Limitations

• Comparing different horizons as if they were the same target. A two-year risk and a one-year risk are not directly comparable. State both landmark and horizon.
• Ignoring survival to follow-up. The patients who reach \(s_1\) event-free are a selected group. The survival factor \(S_i(s_1\mid b)\) contributes information about latent risk.
• Assuming the later prediction must fall. Event-free time can shift latent-risk probabilities down, while new biomarker values can shift them in either direction. Their combined effect determines the update.
• Using a measurement taken after the prediction time. That leaks future information into the prediction and generally makes performance appear better than it is.
• Treating a simplified calculation as a universal joint-model result. The class example assumes fixed class hazards and known measurement likelihoods. Real models may have continuous random effects, time-varying hazards, and more complex trajectories.

The main limitation is dependence on model adequacy and on the information recorded before each landmark. Sparse or irregular measurements, misspecified association between the marker and event process, and informative visit or censoring patterns can undermine the update. A later prediction may be more informed, but it is not necessarily more accurate unless its assumptions and predictive calibration are supported by the data.