When Landmark Predictions Approximate Joint-Model Predictions

When Are the Predictions Comparable?

Landmark and joint models can answer a closely related clinical question: for a patient who is event-free at time \(s\), what is the probability of an event by \(s+\Delta\), given the information available at \(s\)? A landmark model estimates this conditional risk directly from predictors recorded or derived at \(s\). A joint model uses a longitudinal model and an event model together, and conditions on the patient’s observed biomarker history to infer the distribution of the patient’s underlying trajectory.

The predictions can be similar when the landmark predictors retain the parts of that history that matter for future risk, the models target the same population and horizon, and the fitted relationships are adequate. Similarity is not automatic: a landmark model may discard information about the trajectory, while a joint model’s prediction reflects uncertainty about the underlying trajectory as well as observed history.

For illustration, suppose a joint model has longitudinal outcome \(Y_i(t)=m_i(t)+\epsilon_i(t)\), where \(m_i(t)\) is the patient-specific underlying biomarker trajectory. Suppose its event hazard, conditional on the latent trajectory and baseline covariates \(w_i\), is \(\lambda_i(t\mid b_i)=\lambda_0(t)\exp\{\gamma^\mathsf{T}w_i+\alpha m_i(t)\}\), where \(b_i\) denotes the patient-specific trajectory parameters. In a setting with a single event and no competing risks, the joint-model risk can be written as:

The expectation averages over uncertainty about \(b_i\) after accounting for the observed biomarker history and the fact that the patient has remained event-free through \(s\). A landmark model instead estimates a risk function of the form \(g_{s,\Delta}\{X_i(s)\}\). Its predictors might include the last observed value, a change from baseline, or a recent slope. These summaries need not reproduce the full conditional distribution of the latent trajectory.

Conditions That Favor Similar Predictions

A useful way to assess the likely agreement is to ask whether the landmark summaries preserve the information that drives risk in the joint model. Approximation is more plausible when the biomarker is measured close to the landmark, its recent value or trend is a strong summary of the relevant trajectory, and the event association depends mainly on that summary over the chosen horizon. It also helps when the prediction horizon is short enough that uncertainty in the future trajectory has limited influence.

The comparison is meaningful only if both predictions target the same quantity. Both must condition on survival to the same landmark, use the same horizon and covariate information, and refer to a comparable population. The fitted models also need to be adequate for that target. In estimating landmark risks, censoring before the end of the horizon must be handled appropriately; for a simple interpretation of the joint-model formula above, the stated single-event setting applies. If either approach is misspecified or the relevant predictors are poorly supported by the data, apparent agreement does not guarantee accurate prediction.

Why Averaging Hazards Can Change the Risk

Even if the landmark model captures a posterior mean hazard, that mean alone may not capture the joint-model risk. The joint model averages survival probabilities across plausible trajectories. In general, averaging hazards first and then converting the result to risk is not the same as averaging risks. The difference follows directly from the shape of the exponential function.

When Recent History Is Not Enough

A landmark model can only distinguish patients to the extent that its predictors distinguish them. If two patients have the same last biomarker value but different recent trends, their underlying trajectories and future hazards may differ. A landmark model using only the last value assigns them the same prediction, while a joint model may infer different latent trajectories from their full histories. Including a recent slope or another informative summary may reduce the gap, though the summary still needs to represent the information relevant to the joint-model risk.

Diagnostics, Alternatives, and Limitations

Assessing agreement should begin by checking the prediction target and information set, not by comparing model labels. Make sure that predictions use the same landmark, horizon, event definition, and patient information available at the landmark. Then examine whether the landmark summaries distinguish clinically plausible histories that imply different trajectory shapes or levels of uncertainty. Comparing predictions across patients with similar last values but different recent histories can expose a summary that is too coarse.

A landmark model is a direct, horizon-specific approach: its flexibility comes from the predictors and functional form chosen for the conditional risk. A joint model represents the longitudinal and event processes together and can use the full observed history to update the latent trajectory distribution. That structure may be useful when trajectory shape or uncertainty is important, but its predictions depend on the chosen longitudinal and event models. Neither approach is automatically preferable for every prediction task. Agreement in one subgroup or horizon does not establish agreement everywhere, and disagreement alone does not show which prediction is more accurate.

A practical comparison should therefore consider whether each model is suitable for the intended prediction task and evaluate predictions against observed outcomes in an appropriate validation sample. Differences may arise because the landmark predictors omit relevant history, because the models encode trajectory-risk relationships differently, or because either fitted model is inadequate. They may also be small when the omitted information has little effect on risk over the horizon. Prediction agreement is a property of the models, predictors, population, and horizon together.

• Pitfall: treating the last value as the entire biomarker history. The reasoning “the last values match, so the predictions should match” ignores different trends and different uncertainty about the underlying trajectory. The correction is to ask whether the chosen landmark features capture the history relevant to risk.
• Pitfall: averaging hazards and calling the result the joint-model risk. The calculation \(1-\exp\{-E(\lambda)\Delta\}\) generally differs from \(1-E\{\exp(-\lambda\Delta)\}\). The correction is to average survival over the conditional distribution of hazards or trajectories when that distribution is part of the prediction.
• Pitfall: comparing predictions with different conditioning sets or horizons. Such predictions answer different questions. Align the landmark, survival condition, horizon, covariates, and target population before interpreting their difference.