The Event-Free State In Competing-Risks Data

Why the Event-Free State Matters

In competing-risks data, several mutually exclusive types of first event may occur, but a subject can experience only one of them as the first event. A useful way to organize these data is to begin with a shared state: the subject has not yet experienced any event type of interest. Each observed first event then moves the subject out of that state and into the state labelled by its event type.

This representation makes an important distinction clear. A subject who has not yet had any event of interest is in the event-free state, even if an event remains possible. A subject who has experienced one event type is no longer event-free, even if they remain under observation or later experience other health outcomes. The state describes event history up to a time, not a promise about what will happen next.

The definition uses a specified time origin and a specified set of events. For example, “event-free” might mean free of any of three defined first events since treatment assignment; it does not necessarily mean free of every possible health condition. The time origin and event definitions must therefore be clear before assigning states. We assume that subjects enter the analysis event-free at time zero, so \(T>0\). At the time the first event occurs, the subject leaves state \(0\) and enters the state corresponding to \(J\).

One Origin, One Event-Free State, Several Destinations

The state structure for a first-event analysis can be written as \(0\rightarrow 1,\ldots,K\). State \(0\) is shared: it contains everyone who has not yet had any event type of interest. The destination states keep the types separate. A death from cause 1 and a death from cause 2 are different exits, even though both remove the subject from the event-free state.

In this representation, destination states are absorbing with respect to the first-event record: once the first event is recorded, the subject does not return to state \(0\). “Absorbing” here is a feature of how the data are represented, not a claim that the person’s actual health or clinical condition cannot change. If the scientific question concerns events after the first event, or return to an earlier clinical condition, that requires a representation that tracks those later transitions too.

For a randomly selected subject from the population being studied, the probability of occupying state \(0\) at time \(t\) is

This is the event-free probability at \(t\). It is a state-occupancy probability: it describes the chance that no event of interest has happened by that time. It does not, by itself, say which type of event will occur among subjects who eventually leave state \(0\).

A Partition of Event Histories

The event-free state and the destination states divide subjects’ histories at any fixed time \(t\) into mutually exclusive possibilities. Either no event has occurred by \(t\), or a first event has occurred by \(t\) and has exactly one of the \(K\) types. This yields a useful probability identity.

To justify it, the event \(\{T>t\}\) is disjoint from each event \(\{T\leq t,J=j\}\): the first says there has been no event by \(t\), while the latter says a first event of type \(j\) has occurred by \(t\). The events for different \(j\) are also disjoint because the first event has only one type. Together these events cover all possible histories through \(t\), under the stated event definitions. Adding the probabilities of this disjoint and exhaustive partition gives 1.

The identity is a bookkeeping result about state membership. It does not define a rate of leaving state 0, nor does it imply that the separate event types can be treated as unrelated outcomes. Those are different questions. Here the essential point is that any event type in the defined set ends event-free status.

Censoring Is Not an Exit Event

In practice, a subject’s event history may be observed only until follow-up ends. Let \(C\) be the censoring time, \(Y=\min(T,C)\) the observed follow-up time, and \(\Delta=I(T\leq C)\) the indicator that the event is observed by the end of follow-up. If \(\Delta=1\), the observed first-event type is \(J\). If \(\Delta=0\), the subject is known to have remained event-free through \(C\), but their state after \(C\) is not observed.

Censoring is therefore not a transition to another event state. It marks the end of observed follow-up. In particular, a subject censored before \(t\) cannot simply be counted as event-free at \(t\): an event might have occurred after censoring and before \(t\). Nor can that subject be counted as having had an event, since none was observed. The state at \(t\) is unknown from that subject’s record alone.

Choosing and Checking the State Representation

The event-free-state representation is useful when the scientific question concerns the first occurrence of any event in a defined set. It keeps the event-free status common across event types while preserving the type of the first exit. A table of first-event times and types is another way to record the same basic information, but the state representation makes the shared origin and distinct destinations explicit.

It is not suitable without modification when the question requires tracking multiple episodes, transitions after the first event, or return to an earlier state. For example, if recovery after an initial event is part of the target question, treating the first-event destination as absorbing discards relevant transitions. The representation must match the outcome being studied rather than impose a one-way clinical story.

Useful checks when organizing data include:

• Check the event set. State exactly which event types count as exits. An event excluded from the definition does not end this particular event-free state.
• Check the origin and time scale. The same event time has different meaning if measured from diagnosis, treatment assignment, or another starting point.
• Check first-event ordering. Each subject should have at most one recorded first-event type. If event dates tie or the order cannot be determined, use a prespecified rule or acknowledge that the first type is ambiguous.
• Check follow-up status. Distinguish an observed event, observed event-free follow-up to a target time, and censoring before that time.
• Check whether “absorbing” is only a record convention. Do not interpret the first-event state labels as a complete account of later clinical history.

A Common Mistake

A common error is to keep a subject in the event-free state after a competing event because the event of primary interest has not occurred. For instance, if type 1 is the event of interest and type 2 is another event in the defined competing-risks set, it is incorrect to call a subject event-free after a type 2 event. The correction is to separate two questions: whether the subject is free of all defined event types, and whether the subject has had a particular event type. The subject exits the event-free state upon either type 1 or type 2, even though only one of those exits is type 1.

A second error is to treat censoring as if it were another event type. Censoring does not establish that an event occurred; it limits how long the subject’s state is observed. Keeping these distinctions straight is necessary before any analysis of how subjects leave the event-free state.