Why Competing Outgoing Transitions Need Careful Risk Sets
When a process can leave a state in more than one way, the risk set is determined by where subjects are immediately before the time being considered, not by which destination they will eventually reach. For example, a patient in a disease-free state might experience relapse or death. Immediately before either event, the patient is at risk for each transition allowed out of that state. Once either transition occurs, the patient is no longer in that state and no longer belongs to its risk set.
This distinction is essential when estimating transition-specific intensities. The event count in the numerator distinguishes destinations; the risk set in the denominator describes exposure in the origin state. A subject who experiences one outgoing transition is not retained in the denominator for other outgoing transitions after leaving the state.
The notation \(t-\) means just before \(t\). It settles an important boundary question: a subject who leaves \(r\) at time \(t\) belongs to the risk set for that event, because the subject was in \(r\) immediately before it. Immediately after the transition, that subject is no longer in the risk set for \(r\). Conversely, a subject who enters \(r\) at time \(t\) is not in the risk set for outgoing transitions from \(r\) at that same time under the usual single-transition-at-a-time convention; the subject is in \(r\) just after the entry.
For every allowed destination \(s\), the risk set \(Y_{rs}(t)\) is shared. What changes across destinations is the event count \(dN_{rs}(t)\). This does not mean that every subject will eventually experience each type of event. Risk-set membership records current eligibility for an outgoing transition, not a prediction of the subject’s eventual path.
Why the Denominator Is Shared
For a transition \(r\to s\), the transition intensity \(\alpha_{rs}(t)\) describes the instantaneous rate of moving from \(r\) to \(s\), conditional on being in \(r\) immediately beforehand and on the model’s stated history assumptions. In a short interval of length \(dt\), a subject in \(r\) has probability approximately \(\alpha_{rs}(t)dt\) of making that transition, with the approximation understood to first order in \(dt\). A subject outside \(r\) is not currently exposed to that transition.
Suppose, for clarity, that the subjects in the risk set share the same transition intensity at time \(t\), and that there is no more than one jump per subject in the short interval. Then each of the \(Y_r(t)\) subjects contributes exposure to the \(r\to s\) transition. Adding their first-order event probabilities gives the expected event count to first order:
Dividing by the exposure count therefore gives the transition-specific rate contribution \(dN_{rs}(t)/Y_r(t)\), when \(Y_r(t)>0\). In a Nelson–Aalen increment, this is the contribution for transition \(r\to s\). If a subject instead moves from \(r\) to another allowed destination, that event is counted in the other transition’s numerator. It does not turn the subject into someone who was never at risk for \(r\to s\); rather, it ends that subject’s exposure in \(r\) from that time onward.
This argument relies on correctly observed states and transition times, a correctly specified set of allowed transitions, and a well-defined time origin. For a pooled transition-intensity interpretation, subjects contributing to the same risk set must also be appropriately comparable under the model—for example, they share the modeled intensity or relevant covariates are accounted for. The risk-set definition itself is a bookkeeping rule; it does not guarantee that a pooled model is scientifically adequate.
Worked Example: Two Competing Exits From One State
Consider five hypothetical subjects, all initially in state \(A\). State \(A\) has two allowed exits: \(A\to B\) and \(A\to D\). During complete observation, four distinct event times occur: subject 1 moves \(A\to B\) at time 1, subject 2 moves \(A\to D\) at time 2, subject 3 moves \(A\to B\) at time 3, and subject 4 moves \(A\to D\) at time 4. Subject 5 remains in \(A\) through the end of observation. There are no ties or other transitions in this example.
| Time | Event at time | In state \(A\) just before time | \(Y_A(t)\) | \(dN_{AB}(t)\) | \(dN_{AD}(t)\) |
|---|---|---|---|---|---|
| 1 | Subject 1: \(A\to B\) | 1, 2, 3, 4, 5 | 5 | 1 | 0 |
| 2 | Subject 2: \(A\to D\) | 2, 3, 4, 5 | 4 | 0 | 1 |
| 3 | Subject 3: \(A\to B\) | 3, 4, 5 | 3 | 1 | 0 |
| 4 | Subject 4: \(A\to D\) | 4, 5 | 2 | 0 | 1 |
At time 1, the risk set for both \(A\to B\) and \(A\to D\) has size 5. The \(A\to B\) event contribution is \(1/5=0.200\), while the \(A\to D\) contribution is \(0/5=0\). After subject 1 leaves \(A\), the risk set size becomes 4. At time 2, subject 2 is still at risk just before leaving by \(A\to D\). The contributions are \(0/4=0\) for \(A\to B\) and \(1/4=0.250\) for \(A\to D\). The remaining event-time contributions are \(1/3\approx0.333\) for \(A\to B\) at time 3, and \(1/2=0.500\) for \(A\to D\) at time 4.
