Selecting Prediction Horizons for Landmarking

What a Prediction Horizon Specifies

A landmark prediction has two time choices. The landmark time determines when a patient must be eligible and when predictor information is summarized. The prediction horizon determines how far beyond that landmark the prediction concerns. A prediction for the next 30 days and a prediction for the next year can therefore use the same landmark and predictors while answering different clinical questions.

The horizon should correspond to a decision: for example, whether to intensify monitoring over the next month or whether a patient is likely to experience an event within the next year. Choosing a convenient horizon first and attaching a clinical interpretation afterward can produce a mathematically valid risk that is not useful for the decision.

Definition: Let $s$ be a landmark time, $T$ the time of the event of interest, and $h>0$ a prediction horizon measured in time units after the landmark. Among patients who are event-free at $s$, the horizon-specific risk, given landmark information $X(s)=x$, is $$ R(h\mid s,x)=P(T\leq s+h\mid T>s,\ X(s)=x). $$ It is the probability of the event during the interval $(s,s+h]$, conditional on being event-free at $s$ and on the stated predictor information. The horizon $h$ is elapsed time after the landmark; $s+h$ is the corresponding endpoint on the study time scale.

The earlier material on conditional risk estimands at a landmark establishes the importance of naming the conditioning population. Here, the horizon completes that estimand: changing $h$ changes the event interval and generally changes the risk, even when $s$ and $x$ stay fixed. A risk at 90 days after a landmark is not the same quantity as risk by study day 90 unless the landmark is at time zero.

Choose the Horizon From the Decision

A useful horizon is clinically interpretable and supported by the follow-up data. Start by specifying the action that the prediction could inform, the period during which that action is relevant, and the time units and endpoint convention. Then check whether enough eligible patients have observation extending to the proposed horizon. A horizon that is too short may miss the event window relevant to the decision; one that is too long may depend heavily on uncertain extrapolation or on a small number still under observation.

Candidate horizons should be compared on the same landmark cohort and for the same event definition. This makes clear how the target changes with time. Longer horizons include the earlier event interval, so the cumulative risk for a single event type cannot decrease as the horizon grows. However, a longer horizon does not automatically make a prediction more useful: clinical relevance, precision, and follow-up support also matter.

Worked Example: Matching the Horizon to a Follow-Up Decision

Suppose a clinic wants to use a landmark prediction to decide which event-free patients should receive a telephone check after a treatment visit. The proposed decision is whether to arrange an additional check during the next six weeks. The landmark is the treatment visit, and the event of interest is an unplanned admission.

A 14-day horizon would target $R(14\mid s,x)$, the risk of admission in the two weeks after the visit. That may be relevant if the check is intended to address immediate complications, but it does not cover the full six-week period in which the proposed check is meant to help. A 180-day horizon would include many months beyond the decision window and could combine different phases of care.

A six-week horizon is $h=42$ days. It targets $$ R(42\mid s,x)=P(T\leq s+42\mid T>s,\ X(s)=x). $$ If the data record admission times and follow-up through at least 42 days for a reasonable number of landmark-eligible patients, this target aligns with the decision and has direct data support. The choice is justified by the stated clinical window, not by which horizon produces the largest estimated risk.

Before using it, the clinic would still need to check how many patients have follow-up through day 42 and whether censoring before day 42 is substantial. The decision fit alone cannot establish that the risk is estimable with adequate reliability.

Event Status and Censoring at a Fixed Horizon

Let $C$ denote the censoring time, $U=\min(T,C)$ the observed follow-up time, and $\delta=I(T\leq C)$ the event indicator. For a landmark cohort, these quantities can be expressed as times since the landmark; the horizon is then $h$. A patient's event status by $h$ is known if an event is observed by $h$, or if the patient is observed event-free for the entire interval through $h$. If follow-up ends before $h$ without an event, status at $h$ is unknown.

Key rule: For a horizon $h$, an event by the horizon is observed when $U\leq h$ and $\delta=1$. Event-free status through the horizon is observed when $U\geq h$ with no event by $h$. If $U<h$ and $\delta=0$, the patient's event status at $h$ is unknown, not event-free.

