Landmark Prediction With Longitudinal Omics

Why Use a Landmark?

Longitudinal omics measurements can change over time, and a prediction based on a person's observed history may be more informative than one based only on baseline data. But the prediction must answer a clear question: for whom is it being made, at what time, and for what future period? Landmark prediction makes those choices explicit. It selects a fixed time \(s\), uses information available by \(s\), and predicts subsequent event risk among people who are still event-free and under observation at \(s\).

This conditional target differs from predicting survival from study entry. It also differs from a dynamic prediction that updates continuously as new measurements arrive. The earlier material on Dynamic Survival Prediction From Joint Models addressed updating predictions through a joint model. Here the focus is a fixed landmark and a model built from the history available there.

The event-free condition is essential: a person who experienced the event before \(s\) is not someone for whom future event risk from \(s\) can be predicted. Likewise, a person's measurements after \(s\) cannot be used to make a prediction that is supposed to be available at \(s\). The landmark, prediction horizon, eligible population, and definition of available history should be specified before model fitting.

Building Predictors From Available History

Write the observed omics history by the landmark as \(H_i(s)\). A model generally does not use every raw measurement as an unconstrained list of predictors. Instead, a prespecified feature map \(g\) converts the history into a manageable predictor vector \(Z_i(s)=g\{H_i(s)\}\). Depending on the scientific question and the measurement schedule, these features might include a most recent value, a mean over a defined window, a change from baseline, or a slope estimated from measurements before \(s\). The earlier material on repeated-measurement estimands and spline models for omics trajectories provides relevant ways to define and summarize trajectories; the landmark-specific requirement is that the summary use only pre-landmark information.

Feature definitions need to address how much history is required. A slope cannot be estimated from a single measurement. If the last measurement was several months before \(s\), its value may not represent the value at \(s\); time since the last measurement can be recorded, or the analysis can use a prespecified model-based estimate based only on measurements available by \(s\). If a feature is unavailable for some eligible patients, the handling rule must be specified and applied consistently. Silently substituting a post-landmark measurement is not a valid solution.

Landmark Cox Model and Conditional Survival

A common approach fits a Cox proportional-hazards model to follow-up after the landmark. Let \(U_i=T_i-s\) be residual event time after \(s\) for an eligible person, and let \(Z_i(s)\) be the predictor vector formed from history through \(s\). A landmark Cox model specifies the hazard at residual time \(u\), conditional on being event-free up to that residual time, as

Here \(\lambda_0(u)\) is the baseline hazard after the landmark and \(\beta\) is a vector of regression coefficients. The model assumes proportional hazards over post-landmark follow-up for the specified features, a suitable functional form for their effects, and independent censoring conditional on the modeled predictors for the survival estimation being used. It also requires the risk sets and event times to be correctly recorded. As in other Cox models, the baseline hazard is not required to follow a particular parametric distribution.

For a person whose predictor vector is fixed at the landmark, the model-based conditional survival estimate through horizon \(u\) is

where \(\widehat{H}_0(u)\) is the estimated cumulative baseline hazard after \(s\). The corresponding estimated event risk is \(1-\widehat{S}(u\mid Z_i(s))\). A Cox model can estimate the relative effects in \(\beta\) without specifying the baseline hazard's form, but a numerical absolute-risk prediction requires an estimate of that baseline hazard as well.

For example, the Breslow estimate of the cumulative baseline hazard at post-landmark event times \(u_j\) is

where \(d_j\) is the number of events at \(u_j\), and \(R(u_j)\) is the set of eligible people still at risk just before that time. This construction makes clear why people censored before an event time contribute to risk sets only until their censoring time.

Clock, Eligibility, and Assumptions

A model can use residual time \(u=T_i-s\), as above, or retain time from study entry. With residual time, everyone enters the post-landmark risk set at zero. If the model retains the original time scale, people must enter the risk set at \(s\), rather than being treated as at risk before they became eligible. This is delayed entry (left truncation). Either clock can be used when the risk sets and target are defined consistently; mixing the two can create biased risk sets and incorrect predictions.

The landmark analysis conditions on observed eligibility and history. For a fitted conditional survival curve to represent the intended target, the model must have adequate support in the relevant predictor range, and censoring after the landmark must be independent of future event time conditional on the modeled information, or handled with an appropriate censoring adjustment. The proportional-hazards and predictor-effect assumptions should be examined rather than presumed. If proportional hazards fail, a single constant coefficient may not describe the predictor association over the whole prediction horizon.

The feature-construction rule is part of the prediction model. Repeating the process at multiple landmarks creates multiple landmark records per person; those records are not independent. Inference and validation should account for the person as the sampling unit, and predictions at each landmark should be assessed among people eligible at that landmark. The earlier material on concordance, time-dependent ROC curves, calibration, and Brier scores gives established tools for evaluating survival predictions; the evaluation must match the landmark population and time horizon.

How Landmark Prediction Differs From Alternatives

Landmarking is often operationally convenient: it uses a defined data cut-off and can be fitted with standard survival-model tools. Its cost is that it fixes the prediction time and compresses the observed history into chosen features. A joint model can represent longitudinal trajectories and event risk together, while a time-dependent Cox model represents changing covariates in its hazard. These alternatives answer related but not identical questions; landmark estimates should not be described as continuously updated predictions.

Diagnostics, Pitfalls, and Limitations

• Information leakage. A feature calculated using an assay collected after \(s\) uses information unavailable at prediction time. Rebuild features using only data timestamped at or before the landmark.
• Wrong target population. Reporting a landmark risk as if it applied to everyone enrolled at baseline ignores the conditioning on event-free status and observation at \(s\). State the landmark-eligible population with every prediction.
• Inconsistent time origin. Treating people as at risk before their landmark eligibility misstates risk sets. Use residual time after \(s\) or account for delayed entry on the original scale.
• Overinterpreting summaries. A last value, slope, or mean is not the full trajectory. Different histories can yield the same summary, and estimated slopes may be unstable when measurements are sparse or noisy.
• Unexamined model fit. Check whether estimated effects are plausible across the prediction horizon, whether enough events support the number of predictors, and whether predictions are calibrated in the intended population. Sparse events, poor overlap, or a changing hazard association can make predictions unreliable.
• Selective measurement availability. The patients with enough assays to form a feature may differ from those without them. State the eligibility and missing-feature rules; do not assume that restricting to complete histories is harmless.

A landmark model is not automatically causal. Its coefficients describe conditional associations in the selected landmark population, not the effects of intervening on an omics measurement. Nor does a good ranking of patients guarantee accurate absolute risks: discrimination and calibration are distinct properties.