Landmark Outcomes for Delayed Event Endpoints

Why a Fixed Landmark Can Help

When the final endpoint takes time to mature, a MAMS design may use an intermediate measure to compare arms at an interim look. One such measure is whether a participant has experienced the event by a prespecified time after a common origin. This gives a clinically interpretable status at a fixed time, even when the final event-time analysis is not yet available. Unlike a simple binary outcome with complete follow-up, however, some participants may be censored before the landmark, leaving their status at that time unknown.

The key distinction is between no event observed by the landmark and known to be event-free at the landmark. A participant censored early without an event is not known to be event-free at the landmark, and should not automatically be counted either as an event or as event-free. The introductory material on time-to-event intermediate outcomes motivates using early event information; here the focus is the fixed-time status and the effect of incomplete follow-up on arm-selection data.

Definition: Let \(T\) be time from a prespecified origin to the event of interest, and let \(\tau\) be a fixed landmark time. Landmark event-free status is \(Y(\tau)=I(T>\tau)\): it equals 1 if the participant remains free of the event through \(\tau\), and 0 if the event occurs at or before \(\tau\). In arm \(a\), landmark survival is \(S_a(\tau)=P(T>\tau\mid A=a)\), where \(A\) denotes randomized treatment assignment.

The event definition, time origin, and \(\tau\) must be specified consistently across arms. If the event occurs exactly at \(\tau\), the definition above counts it as having occurred by the landmark, so \(Y(\tau)=0\). The measure is specific to the chosen event definition and time point; it does not summarize all features of the event-time distribution.

What Censoring Before the Landmark Means

For each participant, let \(C\) be the censoring time, \(X=\min(T,C)\) the observed time, and \(\Delta=I(T\leq C)\) the event indicator. If \(\Delta=1\) and \(X\leq\tau\), the participant is known to have had the event by the landmark. If \(X\geq\tau\) and no event has occurred by \(\tau\), the participant is known to be event-free at the landmark. But if \(\Delta=0\) and \(X<\tau\), the participant’s landmark status is unknown: the event may or may not have occurred after censoring and before \(\tau\).

A standard way to estimate \(S_a(\tau)\) is the Kaplan–Meier product-limit estimator, calculated separately within each arm. Its interpretation requires more than having recorded follow-up times. Within an arm, censoring must be independent of the event time, or independent conditional on variables used to adjust the analysis. Event times must be measured accurately, censoring must be distinguished from an event, and participants must share a defined time origin and landmark. With covariate-dependent censoring, an unadjusted Kaplan–Meier estimate need not be valid just because censoring is independent after conditioning on those covariates.

At each distinct event time \(t_j\leq\tau\), let \(n_j\) be the number at risk immediately before that time and \(d_j\) the number of events at that time. The estimate multiplies the estimated probability of surviving each event time. Censoring reduces later risk sets, but does not itself count as an event.

\hat{S}_a(\tau) = \prod_{t_j\leq\tau} \left(1-\frac{d_j}{n_j}\right)

This formula does not assign a known landmark outcome to every participant. Rather, under the independent-censoring assumption, it uses the event experience among those still at risk to estimate survival for the arm’s original population. It therefore differs from simply calculating the fraction of participants with known status who are event-free.

Worked Calculations

Worked Example: Kaplan–Meier Estimate With Early Censoring

Consider a hypothetical arm of 8 participants followed toward a landmark \(\tau\). Before the first event, 2 participants are censored. Thus 6 remain at risk when 1 event occurs. After that event, 1 more participant is censored; 4 are then at risk when a second event occurs. The remaining 3 participants are followed event-free through \(\tau\). Assume censoring is independent of event time within this arm and that event status is correctly recorded.

The risk-set counts at the two event times are \(n_1=6,d_1=1\) and \(n_2=4,d_2=1\). Therefore,

\hat{S}_a(\tau) = \left(1-\frac{1}{6}\right) \left(1-\frac{1}{4}\right) = \frac{5}{6}\times\frac{3}{4} = 0.625

The estimated probability of remaining event-free through the landmark is 0.625, or 62.5%, for this arm. This is an estimate, not a claim that the 3 participants followed to \(\tau\) alone represent all 8 participants. The two early censorings and the censoring between events are handled through the risk sets, relying on the stated censoring assumption.

Greenwood’s estimated variance is

\widehat{\operatorname{Var}}\{\hat{S}_a(\tau)\} = \hat{S}_a(\tau)^2 \sum_{t_j\leq\tau} \frac{d_j}{n_j(n_j-d_j)} = 0.625^2 \left\{ \frac{1}{6\times5}+\frac{1}{4\times3} \right\} = 0.04557

The estimated standard error is \(\sqrt{0.04557}=0.2135\). This sizable standard error relative to the estimate reflects substantial uncertainty in this small hypothetical sample. It is not, by itself, a complete confidence interval or a reason to treat the estimate as precise.

Worked Example: Why Complete-Case Proportions Can Differ

Use the same hypothetical data. By \(\tau\), 2 participants are known to have had an event, 3 are known to be event-free, and 3 were censored before \(\tau\) without an event. If an analyst discards the 3 censored participants and divides the 3 known event-free participants by the 5 participants with known status, the result is

\frac{3}{3+2}=0.600

This complete-case proportion is 0.600, compared with the Kaplan–Meier estimate of 0.625. The difference arises because the two calculations use different information and assumptions. A complete-case calculation describes the observed subset with known status; it estimates arm survival only if that subset is representative for the target population with respect to landmark status. Kaplan–Meier instead uses the timing of events and censorings and requires independent censoring. Neither method makes unobserved statuses known, and the numerical proximity here does not establish that the complete cases are representative.

