Why Landmark Cohorts Need Explicit Rules
A landmark analysis asks about people in a defined condition at a particular time. In competing-risks and multi-state data, that condition cannot be reduced to “still in the study.” By the landmark, some individuals may have experienced a competing event, moved to another state, entered observation late, or become unobservable. Whether each person belongs in the cohort depends on both their observed history and the question the analysis is meant to address.
For example, consider an illness-death process with state 0 for alive without illness, state 1 for alive after illness, and state 2 for dead. A person in state 1 is no longer at risk of a first illness transition from state 0. But that person is alive and may be eligible for a prediction cohort about subsequent state occupation or death. Calling both groups “the risk set” without specifying the target leads to incorrect denominators.
The earlier material on the landmark state-occupation estimand and conditioning information at a landmark motivates defining the target population first. This tutorial focuses on translating that target into a reproducible set of individuals from their records.
Definition of a Landmark Cohort
Let \(s\) be a prespecified landmark time, and let \(X_i(s)\) denote individual \(i\)'s state at that time, when it can be ascertained. Let \(E_i\) be the time individual \(i\) enters observation, and let \(C_i\) be the last time their state is known under the observation scheme. Define \(O_i(s)=1\) when the individual has entered observation by \(s\) and their state is ascertained through \(s\). In the usual convention, \(E_i \leq s \leq C_i\); the handling of events tied with censoring at \(s\) must be specified in advance.
For an analysis restricted to people alive at \(s\), \(\mathcal{A}\) is the set of living states. In the illness-death example, \(\mathcal{A}=\{0,1\}\). For a first-event competing-risks question about illness among people who have not yet had illness or death, the eligible state set is instead \(\mathcal{A}=\{0\}\). These are different landmark cohorts, even though they are formed from the same records.
A person who has died before \(s\) is not eligible for a prediction cohort among people alive at \(s\). A person whose follow-up ended before \(s\) with no later state information is also not eligible for a directly observed cohort defined by state at \(s\). The latter person's state is unknown, not necessarily dead or event-free. If the target is the state among people whose status is known at \(s\), they are excluded; other methods may address incomplete observation, but should not silently assign the person a state.
Constructing Membership From Event Histories
A reliable construction follows a fixed sequence. First establish that the individual has entered the observation scheme by \(s\). Next determine whether their state is ascertained at \(s\). Then reconstruct the state at \(s\) from the ordered transitions, applying the prespecified convention for events at exactly \(s\). Finally, apply the state eligibility rule implied by the target.
Include only individuals whose observation began by the landmark. Someone entering after \(s\) has no observed history at \(s\) in that study.
Determine whether follow-up or another valid data source establishes the individual's state through \(s\). A record ending before \(s\) does not establish the state at \(s\).
Use all observed transitions up to the landmark. In an illness-death process, illness before \(s\) places a living person in state 1; death before \(s\) places them in state 2.
Retain the individual exactly when their ascertained state belongs to the prespecified eligible-state set \(\mathcal{A}\).
Worked Example: Alive at the Landmark Versus Event-Free
Suppose ten hypothetical individuals are followed from time 0, and the landmark is \(s=5\). Records establish that three are alive without illness at time 5, two are alive after illness, two died before time 5, two were lost to follow-up before time 5 without later status information, and one entered observation at time 6.
For a cohort of people alive at time 5, the eligible states are 0 and 1. The two living individuals with prior illness are included; the two known deaths are not. The two individuals lost before time 5 have unknown state at the landmark, and the person entering at time 6 was not yet observed. Therefore,
For a first-illness analysis among people still free of illness and death at time 5, only state 0 is eligible, so the cohort size is \(n_{\{0\}}(5)=3\). Equivalently, the five living people are reduced by the two already ill: \(5-2=3\). Both counts follow from the same data; only the target differs.
Derivation: Why Different Targets Give Different Counts
In the illness-death model, states 0 and 1 are living states, while state 2 is death. For an alive-at-landmark cohort, an individual contributes if they are observed at \(s\) and occupy either living state. Since the state indicators are mutually exclusive,
Summing over individuals gives \(n_{\{0,1\}}(s)=n_{\{0\}}(s)+n_{\{1\}}(s)\). This identity is a useful construction check, not a claim that the two groups have the same event risk. A first-illness cohort uses state 0 because illness has already occurred for someone in state 1. A cohort for subsequent mortality among living individuals can include both states because prior illness does not make a person dead or unobserved. Thus the target defines which states belong in the denominator.
This cohort definition is a membership rule, not an estimator of future state probabilities. It requires correctly recorded histories and a clear target-state rule. Estimating outcomes after the landmark raises additional modeling and observation questions; cohort membership alone does not resolve them.
Worked Example: Delayed Entry and Incomplete Follow-Up
At landmark \(s=4\), six hypothetical records are available. Individual A entered at time 0 and is in state 0 at time 4. B entered at time 2 and is in state 1 at time 4. C entered at time 0 and died at time 3. D entered at time 0 but was lost at time 3, with no subsequent state information. E entered at time 5. F entered at time 1, is in state 0 at time 4, and has follow-up through that time.
