Why Counterfactual Event Times Need Two Parts
With competing events, a treatment strategy can affect both whether an event of interest occurs and whether another event happens first. For example, a strategy might change the timing of relapse and death. A description that records only “time to relapse” can therefore be ambiguous: does it mean the time relapse would occur in the actual competing-event process, or the time it might occur if death were prevented? These are different counterfactual questions.
For the first-event question, the outcome under a treatment strategy has two parts: the time to the first event and its type. The type identifies whether that first event is the event of interest or a competing event. This representation describes the event process under the strategy as defined; it does not silently remove a competing event.
The time origin, event definitions, and follow-up period are part of the outcome’s definition. For instance, “death” may need a specified cause or ascertainment rule, and “relapse” may need a clinical definition. If the outcome is defined only through a study horizon \(\tau\), then \(J^a=0\) means no defined event occurred by \(\tau\); it does not assert that the individual can never experience one later.
Suppose there are \(K\) mutually exclusive types of first event, with type \(k\) designated as the event of interest. The other types are competing events for this particular question. Their labels are not intrinsic: changing which event is of interest changes which remaining types are called competing events, but does not change the underlying first-event record.
Encoding the Event of Interest and Competing Events
A convenient event-specific encoding follows directly from the pair \((T^a,J^a)\). For each event type \(j\), define \(T_j^a=T^a\) when \(J^a=j\), and \(T_j^a=\infty\) otherwise. This is a bookkeeping representation of which event was first under strategy \(a\). It is not a set of independent biological clocks, one for each possible event.
If no event occurs, all \(T_j^a=\infty\), \(T^a=\infty\), and \(J^a=0\). If exactly one type is recorded as the first event, then exactly one \(T_j^a\) is finite. A tie between event types requires a prespecified rule—for example, defining a joint type or using a reliable ordering rule—because an unaddressed tie does not yield a unique \(J^a\).
For a fixed time \(t\), the event-of-interest indicator is \(I(T^a\le t,\ J^a=k)\). It equals one only when type \(k\) is the first event by \(t\). An individual who experiences a competing event first does not count as having the event of interest by \(t\), even if one can imagine a different process in which the event of interest might later occur.
Worked Example: Relapse Versus Death
Hypothetical setup. Consider one person and two strategies: \(a=0\) is standard care and \(a=1\) is an enhanced follow-up strategy. The event of interest is first relapse; death before relapse is the competing event. Suppose this person’s potential outcomes are \((T^0,J^0)=(8,\text{death})\) and \((T^1,J^1)=(11,\text{relapse})\), with times measured in months.
Under standard care, the first event occurs at 8 months and is death. Thus \(T_{\text{death}}^0=8\) and \(T_{\text{relapse}}^0=\infty\). Under enhanced follow-up, the first event occurs at 11 months and is relapse, so \(T_{\text{relapse}}^1=11\) and \(T_{\text{death}}^1=\infty\).
By 10 months, the event-of-interest indicator under standard care is \(I(8\le10,\ J^0=\text{relapse})=0\): an event occurred, but it was death. Under enhanced follow-up it is \(I(11\le10,\ J^1=\text{relapse})=0\), because relapse has not yet occurred. These statements encode the two potential first-event outcomes; they do not establish that the strategy caused either outcome for this individual.
Potential Outcomes and the Observed Record
For each individual, the notation \((T^a,J^a)\) refers to the outcome that would arise under strategy \(a\). If there are two strategies, the individual has the conceptual pair \((T^0,J^0)\) and \((T^1,J^1)\). Only the outcome under the strategy actually received is observed. This is the fundamental missing-counterfactual feature of causal inference: the unassigned strategy’s outcome is not recorded for that same individual.
Let \(A\) denote the strategy actually assigned or received. Under consistency, the observed event time and type equal the potential time and type under the strategy actually followed: \(T=T^A\) and \(J=J^A\). This requires that the strategy be specified clearly enough that the observed version corresponds to the potential outcome being defined. Under no interference, one individual’s potential outcome under a strategy does not depend on other individuals’ assigned strategies. These conditions make the link between potential and observed outcomes meaningful; they do not, by themselves, make treatment groups comparable.
If observation ends before an event is observed, the record may be censored. Censoring is not a value of \(J^a\): a person lost to observation before any event has not thereby had “no event,” nor has censoring become a competing event. For example, if the observed time is \(Y=\min(T,C)\), where \(C\) is the censoring time, then an event type is observed only when the event is observed before censoring. Interpreting censored records requires appropriate assumptions about follow-up; the censoring record must remain distinct from the potential event outcome.
Worked Example: What Random Assignment Reveals
Hypothetical data. Four people are randomly assigned to one of two strategies, with complete follow-up through 12 months and no ties. Type R is relapse, the event of interest; type D is death, a competing event. The potential outcomes below are shown to explain the notation. In an actual study, only the outcome in the assigned column would be observed for each person.
| Person | \((T^0,J^0)\), standard care | \((T^1,J^1)\), enhanced follow-up | Assigned strategy | Observed outcome |
|---|---|---|---|---|
| P | (5, R) | (9, D) | 0 | (5, R) |
| Q | (7, D) | (6, R) | 1 | (6, R) |
| R | (∞, 0) | (10, R) | 0 | (∞, 0) by 12 months |
| S | (11, R) | (8, D) | 1 | (8, D) |
For person P, consistency gives \((T,J)=(T^0,J^0)=(5,\mathrm{R})\), because \(A=0\). The person’s enhanced-follow-up outcome \((9,\mathrm{D})\) remains counterfactual and unobserved. For person S, consistency gives \((T,J)=(T^1,J^1)=(8,\mathrm{D})\), while the standard-care outcome is unobserved.
