Omitting Survival to the Landmark From the Update

Why Survival to the Landmark Is Part of the Update

A dynamic prediction at landmark time \(s\) concerns a patient known to be event-free at \(s\), given the biomarker measurements available up to \(s\). The event-free status is information about the patient, just as the biomarker history is. If patients with higher latent risk are more likely to have experienced the event before \(s\), the patients who remain event-free are a selected group. A patient-specific prediction that updates latent risk using only the biomarkers ignores that selection.

The consequence is not merely a technical detail in the likelihood. Omitting survival to the landmark can leave too much posterior weight on latent risk profiles that are less compatible with the patient having remained event-free. The resulting future risk can be too high, sometimes substantially so. The direction and size of the error depend on how survival to \(s\) differs across latent profiles.

The Correct Update and Prediction

Suppose the joint model specifies a prior density \(p(b_i)\), a longitudinal likelihood \(L_Y\{\mathcal{Y}_i(s)\mid b_i\}\), and a hazard \(h_i(t\mid b_i)\). Define the event-free survival function conditional on \(b_i\) by

Assume that, conditional on \(b_i\) and the model covariates, the longitudinal and event processes are modeled jointly in the usual way: the observed marker history contributes \(L_Y\), and the event-free status through \(s\) contributes \(S_i(s\mid b_i)\). Then Bayes' rule gives

The survival factor is a likelihood contribution, not a second future-risk adjustment. Once this posterior has been formed, the probability of an event by \(s+w\), conditional on being event-free at \(s\) and on \(b_i\), is

Averaging this conditional risk over the updated posterior gives the patient-specific prediction:

The denominator \(S_i(s\mid b)\) in the conditional risk matters: the target is risk over the future interval among those already known to be event-free at \(s\), not event probability from time zero. An equivalent expression makes the role of the landmark update especially clear:

Why the Survival Factor Belongs in the Posterior

The update follows directly from conditioning. The joint likelihood contribution for the observed marker history and event-free status, conditional on \(b_i\), is \(L_Y\{\mathcal{Y}_i(s)\mid b_i\}S_i(s\mid b_i)\). Multiplying by the prior density and normalizing yields the posterior above. If the survival factor is left out, the calculation instead produces \(p\{b_i\mid\mathcal{Y}_i(s)\}\): a posterior informed by markers but not by the fact that the patient survived event-free to \(s\).

When latent profiles with greater event hazard also have smaller \(S_i(s\mid b)\), the correct update downweights those profiles relative to the marker-only update. This is a selection effect, not a claim that the patient’s survival caused a change in their latent risk. The prediction conditions on the information observed about the patient.

When the Omission Can Matter Most

The omission is most consequential when survival through \(s\) is quite different across the latent profiles that remain plausible after observing the marker history. A strong marker signal does not guarantee that survival information is negligible. In the next example, the marker initially favors the high-risk profile, but survival to the landmark changes that assessment sharply.

Assumptions, Diagnostics, and Alternatives

The calculation relies on the fitted joint model being an appropriate description of the relevant data. In particular, the hazard \(h_i(t\mid b_i)\), the distribution \(p(b_i)\), and the longitudinal likelihood must be specified coherently. The survival contribution \(S_i(s\mid b_i)\) assumes the patient’s event-free status through \(s\) is known and correctly represented by the event model. If a patient was censored before \(s\), that does not establish \(T_i>s\); the observed censoring information must be handled according to the censoring model rather than treated as known event-free survival through \(s\).

A practical diagnostic is to compare the posterior distribution of the latent effects before and after adding the survival contribution. If survival probabilities differ meaningfully across latent profiles, but the two posteriors are identical, the update may have omitted the event-free likelihood. Also check that the landmark survival is included exactly once: including it in the posterior and then treating survival from baseline as the future conditional risk would not target the stated prediction. Numerical integration methods such as quadrature or Monte Carlo integration approximate the required integrals; they do not correct a missing survival factor.

The principal alternative is a marker-only posterior update, which can be appropriate for a different target that conditions on biomarker history but not on event-free survival to \(s\). It is not the patient-specific prediction for someone known to be event-free at \(s\). A population-level marginal survival probability is different again: it averages over a population distribution and does not, by itself, incorporate this patient’s biomarker history and landmark status. The distinction between marginal survival and conditional risk is useful here, but the key point is that the landmark information must be included in the patient-specific update.