An event of either type reduces the risk set for state \(A\) by one. At time 2, subject 1 is not in the risk set: that subject already left \(A\) at time 1. Subject 5 remains at risk for both outgoing transition types throughout the observed period because subject 5 remains in \(A\), not because either destination is known to be the eventual outcome.
Risk Sets Change With State Occupancy
A subject need not start in the state of interest to enter its risk set. If the subject later moves into that state, the subject becomes eligible for its outgoing transitions after entry. Likewise, a subject who leaves the state exits its risk set, whatever the destination. The risk set is therefore reconstructed from the state histories over time, rather than copied from the baseline cohort.
Worked Example: Subjects Entering the Origin State Later
In a hypothetical study, subjects 1, 2, and 3 start in state \(A\), while subjects 4 and 5 start in state \(C\). Subject 4 moves \(C\to A\) at time 1. Subject 1 moves \(A\to B\) at time 2. Subject 5 moves \(C\to A\) at time 3. Subject 2 moves \(A\to D\) at time 4. Assume these are the only transitions through time 4, with no ties.
Just before time 1, the risk set for transitions out of \(A\) is \(\{1,2,3\}\), so \(Y_A(1)=3\). Subject 4 is not yet in \(A\) immediately before the \(C\to A\) transition. After time 1, subject 4 joins the state-\(A\) risk set.
At time 2, the risk set is \(\{1,2,3,4\}\), so \(Y_A(2)=4\). Subject 1’s \(A\to B\) event has transition contribution \(dN_{AB}(2)/Y_A(2)=1/4\). After this event, subject 1 leaves the state-\(A\) risk set.
Just before time 3, subjects 2, 3, and 4 are in \(A\), so \(Y_A(3)=3\). Subject 5 is still in \(C\) immediately before entering \(A\), and so is not included in this risk set. After the entry at time 3, subject 5 is added. At time 4, subjects 2, 3, 4, and 5 are in \(A\), giving \(Y_A(4)=4\). Subject 2’s \(A\to D\) event contributes \(1/4\) to that transition. The denominator has increased because of entry into \(A\), even though no new subject joined the study at baseline.
This example shows why a fixed baseline denominator is generally incorrect in a multi-state process. Eligibility changes as state transitions occur. The required membership at each event time comes from the states occupied immediately beforehand.
Leaving and Re-Entering the Same State
In a model that allows cycles, a subject can leave a state and later return. The subject then re-enters the risk set for transitions out of that state. Risk-set membership is based on current state, not on whether the subject has ever been in the state before. For a transition count, each observed transition of the specified type is counted; an individual may contribute more than one event if the model permits repeated transitions.
Worked Example: Re-Entry After a Cycle
Suppose four hypothetical subjects start in \(A\), and the allowed transitions include \(A\to B\), \(B\to A\), and \(A\to D\). Events occur in this order: subject 1 moves \(A\to B\) at time 1; subject 2 moves \(A\to D\) at time 2; subject 1 returns \(B\to A\) at time 3; subject 3 moves \(A\to B\) at time 4; and subject 1 moves \(A\to D\) at time 5. There are no other transitions or ties.
At time 1, all four subjects are in \(A\), so \(Y_A(1)=4\). After subject 1 leaves for \(B\), three remain in \(A\). At time 2, \(Y_A(2)=3\); subject 2’s \(A\to D\) event removes that subject from the risk set, leaving subjects 3 and 4 in \(A\).
