Why Condition on a Landmark?
A prediction made at a fixed time \(s\) answers a different question from a prediction made at baseline. At baseline, the population includes people who may experience the event before \(s\). At the landmark, the target population consists of those who have remained event-free through \(s\). Their risk over the next \(h\) time units is conditional on reaching the landmark event-free.
This distinction matters whenever clinical decisions are made at a follow-up visit. For example, a clinician asking about a patient's risk of an event within the next year at a two-year visit does not want the probability of an event during the first two years included in that answer. The time origin for the prediction is the landmark, although event time may still be measured from the original baseline.
The strict inequality \(T>s\) defines event-free status at the landmark. For settings where event times can equal \(s\), the study must state how events recorded exactly at the landmark are classified; the convention must be used consistently in eligibility and in the risk interval. The horizon \(h\) is a duration after \(s\), so the prediction window ends at \(s+h\).
Relationship to Survival Probabilities
For a single event type, define the survival function \(S(t)=P(T>t)\). Provided \(S(s)>0\), the conditional risk has a direct expression in terms of survival probabilities:
The first equality uses the fact that the event \(s<T\leq s+h\) is contained in \(T>s\). The numerator is the probability of surviving past \(s\) but not past \(s+h\); dividing by the probability of surviving past \(s\) conditions on being at risk at the landmark. Thus the ratio \(S(s+h)/S(s)\) is conditional survival through the next \(h\) time units, and its complement is conditional event risk.
The same identity can be stated for a landmark covariate value \(z\), using the conditional survival function \(S(t\mid z)=P(T>t\mid Z(s)=z)\), when the conditioning is well-defined and \(S(s\mid z)>0\):
This is a definition of the target quantity, not a claim that a particular model or estimation method is appropriate. The conditional probabilities must refer to the same target population, event definition, and covariate information. In particular, covariates used for a prediction at \(s\) must be available by \(s\), not measured later.
Worked Example: Converting Survival Probabilities to Landmark Risk
Suppose a hypothetical cohort has \(S(2)=0.72\) and \(S(3)=0.54\). Find the risk of the event between year 2 and year 3 among people event-free at year 2.
Here \(s=2\) years, \(h=1\) year, and \(S(s)>0\), so the single-event formula applies:
The landmark conditional risk is 0.25, or 25%, over the year after the two-year landmark. As a check, the unconditional probability of an event in that interval is \(S(2)-S(3)=0.72-0.54=0.18\). Dividing by the probability of being event-free at year 2 gives \(0.18/0.72=0.25\). The 18% interval probability and the 25% conditional risk have different denominators and answer different questions.
Three Risks That Should Not Be Confused
A risk is not fully described by a number alone. Its time origin, target population, conditioning set, and event definition all matter. Three quantities are often confused:
| Quantity | Definition | Population and question |
|---|---|---|
| Baseline cumulative risk by \(s+h\) | \(P(T\leq s+h)\) | Everyone at baseline; event by the specified time from baseline. |
| Unconditional interval probability | \(P(s<T\leq s+h)\) | Everyone at baseline; event occurs in the interval, without conditioning on reaching \(s\). |
| Landmark conditional risk | \(P(s<T\leq s+h\mid T>s)\) | Those event-free at \(s\); event occurs in the next \(h\) time units. |
For a single event type, the unconditional interval probability equals \(S(s)-S(s+h)\), whereas landmark risk divides this probability by \(S(s)\). Baseline cumulative risk by \(s+h\) also includes events before or at \(s\), so it is not an estimate of the next-horizon risk among landmark survivors.
Worked Example: Separating Baseline and Landmark Questions
In a hypothetical study, 1,000 people begin follow-up. By year 2, 300 have had the event, leaving 700 event-free. Between years 2 and 3, a further 140 have the event. Assume complete event ascertainment for this arithmetic example.
The unconditional probability of an event in years 2 to 3 is \(140/1{,}000=0.14\), or 14%. The conditional risk among those event-free at year 2 is
Thus, the landmark risk is 20% over the year after year 2. The baseline cumulative risk by year 3 is \((300+140)/1{,}000=0.44\), or 44%. These numbers are consistent: \(0.14=0.70\times0.20\), and 0.44 includes the 30% who experienced the event before the landmark. Reporting 44% as the risk faced by a person event-free at year 2 would answer the wrong question.
Conditional Risk Given Landmark Information
Often the target is not one risk for all landmark survivors, but a risk conditional on characteristics known at \(s\). These may include baseline features and measurements collected up to the landmark. The target \(R(s,h\mid z)\) describes the event probability over \((s,s+h]\) for eligible people with the specified landmark information \(z\). It is a conditional probability, not a causal effect of changing \(z\).