For a cohort of living people at time 4, A, B, and F meet the entry and ascertainment requirements and occupy living states. C is known to be dead; D's state at time 4 is unknown; E has not yet entered. Hence the cohort size is \(3\), consisting of two people in state 0 and one in state 1:
For the state-0-only cohort, B is excluded because illness occurred before the landmark. The count is \(n_{\{0\}}(4)=2\), comprising A and F. Delayed entry does not itself exclude F: F entered before \(s\) and is observed at \(s\). Conversely, E cannot be included merely because they are observed later.
The example assumes that “entered” and “follow-up through time 4” correctly describe the observation design. If entry, event, or censoring times are tied at 4, the data protocol must specify their ordering; changing that convention can change membership.
Worked Example: Illness-Death Risk Sets at One Landmark
Consider twelve hypothetical individuals whose state is fully ascertained at \(s=6\). Two are dead, five are alive without illness, and five are alive after illness. For an alive-at-landmark cohort, states 0 and 1 are eligible, giving
For a first-illness analysis among those still in state 0, the eligible count is \(n_{\{0\}}(6)=5\). For a question about subsequent outcomes among people currently alive, all ten living people are eligible under this state-based rule, including those with prior illness. The two deaths are excluded from both living cohorts.
These counts do not assert that the two living groups have equal future risks, or that prior illness is irrelevant. They show only how current state determines membership for each stated target. A covariate or history restriction could further narrow a cohort, but it would need to be specified as part of the target rather than added after inspecting outcomes.
Assumptions, Diagnostics, and Failure Modes
A cohort can be enumerated without assuming a Markov model or a particular transition-intensity model. It does require a well-defined landmark, consistent state definitions, and sufficiently accurate event and observation records to classify state at that time. If the aim is to interpret the observed cohort as representative of a broader population, the sampling and follow-up design also matter. In particular, simply excluding people censored before \(s\) does not guarantee that the remaining observed individuals represent everyone who would have been eligible had their state been known.
Useful checks include the following:
- Reconcile state counts. Among individuals with ascertained states, the counts in mutually exclusive states should sum to the total with ascertained states. In the illness-death example, the living count should equal the state-0 count plus the state-1 count.
- Audit transitions. Check that state sequences follow permitted transitions, such as 0 to 1 to 2 or 0 to 2, and that no person is classified as alive after a recorded death.
- Check the time boundary. Tabulate or inspect events, entry, and censoring at exactly \(s\). An undocumented tie convention can produce inconsistent cohort membership.
- Compare exclusions with records. Separate known deaths, follow-up ending before \(s\), and entry after \(s\). These are distinct reasons for exclusion and must not be merged into “not at risk.”
- Verify the target against the state rule. If the question is about first illness, people already ill should not remain in the state-0 risk set. If the question is about outcomes among living people, excluding everyone previously ill would change the target population.
A common mistake is to subtract every person with a prior event from every landmark cohort. The reasoning might be: “An event has already happened, so that person is no longer at risk.” The correction is to name the event and the target. Prior illness removes a person from a first-illness risk set, but not from an alive-at-landmark cohort for questions about subsequent state occupation or death.
Another mistake is to treat a person lost before the landmark as event-free through \(s\). Loss to follow-up establishes that observation stopped; it does not establish the person's later state. A directly observed landmark cohort includes people whose state is known at \(s\). Other approaches can address incomplete status under additional assumptions, but imputing a state without justification changes the cohort.
How This Cohort Differs From Other Denominators
| Denominator | Who is included at \(s\)? | Typical distinction |
|---|---|---|
| Alive-state landmark cohort | Observed individuals in any living state | Includes state 0 and state 1 in the illness-death model |
| First-event event-free cohort | Observed individuals still in the event-free state | Excludes prior illness and competing death |
| Baseline cohort | Individuals eligible at study entry | Does not automatically represent those eligible at a later landmark |
| Observed-at-landmark set | Individuals whose state is ascertainable at \(s\), before any target-state restriction | May include dead state 2 when the status is known |
These denominators answer different questions. A baseline cohort can include people who later die or become unobserved before \(s\), while an observed-at-landmark set can include people known to be dead. Neither is automatically the cohort of living people at the landmark. State-based eligibility is therefore essential whenever the target population is defined by current state.
Check Your Understanding
Answer each question using an explicit eligibility rule. Solutions follow the questions.
- At a landmark, four observed individuals are in state 0, three are in state 1, and two are known to be dead. What is the alive-at-landmark cohort size? What is the state-0-only cohort size?
- An individual is lost to follow-up before the landmark, with no later registry information. Should they be classified as dead, event-free, or observed in their last known state at the landmark?
- Why might someone with prior illness belong in an alive-at-landmark cohort but not a first-illness cohort?
- Five people are known to be in state 0 and four in state 1 at the landmark. Verify the alive-state count and state which additional fact is needed before treating it as the full target cohort size.
Solutions. (1) The alive cohort is \(4+3=7\); the state-0-only cohort is \(4\). (2) None of those classifications is justified: their state at the landmark is unknown, so they are not in a directly observed cohort defined by state at that time. (3) Prior illness excludes the person from a first-illness risk set because illness has already occurred, but the person remains alive in state 1 and can be eligible for a target covering living states. (4) The two living-state counts sum to \(5+4=9\). To call 9 the full cohort size, one must also establish that these counts include every eligible individual with an ascertained living state at the landmark, with no omitted eligible records.