This rule defines what the records tell us; it does not by itself specify how to estimate risk with censored observations. A raw fraction of events among patients whose status happens to be known can be biased if being observed through $h$ is related to event risk. For example, patients with worsening health may be more likely to leave observation early. Methods that use censored follow-up require assumptions about censoring, commonly that censoring is independent of the event time conditional on the information used in the analysis. That assumption must be scientifically plausible and should be examined where possible.

The Kaplan–Meier estimator is one standard way to estimate event-free survival in a single event setting under independent censoring. At each observed event time, it multiplies the previous survival estimate by the fraction of the risk set that remains event-free. The estimated horizon risk is one minus the estimated survival at $h$. With predictors, corresponding conditional survival estimates require an appropriate model; the same caution about censoring applies.

Worked Example: Classifying Follow-Up at a 30-Day Horizon

Consider 20 hypothetical patients who were eligible at the landmark, with $h=30$ days. By day 30, three have a recorded event on days 8, 19, and 27. Two are censored on days 12 and 24 without a recorded event. The other 15 are observed event-free through day 30.

The three recorded events are known events by the horizon because each event time is at most 30 days. The 15 patients observed event-free through day 30 have known event-free status at the horizon. The two censored before day 30 have unknown status at day 30: either could have experienced the event after censoring.

Thus 18 of 20 patients have known status at day 30, while two do not. The calculation $3/18=0.167$ is the fraction of events among patients with known status, not automatically an unbiased estimate of the 30-day risk in all 20 patients. It would require an additional justification that excluding the two censored patients does not distort the event fraction. Instead, a survival method can use their partial follow-up under its censoring assumptions.

If one of the two patients had instead been observed event-free through day 30, that patient's status would be known as event-free, even if follow-up ended immediately after day 30. Follow-up ending before the horizon is the critical distinction.

Comparing Risks Across Candidate Horizons

Comparing risks at several horizons can reveal whether event burden is concentrated early or accumulates gradually. The comparison should use the same eligible landmark cohort, predictor definition, event type, and estimation approach. Report each horizon with its units and explain which decision it addresses. A difference between horizon-specific risks is not evidence that one horizon is inherently better; it is a difference in the target quantity.

For a single event type with a consistently defined event time, the conditional risks are nondecreasing in $h$. This follows because if $h_1<h_2$, then $\{T\leq s+h_1\}$ is a subset of $\{T\leq s+h_2\}$, among patients with $T>s$. Therefore, $$ R(h_1\mid s,x)\leq R(h_2\mid s,x). $$ A reported decrease can indicate sampling variation in estimates, different analysis populations or methods, or a coding or definition problem; it is not the expected ordering of the underlying cumulative risks.

Here is the survival-based calculation behind a horizon risk. Let $S_s(h\mid x)=P(T>s+h\mid T>s,X(s)=x)$ be conditional event-free survival for $h$ time units after the landmark. The event by $s+h$ and remaining event-free beyond $s+h$ partition the landmark-eligible outcomes, so $$ R(h\mid s,x)+S_s(h\mid x)=1, \qquad R(h\mid s,x)=1-S_s(h\mid x). $$ Equivalently, if $S(t\mid x)$ is survival from the study origin under the same covariate conditioning, the conditional survival ratio is $S(s+h\mid x)/S(s\mid x)$, when the denominator is positive. The ratio conditions on having survived to $s$; it is not the unconditional survival probability from the study origin.

Worked Example: Comparing Two Horizons With Censoring

In a hypothetical landmark cohort of 10 event-free patients, follow-up after the landmark gives one event at day 2, one censoring at day 3, two events at day 4, one censoring at day 6, and one event at day 8. The remaining four are observed event-free through day 10. Assume, for this illustration, independent censoring and a single event type.

For the Kaplan–Meier estimate, the risk set begins with 10. At day 2, one of 10 has the event, so the survival multiplier is $9/10$. The day-3 censoring removes one patient from later risk sets but does not create an event multiplier. At day 4, eight remain at risk and two have the event, giving multiplier $6/8$. Thus through day 5, $$ \widehat S(5)=\frac{9}{10}\times\frac{6}{8}=0.675, \qquad \widehat R(5)=1-0.675=0.325. $$ The estimated event risk by day 5 after the landmark is 32.5%.