For arm selection, replacing early-censored participants with failures would be equally unjustified absent a prespecified estimand that defines them that way. It would change the outcome being analyzed rather than solve the missing-status problem.

Worked Example: Informative Censoring Can Defeat the Estimate

Suppose, hypothetically, that 10 participants in one arm consist of 5 high-risk and 5 low-risk individuals. All 5 high-risk individuals would experience the event at time 2, before a landmark at \(\tau=3\), but each is censored at time 1. The 5 low-risk individuals are followed to \(\tau\) without an event. In the underlying full data, landmark survival is \(5/10=0.500\).

In the observed data, all 5 censorings occur before any observed event, and the 5 remaining participants have no events. The Kaplan–Meier product consequently contains no event-time factor and equals 1.000. It substantially overestimates the true landmark survival of 0.500 because censoring is related to the participants’ event risk. The independent-censoring assumption fails: the censored participants would have had different event outcomes from those who remained observed.

A risk-set calculation cannot recover outcomes that are systematically missing in this way merely by applying the product-limit formula. Useful responses include investigating why follow-up ended, measuring predictors of censoring, using methods that appropriately account for measured predictors when their assumptions are tenable, and conducting sensitivity analyses for departures from those assumptions. If the reasons for censoring are not understood, arm-selection conclusions based on the estimate should be treated cautiously.

Using Landmark Estimates for Arm Selection

At a selection look, landmark estimates provide an arm-specific summary on a common, interpretable scale: the estimated proportion event-free at \(\tau\). For example, if hypothetical estimates are \(\hat{S}_A(\tau)=0.80\) and \(\hat{S}_0(\tau)=0.70\) for an experimental arm and control, the estimated difference is

\hat{S}_A(\tau)-\hat{S}_0(\tau)=0.80-0.70=0.10

This corresponds to an estimated 10 percentage-point higher event-free probability at the landmark in arm A. It does not establish that the difference is statistically reliable, that it will persist beyond \(\tau\), or that the arm will meet a final-endpoint objective. In a MAMS design, the prespecified selection procedure must account for uncertainty and for the consequences of making a selection from interim data; the landmark estimate itself is only one input to that procedure.

The main alternative is to retain more of the event-time information rather than reduce it to a fixed-time status. That can be useful when the timing of events matters, but it answers a different question and may be harder to interpret as a simple arm-selection summary. A complete-follow-up binary proportion is straightforward when status at \(\tau\) is available for everyone. With censoring before \(\tau\), however, a raw proportion among observed participants is not a general substitute for an appropriate time-to-event estimate.

A landmark also has limitations. Estimates can be unstable when few participants remain at risk near \(\tau\), and they may be sensitive to the chosen time point. Different arms can have different follow-up maturity at the interim look, so apparent differences in the amount or timing of censoring deserve attention. The measure also does not distinguish event types or describe what happens after the landmark. Those are not reasons to avoid landmark outcomes, but they limit what conclusions selection data can support.

Diagnostics and Common Pitfalls

Before using landmark estimates as arm-selection data, inspect event and censoring counts, numbers at risk over time, and how much follow-up each arm has accumulated by \(\tau\). Review reasons for censoring and whether they differ by arm or are associated with prognosis. These checks can reveal concerns, but similar censoring percentages across arms do not prove that censoring is independent of event time within each arm. Independence is an assumption about the relationship between event and censoring processes, not simply a balance in their overall totals.

A related issue is sparse risk sets. If very few participants remain at risk just before \(\tau\), the estimate can be highly uncertain even if its point value looks favorable. Greenwood’s standard error is a useful uncertainty summary under its assumptions, but a normal approximation based on that standard error can behave poorly near the boundaries 0 and 1 or in small samples. Any interval used in a selection procedure should be suitable for survival probabilities and the design’s prespecified analysis plan.

Common mistake: “No event was recorded before this participant’s early censoring, so count the participant as event-free at \(\tau\).” This is incorrect: absence of an observed event after follow-up ends does not establish \(T>\tau\). The correction is to treat the landmark status as unknown and use a method whose censoring assumptions are defensible, or to acknowledge that the available data do not identify the desired arm-specific status without further assumptions.

Key takeaway: A landmark outcome is a well-defined fixed-time survival probability, but an individual censored before that time has unknown status. Kaplan–Meier estimation accounts for such censoring through risk sets only when its assumptions are plausible; arm-selection decisions must reflect the estimate’s uncertainty and limitations.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

For a landmark at \(\tau=6\) months, a participant has an event at 5 months. What is the landmark status? What if the participant is censored event-free at 4 months?
At an event time, 10 participants are at risk and 2 events occur. What factor does that event time contribute to the Kaplan–Meier estimate?
Why is it generally not valid to divide the number event-free at the landmark by the number whose status is observed when some participants were censored early?
In a hypothetical arm, \(\hat{S}_A(\tau)=0.72\) and \(\hat{S}_0(\tau)=0.64\). State the estimated difference in context and give one conclusion it does not establish.
What does it suggest if high-risk participants are more likely to be censored before the landmark?

Solutions. (1) The participant with an event at 5 months has \(Y(6)=0\). The participant censored at 4 months has unknown status at 6 months, not status 1 or 0. (2) The factor is \(1-2/10=0.80\). (3) The observed-status subset may differ systematically from the full arm; its proportion estimates the target only under an appropriate representativeness assumption. (4) The estimated difference is \(0.72-0.64=0.08\), or 8 percentage points higher event-free probability in arm A at \(\tau\). This alone does not establish statistical reliability or final-endpoint benefit. (5) Censoring may be informative; the Kaplan–Meier estimate can be biased because the censored participants’ event risks differ from those remaining under observation.