Why the Outcome Definition Matters
A prediction for one outgoing transition is not a prediction that the subject will simply be in its destination state at a later time. Before the evaluation horizon, a subject may take a different outgoing transition, remain in the origin state, or take the target transition and then move again. These histories imply different labels for an edge-specific prediction.
The key is to define one endpoint before assigning cases and non-cases. In this tutorial, the endpoint is whether the first transition out of the prediction origin state is the target edge and occurs by the evaluation horizon. A different endpoint, such as occupying the destination state at the horizon, answers a different question.
The prediction for subject \(i\) must target this same quantity: \(p_i(\tau)=P(T_i\leq\tau,\ J_i=s\mid\) information available at the prediction origin\()\). The conditioning information and the origin state are fixed when the prediction is made. A prediction of \(P(X(\tau)=s)\), the probability of occupying state \(s\) at the horizon, is not interchangeable with this edge probability.
Labeling the First Exit
For the first-exit endpoint, the relevant outcomes are mutually exclusive. By the horizon, the subject either first exits along the target edge, first exits along another outgoing edge, or has not exited the origin state. The latter two outcomes are non-cases for this binary endpoint, although they are substantively different histories.
This partition assumes the set of possible first destinations is specified well enough to identify the target edge and the other outgoing transitions. Include relevant transitions such as death if they can occur before the target edge. If the state record omits a possible exit or its timing, the label may be wrong or unclassifiable.
Worked Example: A Competing Exit Before the Horizon
Suppose a hypothetical model starts subjects in state A and predicts the probability of the edge A→B within 12 months. State A also has outgoing edges to C and D. Consider three subjects with complete follow-up through 12 months.
Subject 1 first moves from A to B at month 7. The first exit is along the target edge and \(7\leq12\), so \(Y_{A\to B}(12)=1\): this subject is a case.
Subject 2 first moves from A to C at month 4. Since \(C\neq B\), this is a competing-transition non-case, even if the subject later enters B from C. That later pathway does not change which edge was taken first out of A.
Subject 3 remains in A through month 12. Here \(T>12\), so the subject is a no-exit non-case. The subject is not a case merely because the target transition might occur after the evaluation horizon.
All three labels concern the same endpoint: first exit from A along A→B by month 12. The prediction is not a prediction of whether B is ever visited.
What Happens After a Transition?
Once the target edge has occurred by the horizon, later movement does not undo the case label. Conversely, a competing first exit cannot be reclassified as a case merely because a later path reaches the target destination. This follows directly from the definition: both \(T\) and \(J\) refer to the first departure from the origin state.
Worked Example: A Target Edge Followed by Another Move
In hypothetical follow-up from A, a subject moves A→B at month 3 and B→C at month 9. The target is A→B by a 12-month horizon. The first exit time is \(T=3\) months and its destination is \(J=B\), so $$ Y_{A\to B}(12)=I(3\leq12,\ B=B)=1. $$ The subject is a case even though the subject is in C, not B, at month 12. The later B→C transition changes the horizon state but not the fact that A→B occurred first within the horizon.
Now consider a subject whose first move is A→C at month 3, followed by C→B at month 9. The first exit is still A→C, so \(J=C\neq B\) and \(Y_{A\to B}(12)=0\). Arrival in B along a later path is not the target edge A→B.
This first-exit convention is appropriate when the prediction question is about which outgoing transition occurs first from the state at the prediction origin. If the scientific question instead concerns any occurrence of A→B after possible returns to A, that requires a different outcome definition and matching prediction. Do not silently switch between these endpoints.
Horizon Boundaries and Incomplete Follow-Up
Use a prespecified horizon and a consistent boundary convention. With the definition above, an observed target transition at exactly \(\tau\) counts as a case because the event condition is \(T\leq\tau\). In a setting where event times are recorded only in intervals, specify how an interval spanning the horizon is handled; the data may not determine whether the transition occurred by \(\tau\).
Worked Example: Censoring Is Not a Non-Case
Suppose the target is A→B by 12 months. In hypothetical records, one subject moves A→C at month 5 and is observed through the horizon; this is a competing-transition non-case. A second subject remains in A and is observed continuously through month 12; this is a no-exit non-case. A third subject is last known to be in A at month 7 and then has no further observation.
