What Population Does a Landmark Prediction Describe?
A risk prediction is meaningful only after specifying for whom and over what interval it applies. In landmark analysis, the prediction is made at a chosen time \(s\), using information available by then. Its target is not everyone who entered the study. It is the population that has remained free of the event through \(s\).
For example, a prediction made at month 6 about an event during the following year applies to people who are event-free at month 6. It does not describe their risk during the first six months, and it does not apply unchanged to people who experienced the event before month 6. The time origin for the predicted interval is the landmark; the calendar-time endpoint is \(s+\tau\), where \(\tau\) is the prediction horizon.
Here, \(T>s\) represents having remained event-free through the landmark. In practice, the event-free status and the covariates must be ascertainable at \(s\). Someone censored before \(s\) has unknown event-free status at that time and cannot simply be treated as a confirmed landmark survivor. The precise event definition also matters: if the event is the first of several possible events, the prediction concerns that specified first-event outcome.
Reading a Landmark Cox Prediction
In a landmark Cox model, the hazard is modeled over residual time after the landmark. Write \(u\) for time elapsed since \(s\). For a fixed landmark, a typical model is
The prediction uses the cumulative baseline hazard \(H_{0,s}(\tau)\) over the next \(\tau\) time units. Under this model, the conditional survival probability and risk are
Thus \(R_s(\tau\mid Z(s))\) is the estimated probability of the event by \(s+\tau\), among people event-free at \(s\) with the specified covariates. It is not the probability of an event by \(s+\tau\) from study entry. The model’s proportional-hazards assumption is that covariate effects multiply the hazard by a constant factor over residual time \(u\); other assumptions needed for a useful estimate include an adequately specified model and censoring that is independent of the event time conditional on modeled information. The horizon should also be supported by follow-up in the landmark risk set.
Worked Example: Interpreting a Predicted Risk
Suppose a hypothetical landmark model at month 6 predicts conditional survival of \(0.82\) over the next 12 months for a patient with a specified covariate profile. Interpret the prediction.
The conditional risk is $$ 1-0.82=0.18. $$
The interpretation is: among comparable patients who are event-free at month 6 and have the stated covariate profile, the model estimates an 18% probability of experiencing the event during months 6 through 18. The 18% is not a prediction of the patient’s risk from study entry, nor does it include any event that could have occurred before month 6. This interpretation assumes the patient belongs to the landmark target population and that the model and follow-up support the 12-month prediction.
Worked Example: Calculating Risk From a Landmark Cox Model
Consider a hypothetical fitted landmark Cox model with \(H_{0,s}(12)=0.12\) over the next 12 months. A binary covariate \(Z(s)\) is 1 for the patient, and its estimated coefficient is \(\log(1.5)\). Calculate and interpret the predicted risk.
The linear predictor is \(\log(1.5)\), so the relative hazard multiplier is $$ \exp\{\log(1.5)\}=1.5. $$
The estimated cumulative hazard for this covariate value is \(0.12(1.5)=0.18\). Therefore, $$ S_s(12\mid Z(s)=1)=\exp(-0.18)\approx 0.8353, \qquad R_s(12\mid Z(s)=1)=1-0.8353=0.1647. $$
The estimated 12-month event risk is about 16.5% among patients with \(Z(s)=1\) who are event-free at the landmark. For comparison, at \(Z(s)=0\), the model gives \(1-\exp(-0.12)\approx0.1131\), or 11.3%. This calculation uses the fitted model’s proportional-hazards form and its estimated baseline cumulative hazard; it is not a causal comparison between covariate groups.
How Survivor Risk Relates to Population Risk
Landmark risk and population-wide cumulative risk answer different questions, but they can be related when the target population and time scale are made explicit. For a single event, the probability of an event by \(s+\tau\) from study entry can be separated into events occurring by \(s\) and events occurring after \(s\):
This identity follows by dividing the event \(T\leq s+\tau\) into two disjoint cases: \(T\leq s\), or \(s<T\leq s+\tau\). For the second case, the definition of conditional probability gives $$ P(s<T\leq s+\tau)=P(T>s)P(s<T\leq s+\tau\mid T>s). $$ Adding the two probabilities yields the displayed identity. If covariates are included, the conditional risk must be averaged over the covariate distribution of the event-free population at \(s\) to obtain its overall post-landmark risk. A prediction for one specified covariate profile is not automatically that average.