At time 3, subject 1 is in \(B\) immediately before returning to \(A\), so \(Y_A(3)=2\). The \(B\to A\) event does not count as an outgoing event from \(A\). After the return, subject 1 joins the risk set for \(A\). Thus \(Y_A(4)=3\), consisting of subjects 1, 3, and 4. Subject 3’s \(A\to B\) event leaves two in \(A\), so \(Y_A(5)=2\). Subject 1’s later \(A\to D\) event is counted at time 5 with contribution \(1/2\).
Subject 1 contributed once to the \(A\to B\) event count at time 1, returned to \(A\), and later contributed to the \(A\to D\) event count. Each risk-set membership and each transition count follows the observed path in time; the subject is not permanently excluded just because of an earlier exit from \(A\).
Diagnostics, Alternatives, and Limits
A risk-set table is a practical diagnostic. At each distinct event time, list who is in the origin state just before the event, identify the destination of the event, and then update state membership. The event count for one transition should never be used to determine how many subjects leave the origin state: all observed outgoing transitions matter for that update.
Common warning signs include a denominator that stays fixed despite events leaving the state, a person counted at risk before entering the state, or a person retained after leaving it. Another warning sign is defining the \(r\to s\) risk set as only those who will eventually experience \(r\to s\). That uses future outcomes to define past exposure and excludes subjects who were genuinely at risk but experienced another destination instead.
The main alternative is not a different risk-set definition but a different modeling question. A transition-specific intensity uses the origin-state risk set and an event count specific to the destination. An intensity for all exits combined instead uses the same origin-state risk set and counts every outgoing event. A competing-risks analysis may focus on event types from a single starting state, while a multi-state analysis tracks transitions among several states, including possible returns. These approaches answer related but distinct questions; the risk set must match the transition or aggregate rate being studied.
The simple bookkeeping described here assumes that state histories and transition times are observed well enough to determine membership just before each event, that the state diagram correctly identifies allowed transitions, and that events are represented according to the study’s time convention. The examples use complete observation and distinct event times so that censoring and simultaneous-event conventions do not obscure the core rule. In actual data, observation ending can also remove a subject from an observed risk set, and tied times require an explicit handling convention. Those issues should be addressed rather than silently treated as transitions from the origin state.
A correctly constructed risk set is necessary, but it does not by itself establish that the transition-intensity model is appropriate. If subjects in the same state have systematically different transition rates because of measured or unmeasured characteristics, a single pooled rate may not describe them adequately. The state definition and any history assumptions must also match the scientific question. Risk-set bookkeeping gets the exposure group right; it cannot repair a misspecified state space or an unsuitable model.
Check Your Understanding
Answer each question using the immediately-before-event convention. Solutions follow the questions.
- Four subjects are in state \(A\) just before time \(t\). One moves \(A\to D\) at \(t\). What are the risk-set size and the \(A\to B\) event count at \(t\), if \(A\to B\) is also allowed?
- A subject moves \(C\to A\) at time \(u\). Is the subject in the risk set for an outgoing transition from \(A\) immediately before time \(u\)? When does the subject become eligible for those transitions?
- A subject moves \(A\to B\), later returns \(B\to A\), and eventually moves \(A\to D\). How many times can this subject enter the risk set for transitions out of \(A\), and which events count toward \(N_{AD}\)?
- Explain the error in defining the \(A\to B\) risk set as only subjects who eventually experience \(A\to B\).
Solutions. (1) The risk-set size is 4 because all four subjects are in \(A\) immediately before \(t\). The \(A\to B\) event count is 0; the observed event contributes to \(A\to D\). After the event, the risk-set size falls to 3. (2) The subject is not in the \(A\) risk set immediately before \(u\), because the subject is still in \(C\). The subject becomes eligible immediately after entering \(A\), and is at risk at subsequent times while in \(A\). (3) The subject enters the risk set for \(A\)-outgoing transitions twice: once at the beginning of the first stay in \(A\), and again upon returning from \(B\). The eventual \(A\to D\) event counts once toward \(N_{AD}\). The earlier \(A\to B\) event counts toward \(N_{AB}\), not \(N_{AD}\). (4) That definition uses future destination information to exclude subjects who were in \(A\) and at risk for \(A\to B\) but instead experienced another allowed exit. The correct risk set includes everyone in \(A\) immediately beforehand; only the event count is destination-specific.