By 12 months, the observed records include two first relapses, one first death, and one person with no event recorded by the horizon. Those counts describe the observed assignments and outcomes; they do not reveal both potential outcomes for anyone. Random assignment supports comparisons across groups at the population level under its design assumptions, but it does not make each individual’s missing outcome observable.
A Short Derivation: Why a Competing Event Is Not a Hidden Event Time
The distinction between a first-event outcome and a latent event time becomes clear by applying the definition of \(T_j^a\). Suppose under strategy \(a\), death occurs first at month 4, before the event of interest. Then \(J^a=\text{death}\) and \(T^a=4\). By the event-specific definition,
The last equality follows because the type condition is false: the first event was death, not the event of interest. Assigning \(T_{\text{interest}}^a=9\) instead would assert that the event of interest occurs at month 9 under the same strategy, despite death at month 4. That assertion does not follow from the observed first-event process. It would require defining a different hypothetical intervention or model in which the competing event is prevented and specifying how that intervention affects the event of interest.
Worked Example: Checking a Proposed Event-Time Record
Hypothetical record. Under a specified treatment strategy, an individual has a first event at 4 months, adjudicated as death; the event of interest is relapse. A proposed data record lists \(T^a=4\), \(J^a=\text{death}\), and \(T_{\text{relapse}}^a=9\).
The first two entries imply \(T_{\text{relapse}}^a=\infty\) under the event-specific bookkeeping definition, because relapse was not the first event. Therefore the proposed value 9 is inconsistent with the defined record. The corrected record is \(T_{\text{death}}^a=4\) and \(T_{\text{relapse}}^a=\infty\).
A value of 9 could be meaningful only if it were explicitly defined as a time under another counterfactual process, such as a specified intervention preventing death. It must not be presented as the person’s relapse time under the original strategy.
Conditions, Diagnostics, and Alternative Representations
Potential event outcomes require a well-defined treatment strategy, a common time origin, an operational definition for each event type, and a rule for ties. If treatment is described only as “usual care,” but its versions differ substantially across settings, consistency may be unclear. If cause classification is incomplete or changes during follow-up, \(J^a\) may not be well-defined. The analysis should also specify the follow-up horizon when “no event” means no event observed by a particular time.
For causal interpretation of population comparisons, further assumptions are needed beyond the definitions. In a randomized study, random assignment makes assignment independent of the potential outcomes by design, subject to the actual randomization scheme and implementation. In an observational study, identification generally requires conditional exchangeability given measured covariates and positivity: within the covariate groups used for comparison, each strategy must have a nonzero probability of being received. Consistency and no interference are also required. These assumptions concern whether observed groups can identify distributions of potential outcomes; they do not change what \((T^a,J^a)\) means.
Useful checks include verifying that every observed event has one permitted type, that the recorded type corresponds to the earliest qualifying event, and that no-event status is limited to the stated horizon. Review dates for event and censoring, and check whether a purported event-specific time is finite only when its event type was first. An impossible combination, such as a competing event first plus a finite \(T_k^a\) under the bookkeeping definition, signals either a coding error or a change in the counterfactual question.
Other summaries can answer different questions. A composite outcome can combine several event types into “any event,” but then it no longer distinguishes the event of interest from its competitors. A cause-specific hazard analysis focuses on event rates among those still event-free, while the pair \((T^a,J^a)\) records the first event and its type. A hypothetical analysis that eliminates a competing event asks about a different intervention and needs additional definitions and assumptions. None of these representations is a substitute for clearly stating which counterfactual outcome is intended.
Common Mistakes and Limitations
- Treating a competing event as ordinary censoring. Death before relapse is an observed event type, not simply an unobserved relapse time. Relabeling it as censoring changes the question.
- Calling the event-specific \(T_k^a\) a latent clock. Under the definition here, it is infinite when another event occurs first. A finite time for event \(k\) after a competing event belongs to a different hypothetical setup.
- Equating “no event observed” with “no event ever.” If the outcome is defined through 12 months, \(J^a=0\) means no event by 12 months, not necessarily no event over the person’s lifetime.
- Assuming consistency reveals both potential outcomes. Consistency identifies the outcome corresponding to the strategy actually received; the other strategy’s outcome remains counterfactual for that individual.
The representation is deliberately limited to the first event. It does not encode later transitions after that event, recurrent events, or the effects of an intervention that removes one event type. Defining those questions requires a different outcome structure and explicit assumptions.
Check Your Understanding
Answer each question using the first-event definitions above.
- Under strategy \(a\), death occurs first at month 6 and relapse is the event of interest. What are \(J^a\), \(T_{\text{death}}^a\), and \(T_{\text{relapse}}^a\)?
- What does \(J^a=0\) mean when follow-up is defined through month 18?
- A person assigned to strategy 1 has an observed first relapse at month 7. Under consistency, which potential outcome is observed, and which remains unobserved?
- Why is censoring at month 5 not equivalent to recording a competing event at month 5?
Solutions. (1) \(J^a=\text{death}\), \(T_{\text{death}}^a=6\), and \(T_{\text{relapse}}^a=\infty\) under the event-specific definition. (2) No defined event occurred by month 18; the statement does not rule out an event later. (3) The observed outcome is \((T^1,J^1)=(7,\text{relapse})\); \((T^0,J^0)\) remains unobserved for that person. (4) Censoring means the event status after month 5 was not observed, whereas a competing event is an observed first event with a defined type; the two records have different meanings.