Worked Example: Combining Survivor Risk With Earlier Events
In a hypothetical cohort of 1,000 people, 280 experience the event by month 6, leaving 720 event-free at that landmark. Suppose the average estimated risk over the following 12 months among those 720 survivors is 15%. Estimate the cumulative event risk by month 18 for the original cohort, assuming the counts and average risk refer to the same complete target cohort.
The event-free proportion at month 6 is \(720/1000=0.72\). The 12-month risk among those survivors is \(0.15\), so the expected number of additional events is $$ 720(0.15)=108. $$
The total expected number of events by month 18 is \(280+108=388\), giving $$ \frac{388}{1000}=0.388. $$
Thus the estimated cumulative event risk by month 18 in the original cohort is 38.8%. The 15% landmark risk remains the relevant estimate for people who were event-free at month 6; it is not the original cohort’s risk by month 18. This calculation treats the 15% as the average conditional risk for the landmark survivors, rather than the prediction for just one covariate profile.
Conditional Risk Is Not Population-Wide Prognosis
The distinction can be summarized by the target population and time interval:
| Quantity | Population and interval | Interpretation |
|---|---|---|
| Landmark risk \(R_s(\tau\mid Z(s))\) | Event-free at \(s\), with specified covariates; next \(\tau\) time units | Risk of the event during \((s,s+\tau]\), conditional on reaching \(s\) event-free |
| Population cumulative risk \(P(T\leq s+\tau)\) | Original study population; from study entry through \(s+\tau\) | Risk accumulated both before and after the landmark |
| Average survivor risk | Event-free at \(s\); next \(\tau\) time units | Average of conditional predictions over the covariate distribution among landmark survivors |
A fixed-horizon landmark logistic model and a landmark Cox model can both target risk among people event-free at \(s\). A logistic model directly estimates the probability of an event by the specified horizon; a Cox model estimates residual-time survival and can produce predictions at horizons supported by its fitted survival model. The model form differs, but neither model’s prediction should be described as population-wide prognosis unless the target population and calculation have been changed accordingly.
Likewise, an average of individual landmark predictions is not necessarily the risk in the original cohort. To describe the original cohort through \(s+\tau\), earlier events must be included, and the conditional risk after \(s\) must represent the survivor population being considered. A single patient-specific prediction cannot be substituted for that population average without justification.
Interpretation Checks and Limitations
Before reporting a prediction, check that its description includes the landmark, the horizon, the event, and the conditioning population. In particular, verify that the event-free risk set at \(s\) was defined as intended, that covariates were available by \(s\), and that patients censored before \(s\) were not incorrectly counted as known survivors. A horizon with little follow-up or few patients remaining at risk can make a numerical prediction unstable, even when the calculation is correct.
A frequent mistake is to write, “This patient has a 16.5% risk of the event by month 18,” without saying that the prediction is conditional on being event-free at month 6. The correction is: “Among patients with this covariate profile who are event-free at month 6, the estimated risk during the next 12 months is 16.5%.” The omitted condition changes the population and can change the meaning of the number.
These predictions are model-based estimates, not guarantees for individual outcomes. Their interpretation relies on the model, measurement, and censoring assumptions, and their precision depends on the information available in the landmark risk set. They also do not establish that changing a covariate would change a patient’s risk: a predictive association is not, by itself, a causal effect. If other event types can prevent observation of the event of interest, the target must specify how those competing events are handled; simply using \(1-S_s(\tau)\) may not represent the desired event-specific probability.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A model predicts a 9-month risk of 0.12 at month 3. What population and time interval does this describe?
- At a landmark, a Cox model has \(H_{0,s}(6)=0.08\) and a patient’s hazard multiplier is 2. Calculate the six-month risk and state its interpretation.
- In an original cohort, 30% experience the event by the landmark. The average risk over the next horizon among landmark survivors is 20%. What is the cumulative risk by the end of that horizon for the original cohort?
- Why is a person censored before the landmark not automatically eligible as a confirmed landmark survivor?
Solutions. (1) It describes the estimated risk during months 3 through 12 among people event-free at month 3 who have the specified covariate information. (2) The cumulative hazard is \(0.08(2)=0.16\), so risk is \(1-\exp(-0.16)\approx0.1479\), or 14.8%. This is the estimated risk during the next six months among patients with that covariate profile who are event-free at the landmark. (3) The proportion event-free at the landmark is \(1-0.30=0.70\). The overall risk is \(0.30+0.70(0.20)=0.44\), or 44%. (4) Censoring before the landmark means the event-free status at the landmark is unknown; the person cannot be classified as having reached it event-free without additional information or an appropriate method.