For the third subject, the record establishes neither a target transition nor the absence of one by month 12. The correct status is unknown at the horizon, not a non-case. Assigning \(Y=0\) would assert \(T>12\) or a known competing first exit, neither of which follows from the observed follow-up.
If follow-up instead confirms that the third subject remained in A through month 12, the subject is a no-exit non-case. If a target transition is observed at month 6 before follow-up ends, the subject is a case for this endpoint even if later status is unobserved.
Outcome labeling itself does not require an assumption about why observation ended. However, calculating observed validation summaries when some outcomes are unknown requires an appropriate method for incomplete follow-up. Treating all early losses as non-cases is generally unjustified; the assumptions and methods for censoring-adjusted validation are a separate issue.
Why the Competing Transition Is a Non-Case
The edge-specific binary outcome can be derived by partitioning first-exit histories. For a subject with complete horizon follow-up, the event that the target edge occurs first by \(\tau\) is disjoint from the event that another edge occurs first by \(\tau\), and both are disjoint from no first exit by \(\tau\). Therefore their indicators sum to one:
The target probability is consequently one component of a set of mutually exclusive outcomes, not the probability obtained by pretending that competing exits never happened. A competing exit is a non-case because the defined first-exit target did not occur; it is not missing follow-up. Keeping the competing outcome distinct from no exit remains useful for describing what generated the non-case label.
Record the state occupied at prediction time, the specified outgoing edge, and the information available then.
For a first-exit target, use the first departure from the origin state and state whether the boundary is inclusive.
A target first exit by the horizon is a case; a different first exit or confirmed no exit is a non-case.
If follow-up ends before the outcome is determined, mark it unresolved rather than forcing a binary label.
Alternatives and Common Mistakes
| Choice of endpoint or handling | What it means | Why it differs from the target here |
|---|---|---|
| First exit along the target edge by \(\tau\) | Target edge occurs first from the origin state within the horizon | This is the defined edge-specific endpoint |
| Occupying destination \(s\) at \(\tau\) | Subject's state at the horizon is \(s\) | A target edge followed by another move may not end in \(s\); another path may end there |
| Any occurrence of the edge by \(\tau\) | Target edge may occur after leaving and returning to the origin | This is not the first-exit endpoint |
| Remove competing exits as if censored | Analyze a hypothetical setting without those exits | It does not label the observed first-exit outcome and changes the estimand |
A common error is to label a competing transition as censored because it prevents the target edge from occurring later as the first exit. That reasoning confuses an observed alternative outcome with loss of observation. For this endpoint, a known competing exit is a non-case; censoring means the horizon outcome cannot be determined from the available record.
Another error is to label every subject who reaches destination \(s\) by the horizon as a case. The correction is to check the actual edge and the first-exit rule: reaching \(s\) by a different route is not evidence that \(r\to s\) occurred. Similarly, a subject who took \(r\to s\) and then left \(s\) remains a case, even if the horizon state differs from \(s\).
The definition has limits. It depends on a meaningful prediction origin, accurate transition timing and destinations, and an edge set that captures relevant exits. It also deliberately collapses competing exits and no exit into one non-case label for a binary edge outcome. Retain the separate history categories when interpreting or auditing results, and ensure every prediction and observed label refer to the same origin, edge, horizon, and endpoint.
Check Your Understanding
For each question, use the first-exit definition and assume all stated follow-up is complete unless the question says otherwise.
- The target is X→Y by 6 months. A subject moves X→Z at month 2 and Z→Y at month 5. What is the label, and why?
- The target is X→Y by 6 months. A subject moves X→Y at month 4 and Y→Z at month 5. What is the label?
- A subject is last observed in X at month 3, with no later record, for a 6-month horizon. Is this a non-case?
- Explain why a subject who is in Y at month 6 need not be a case for target edge X→Y.
- For a 6-month horizon, a subject's first exit from X is the target edge at exactly month 6. Under the stated inclusive convention, what is the label?
Solutions. (1) Non-case: the first exit is X→Z, not X→Y; the later route to Y does not change the first exit. (2) Case: X→Y is the first exit and occurs by month 6; the later move does not undo it. (3) Unresolved, not a non-case: observation ends before the horizon and the outcome is unknown. (4) The subject may have reached Y along another path, such as X→Z→Y, without ever taking X→Y as the first exit. (5) Case, because the boundary rule includes \(T=\tau\), so \(T\leq\tau\).