Repeated Landmark Predictions for One Patient

Why a Patient Can Have More Than One Risk Prediction

A prediction made today may no longer be the most useful prediction after a patient returns for another visit and a new biomarker result is recorded. Landmarking provides a way to update the forecast: at each selected time, estimate the patient’s risk over a specified future horizon using the information available by that time. The result is a sequence of predictions, not one forecast that remains fixed throughout follow-up.

The key qualification is that each prediction is conditional on the patient being event-free and eligible at its landmark. A patient who has had the event cannot receive a later event-risk prediction of this kind. Nor should a patient who is no longer under observation be treated as known to be event-free. Thus the set of patients eligible to inform the model can change from one landmark to the next.

Write \(H_i(t_k)\) for the patient information available by landmark \(t_k\), including the biomarker information used by the fitted model. The estimand for that patient is

This is a conditional risk for the next \(\Delta\) time units, not the patient’s probability of ever having the event and not, by itself, the probability of the event by \(t_k+\Delta\) measured from study entry. The conditioning information changes as follow-up proceeds. For this reason, two predictions for the same person can differ even when both are correctly calculated.

How the Sequence Is Produced

A pooled landmark model uses eligible patient-landmark records from multiple landmark times to estimate a shared prediction rule. In a fixed-horizon logistic version, the outcome for a record is whether the event occurs within the horizon after its landmark. The model may include landmark time as well as the covariates available at that time. For illustration, let \(x_i(t_k)\) be a biomarker measured by \(t_k\). A simple model is

This model assumes that the specified linear predictor adequately describes the log odds of the event during the horizon among eligible records, that the fitted probabilities are applicable to the patient’s setting, and that outcomes for the horizon are handled appropriately. In particular, a record censored before the horizon ends does not automatically have a known event-free outcome for the whole horizon; the analysis must use an appropriate censoring strategy. The model also requires that the biomarker value used at a landmark was actually available by that time. Using a later measurement would leak future information into the prediction.

At each landmark, the model is evaluated using that time’s covariate information. The same patient may therefore contribute more than one eligible record to model fitting, but these records are not independent people. Inference and validation should account for repeated records from the same patient, for example by resampling patients rather than individual rows. A predicted probability for one patient is still a model-based conditional prediction; it is not made more certain merely because that patient appears at several landmarks.

Why the Risk Set Changes

Let the landmark risk set consist of patients known to be event-free and uncensored at that landmark, with the required prediction information available. As time advances, some patients experience the event, are censored, or lack information needed for prediction. They cannot simply be carried forward as if they remained eligible. The patient-specific sequence is therefore defined only at landmarks where the patient is eligible.

The same conditioning applies to the model’s training data. A record at \(t_k\) describes a patient who has survived event-free and remained observed to \(t_k\). A later landmark’s records describe a different, selected risk set: patients who have also remained event-free and observed for longer. Consequently, a risk at \(t_{k+1}\) is conditional on reaching \(t_{k+1}\), not a re-expression of the risk that was calculated at \(t_k\).

Deriving Risk from a Landmark Cox Model

A landmark Cox model can instead describe the hazard after a landmark. Let \(s\) denote elapsed time since \(t_k\), and suppose the model for an eligible patient is

This assumes proportional hazards over residual time \(s\), with the stated covariate effect and baseline cumulative hazard \(H_{0k}(\Delta)=\int_0^\Delta h_{0k}(s)\,ds\). Integrating the hazard gives the conditional survival probability to the end of the horizon. The conditional risk is one minus that survival probability:

The baseline cumulative hazard may be specific to the landmark, depending on how the landmark Cox model was fitted. The formula makes the conditioning explicit: it predicts survival over residual time among patients who have reached the landmark event-free. It does not use the survival probability from study entry as though the patient were still in the original risk set.

Checking a Prediction Sequence and Choosing an Approach

A sequence is useful only when each prediction is attached to the right patient state and time interval. Before interpreting it, verify that the patient was eligible at every stated landmark, that each biomarker value was available by that landmark, and that the horizon and event definition stayed consistent. Check that the fitted model includes an adequate representation of landmark time and that its predictions are calibrated at the landmark times and horizons relevant to use. A pooled average calibration result can conceal poor calibration at a particular time.

Pooled landmark models use records from several times to estimate a common structure and can make efficient use of data when the model is suitably specified. Fitting a separate model at every landmark allows more time-specific behavior but may be unstable when few patients remain at later landmarks. A single baseline prediction avoids repeated updating but cannot use measurements that arrive later. These approaches answer different operational questions; none makes the risk sets identical.

A common mistake is to say, “The patient’s risk increased from 5.73% to 13.70%, so the probability of the event over the whole six months is 13.70%.” That reasoning treats the month-3 prediction as if it were an unconditional baseline probability. It is not: 13.70% is conditional on being event-free and eligible at month 3 and refers only to the next three months. The correction is to report the landmark and horizon with every risk, and not to combine conditional predictions into a baseline cumulative risk unless a model and calculation specifically justify that target.

Landmark predictions also have limits. They describe risk among patients who reach each landmark and can therefore differ across times partly because the eligible patients have changed. They depend on the model, the measured information, outcome ascertainment, and the handling of censoring. A sequence is a set of conditional predictions, not a causal account of what a biomarker change does to an individual.