Why Event-Time Records Need More Than a Number
In a study of time to an event, some participants experience the event while they are being followed; others leave observation or reach the study’s end without an event being observed. The recorded duration alone cannot distinguish these situations. A duration of 8 months might mean an event occurred at month 8, or it might mean the participant was known to be event-free through month 8 and follow-up then ended. Those records contain different information.
This distinction matters in joint modeling because an individual’s event follow-up is paired with longitudinal measurements. The event-time record describes both the duration observed and whether the event was seen. A risk set at time \(t\) identifies the subjects still under observation and event-free just before that time. It is a way to keep track of which subjects can contribute information at each event time, without pretending that a censored subject’s unobserved future is known.
Exact Event Times and Right-Censored Times
An exact event time means that the event was observed and its occurrence time is recorded, to the available measurement precision. For example, if a participant experiences the event at month 8, the data record an event at month 8. It does not mean that the event time is measured without any practical rounding; a date may be recorded to the nearest day, for instance.
A right-censored observation gives a lower bound on the event time rather than the event time itself. If a participant is confirmed event-free through month 8 and then lost to follow-up, the record is \(Z_i=8\) with \(\delta_i=0\). The event may occur later, or it may never occur; this record does not say which. It says only that no event was observed by the end of follow-up.
Administrative censoring occurs when observation ends at a planned study close, even if a participant could have been followed longer. Loss to follow-up is another common reason, but its consequences require care: participants lost to follow-up may differ in event risk from those who remain. Censoring describes what is observed; by itself, it does not guarantee that the resulting analysis is valid.
Worked Example: Reading Event Indicators
Suppose a hypothetical study records time to a first clinical event in months. Three participants have these records: A has \(Z_A=5\), \(\delta_A=1\); B has \(Z_B=5\), \(\delta_B=0\); and C has \(Z_C=9\), \(\delta_C=0\). Translate each record into what is known.
For A, \(\delta_A=1\), so an event was observed at month 5: \(T_A=5\). The record identifies an event time, subject to the study’s time precision.
For B, \(\delta_B=0\), so B was known to be event-free through month 5, when follow-up ended. The record implies \(T_B>5\) under the usual convention that an event at the recorded time would be counted as observed. It does not imply \(T_B=5\).
For C, \(\delta_C=0\) similarly means the event was not observed through month 9. The known lower bound on C’s event time is later than the lower bound for B’s, but C’s actual event time is still unknown. The equal recorded durations for A and B do not make their records equivalent: A had an event, whereas B was censored.
Risk Sets at a Given Time
Assume for now that every subject begins event follow-up at time zero and that there is no delayed entry. At a time \(t\), the risk set consists of subjects whose observed follow-up reaches at least \(t\). With the convention that the risk set is defined immediately before events at \(t\), this is
This definition follows from the information recorded for each subject. If \(Z_i<t\), follow-up ended before \(t\), either because the event occurred earlier or because censoring occurred earlier. Subject \(i\) is therefore not known to be event-free and under observation immediately before \(t\). If \(Z_i\geq t\), the subject is still being followed and has not had an observed event before \(t\), so the subject belongs in the risk set just before \(t\).
A subject who experiences an event at \(t\) is in \(R(t)\): the subject was at risk immediately before the event. A subject censored at \(t\) is also included in \(R(t)\) under this convention, then is no longer under observation after \(t\). If event and censoring times are recorded as tied, the data’s timing precision and the analysis’s tie-handling convention must be considered. When the ordering cannot be determined from the records, the convention should be stated rather than treated as a known ordering.
Worked Example: Constructing Risk Sets From Follow-Up Records
Consider six hypothetical participants with follow-up in months: A has an event at 2; B is censored at 3; C has an event at 4; D is censored at 4; E has an event at 6; and F is censored at 7. All six enter at time zero. Find the risk set just before each event time.
At month 2, no subject has yet had an event or been censored. Therefore, \(R(2)=\{A,B,C,D,E,F\}\), with 6 subjects. A is included because the event at month 2 occurs after A was at risk immediately before that time.
After A’s event at month 2, A is no longer in later risk sets. At month 4, B has also left follow-up because B was censored at month 3. Thus \(R(4)=\{C,D,E,F\}\), with 4 subjects. C is included because C has the event at month 4; D is included under the convention that a subject censored at \(t\) is in \(R(t)\).
At month 6, A and C have had events, and B and D have been censored. Only E and F remain under observation and event-free just before month 6, so \(R(6)=\{E,F\}\), with 2 subjects. E experiences the event at month 6, while F remains in observation until censoring at month 7.
The sizes of the risk sets at the event times are therefore 6 at month 2, 4 at month 4, and 2 at month 6. They shrink when events or censoring remove subjects from subsequent follow-up; censoring is not itself an event.
How Censoring Determines What Is Observed
A subject with an event at time \(t\) contributes observed follow-up up to the event, and the event itself is recorded. A subject censored at time \(c\) contributes observed event-free follow-up only through \(c\). After \(c\), that subject cannot be counted as observed event-free, included in later risk sets, or assigned an event time from the available record. For example, someone censored at month 3 contributes information about follow-up before month 3 but is not in \(R(4)\).