By day 9, the event at day 8 also contributes. After the events at days 2 and 4 and censorings at days 3 and 6, five patients remain at risk immediately before day 8. One has the event, so $$ \widehat S(9)=\frac{9}{10}\times\frac{6}{8}\times\frac{4}{5}=0.540, \qquad \widehat R(9)=1-0.540=0.460. $$ The estimated risk by day 9 is 46.0%. The day-6 censoring is handled by reducing the later risk set; it is not counted as an event or as event-free through day 9. These estimates depend on the independent-censoring assumption and are imprecise in a cohort of only 10. The longer horizon includes an additional event time and has a higher estimated cumulative risk, but whether five or nine days is clinically useful depends on the decision being considered.

Alternatives, Diagnostics, and Limitations

A fixed-horizon landmark logistic model directly targets event status by a chosen horizon when that status is observed or when censoring is handled appropriately. Its advantage is that the target can be closely matched to a specific decision. Its limitation is that a binary outcome at one horizon does not describe the timing of events within the interval, and censoring before the horizon leaves the binary outcome unknown.

A landmark Cox model for residual survival uses follow-up time after the landmark and can represent event timing rather than reducing it to one binary outcome. A survival estimate can then be evaluated at a chosen horizon. This approach avoids requiring a separate binary outcome for every candidate horizon, but it still requires suitable model and censoring assumptions. Neither approach makes an arbitrary horizon clinically meaningful. Comparing several horizons from a pooled landmark analysis is another option, but the target and support at each horizon still need to be stated.

Some practical checks help identify trouble before interpreting a horizon-specific risk:

Tabulate follow-up support. At each candidate horizon, report how many landmark-eligible patients are known to have an event, are observed event-free through the horizon, or are censored earlier.
Inspect risk-set size over time. A long horizon with few patients remaining under observation can produce an unstable estimate, even if the total cohort was initially large.
Check horizon-specific event counts. Few events near or before a horizon limit precision and make fitted risks sensitive to individual observations.
Assess censoring plausibility. If loss to follow-up may depend on event risk, an estimator relying on independent censoring may not represent the target cohort risk.
Check consistency of definitions. Verify that the event, time origin, landmark eligibility rule, horizon units, and endpoint inclusion convention are the same across comparisons.

A further limitation arises when a different event can prevent the event of interest. For example, death may prevent a later nonfatal event from occurring. If the clinical target is the probability of the event of interest before such a competing event, treating the competing event as ordinary censoring does not estimate that probability; a competing-risks estimand and method are needed. The event definition and the clinical question therefore determine whether ordinary event-free survival is adequate.

Common pitfall: “The patient was censored before day 30, so there was no event by day 30.” This is wrong because the patient’s later event status was not observed. The correction is to mark status as unknown at day 30 and use an analysis that handles censored follow-up under stated assumptions, rather than coding the patient as event-free.

Horizon choice also does not guarantee good predictive performance. A clinically aligned horizon can still yield poorly calibrated or weakly discriminating predictions, and a large estimated risk does not by itself establish that acting on the prediction improves outcomes. Those are separate questions from defining the horizon-specific target.

Check Your Understanding

Answer each question using the distinction between the landmark, the horizon, and observed follow-up.

A landmark is at study day 120 and the prediction horizon is 60 days. What calendar-time endpoint does the horizon represent, and what event interval is targeted?
A patient is event-free at the landmark and censored 20 days afterward. Is event status known at a 30-day horizon? Explain.
For one event type, the estimated risk is 0.18 by 30 days after the landmark and 0.31 by 90 days. Is the ordering coherent? What would a decrease suggest?
Why is the fraction of events among patients with known status not necessarily an unbiased estimate when some patients are censored before the horizon?
A competing event prevents the event of interest. Why might treating that competing event as ordinary censoring answer the wrong question?

Solutions. (1) The endpoint is study day $120+60=180$, and the target interval is after day 120 through day 180, conditional on being event-free at day 120. (2) No. Because follow-up ends before day 30 without an observed event, status at the horizon is unknown. (3) Yes. Cumulative risk for a single event type cannot decrease with a longer horizon. A decrease in estimates can reflect sampling variation, differing populations or methods, or a coding or definition error. (4) Early-censored patients may differ in event risk from those whose status is known. Excluding them can therefore change the composition of the analyzed group; an appropriate censoring method and its assumptions are needed. (5) Ordinary censoring treats the competing event as if the person could remain capable of experiencing the event of interest later. If the competing event makes that impossible, this does not estimate the actual probability of the event of interest before the competing event.