Worked Example: Comparing Risks in Two Landmark Groups
Suppose a hypothetical group of people who are event-free at year 1 is divided by a clinical measurement recorded at year 1. Among 200 eligible people in group A, 30 have the event by year 2; among 100 eligible people in group B, 25 have the event by year 2. Assume complete follow-up through year 2 for this example.
With \(s=1\) year and \(h=1\) year, the observed conditional risks are:
The observed risks are 15% and 25% over the year after the landmark. The difference is 10 percentage points, and the ratio is \(0.25/0.15=1.67\). These are descriptive comparisons of risk among the respective eligible groups. They do not establish that the measurement causes the difference, and they do not compare baseline risks for all members of the original cohort.
If a single risk for all 300 landmark survivors is wanted, the groups must be combined using their eligible-population sizes: \((30+25)/(200+100)=55/300\approx0.1833\), or 18.3%. Taking an unweighted average of 15% and 25% would give 20%, which is not the overall risk here because the group sizes differ.
Assumptions, Event Definitions, and Failure Modes
The probability definition itself requires a clearly defined event, landmark, horizon, and eligible population; it does not require a particular statistical model. Interpreting an estimate as this probability additionally requires that the data and analysis support the target. For example, people must be classified as event-free at \(s\) using information that covers follow-up through \(s\), and the event status over \((s,s+h]\) must be ascertained or handled appropriately.
Censoring before \(s+h\) is not the same as remaining event-free through \(s+h\). When follow-up is incomplete, a simple proportion of observed events among eligible people need not estimate the conditional risk. Estimation must account for censoring under suitable assumptions, commonly that censoring is independent of the event time conditional on the variables used in the analysis, together with adequate follow-up support. The target probability remains the same; the method needed to estimate it depends on the observation process.
For competing events, specify whether the target is any first event or a particular event type. If the target is a particular event type and another event can occur first, the probability of the target event by \(s+h\) must account for competing events as actual outcomes that prevent the target first event. It is not generally obtained by taking one minus a survival probability calculated from a cause-specific hazard while treating competing events as ordinary censoring.
A small eligible risk set can make group-specific risk estimates unstable or leave some covariate profiles unsupported. Also check for time-origin errors: a one-year horizon after a year-2 landmark ends at year 3, not year 1. The risk set, denominator, and event window should be reported explicitly so the estimand can be understood without inference from a label such as “one-year risk.”
Why the Landmark Estimand Matters
A baseline model and a landmark prediction target different populations and time questions. A baseline prediction may be useful for initial prognosis, but it can become less relevant after a person has remained event-free and acquired new clinical information. Landmark conditional risk makes the updated target explicit: among those still eligible at \(s\), what is the probability of the event in the next \(h\) time units?
The principal alternative is to report an unconditional risk from baseline, which may be appropriate when the decision is made at baseline but is not interchangeable with a conditional risk at a later time. A second alternative is to report the unconditional probability of an event in the interval. That quantity is useful for describing the original cohort's event burden in that interval, but it does not describe the probability for a person known to have reached the landmark event-free. Choose among these quantities by matching the denominator and time origin to the actual question.
Check Your Understanding
Use the definitions above. Each answer includes the calculation or reasoning needed to identify the target quantity.
- Suppose \(S(4)=0.60\) and \(S(5)=0.45\). What is the conditional risk from year 4 to year 5 among people event-free at year 4? Solution: \(1-S(5)/S(4)=1-0.45/0.60=0.25\), or 25% over the year after the landmark.
- In the same setting, what is the unconditional probability of an event between years 4 and 5? Solution: \(S(4)-S(5)=0.60-0.45=0.15\), or 15% of the baseline cohort. It differs from 25% because it is not conditioned on being event-free at year 4.
- Why is \(P(T\leq s+h)\) not generally the landmark risk? Solution: It includes events occurring by the landmark among the baseline population. Landmark risk conditions on \(T>s\) and concerns only events in \((s,s+h]\).
- At a landmark, group A has 80 eligible people and 12 events during the horizon; group B has 120 eligible people and 30 events. Find the combined observed risk. Solution: The combined risk is \((12+30)/(80+120)=42/200=0.21\), or 21%. It is a size-weighted combination, not the unweighted average of the two group risks.
- If a person experiences a competing event before the target event during the horizon, may that person simply be counted as event-free at the horizon for a target-specific first-event risk? Solution: No. A competing event prevents the target first event and must be handled as a competing outcome, not treated as ordinary event-free follow-up.