One justification for the risk-set rule is to check when a subject’s record establishes observation and event-free status. For a subject with \(Z_i\geq t\), neither an event nor censoring has ended follow-up strictly before \(t\). For \(Z_i<t\), one of those events has ended follow-up before \(t\). Therefore the indicator of membership is \(I(Z_i\geq t)\), and summing these indicators gives the risk-set size:
This is a bookkeeping result about observed follow-up, not a claim that every subject in the risk set has the same event probability. Covariates, clinical history, and other modeled features can distinguish subjects. The risk set answers the more basic question: who was still observable and event-free immediately before this time?
Worked Example: The Effect of an Earlier Censoring Time
Suppose two hypothetical subjects, G and H, have both been event-free at the start of follow-up. G experiences the event at month 4. In one version of the records, H is censored at month 2; in another, H is followed and censored at month 5. Compare whether H belongs to the risk set at month 4.
With censoring at month 2, \(Z_H=2\), so \(Z_H\geq4\) is false. H is not in \(R(4)\). The record confirms H was event-free only through month 2; it provides no observation of H’s status at month 4.
With censoring at month 5, \(Z_H=5\), so \(Z_H\geq4\) is true. H is in \(R(4)\), and remains observed through month 4. Since G has an event at month 4, the risk set just before that event contains G and H, assuming there are no other subjects. H is not an event at month 5: that is the time follow-up ends.
The difference between these records is not a change in H’s assumed event time. It is a change in how long H’s event-free status is observed. Censoring determines the end of the known follow-up interval, and therefore determines membership in later risk sets.
Assumptions, Diagnostics, and Failure Modes
Constructing a risk set from correctly recorded times and event indicators is a direct data operation. Drawing valid statistical conclusions from censored data requires additional assumptions about how censoring occurs. A common condition is that censoring is independent of the event time, possibly conditional on recorded covariates or observed history used in the analysis. In practical terms, after accounting for those variables, the fact that follow-up ended should not carry additional information about when the event would have occurred.
This condition can fail. For example, if participants with worsening health are more likely to leave follow-up and also more likely to experience the event soon, their censoring is related to their unobserved event times. Treating those censoring times as ordinary noninformative endpoints can distort conclusions. The condition is not established simply by observing a low percentage of censored participants. It depends on the study process and the information available about participants who leave.
Useful checks begin with the data and follow-up process:
- Verify that the event indicator agrees with source records and that event times are not mistakenly coded as censoring times.
- Summarize censoring times and reasons, and compare observed characteristics of censored and retained participants.
- Check that recorded event times occur no later than the recorded end of follow-up, and investigate impossible or inconsistent dates.
- Review tied times and the recording precision. If multiple events and censorings share a recorded time, ensure the analysis’s ordering convention is appropriate and documented.
- Ask whether time zero has the same meaning for every subject. If subjects can enter observation after that origin, this simple risk-set definition needs modification; delayed entry is a separate issue.
These checks can reveal data errors or reasons to question the censoring assumption, but they cannot prove that censoring is independent of event time. In particular, similarity on measured variables does not rule out differences in unmeasured factors.
Common Mistakes and Alternatives
| Approach or mistake | What it does | Why it matters |
|---|---|---|
| Count a censored duration as an event time | Labels follow-up ending as if the event occurred then | It invents an event that was not observed and changes the event record. |
| Exclude every censored subject | Uses only subjects with observed events | It discards their known event-free follow-up and may select a nonrepresentative subset. |
| Keep a subject in every later risk set | Uses follow-up after the recorded censoring time | It treats an unobserved future as if event-free status were known. |
| Use the observed time and event indicator | Retains the known follow-up and distinguishes event from censoring | This preserves the information actually available, subject to the assumptions of the chosen analysis. |
The most common conceptual error is to treat a censoring time as an event time. If B is censored at month 5, the data do not say B experienced the event at month 5; they say B was observed event-free through month 5. The correction is to retain both pieces of information—the duration and the event indicator—and use the duration only as an endpoint of known follow-up.
Complete-case analysis, which discards censored subjects, is not a general substitute for accounting for censoring. It throws away the time during which those subjects were observed event-free, and the remaining event-observed subjects can differ systematically from all enrolled subjects. Conversely, retaining censored subjects beyond their last observed time asserts information the data do not provide. Appropriate survival analyses use event status and observed duration together; the particular estimator or model is a separate choice.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A participant is event-free through month 7 and then lost to follow-up. State the observed time and event indicator, and explain what is known about the event time.
- Subject P has an event at month 3, Q is censored at month 3, and R is censored at month 5. Under the convention used here, which subjects are in \(R(3)\), and which are in \(R(4)\)?
- A colleague says, “Everyone censored at month 6 had the event at month 6, so we can count them as events.” What is wrong with this reasoning?
- Why might excluding all censored participants produce a misleading analysis, even when their follow-up durations are recorded accurately?
Solutions. (1) \(Z=7\) months and \(\delta=0\); the participant is known to be event-free through month 7, but the later event time is unknown. (2) \(R(3)=\{P,Q,R\}\), since all three have \(Z_i\geq3\). \(R(4)=\{R\}\): P’s event and Q’s censoring occurred before month 4, while R remains under observation. (3) Censoring at month 6 marks the end of observed follow-up, not an observed event; assigning an event invents information. (4) It discards known event-free follow-up and can select a subset whose event experience differs from that of the enrolled group.