Why Event Order and Follow-Up Belong in the Data
A subject can experience an event more than once: a patient may be hospitalized several times, for example, or an equipment unit may fail repeatedly after repair. A record containing only the first event time cannot describe later events. A record that lists recurrence times but omits how long each subject was observed is also incomplete: no recorded event after a subject’s last follow-up does not mean the subject remained event-free indefinitely.
The basic task here is to represent each subject’s ordered event times alongside the observation period in which those times were ascertained. This is a data description, not yet a choice of recurrent-event estimand or model. In particular, recording censoring does not by itself establish that censoring is independent of future event experience.
The time origin must be defined in context: it might be treatment initiation, a procedure, or entry into a study. For simplicity, the examples use time zero as the start of observation for every subject. If subjects enter observation later, their entry times must also be recorded; time before entry is not observed follow-up. Event times should use a consistent time scale and units, and the event definition should be applied consistently across subjects.
Ordered Event Times and Counts
The subject-level record consists of the event-time sequence and the end of observed follow-up. A compact summary of that sequence is the event count by time \(t\):
Here \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise. Thus \(N_i(t)\) counts recorded recurrent events by time \(t\), within the observed follow-up. It does not assert anything about events after \(C_i\). The final observed count is \(N_i(C_i)=K_i\), subject to the protocol’s convention for events recorded exactly at the follow-up boundary.
When event times are distinct, the ordering of the \(T_{ij}\) determines event order. If two events are recorded at the same time because the time scale is coarse, the data may not establish which came first. Do not assign an order without information or a prespecified convention that is supported by the event definitions. If actual event times are known only to lie between assessments, record that interval information rather than silently replacing it with an exact time.
Worked Example: Preserving More Than the First Event
Suppose three hypothetical subjects are observed from time zero. Subject A has recorded events at 2.0, 5.5, and 9.0 months and is followed through month 12. Subject B has an event at month 4 and is censored at month 7. Subject C has no events and is followed through month 10. Assume these are exact event times, and that each subject’s follow-up end is known.
For A, the record is \((T_{A1},T_{A2},T_{A3})=(2.0,5.5,9.0)\), \(K_A=3\), and \(C_A=12\) months. At month 6, \(N_A(6)=I(2.0\le6)+I(5.5\le6)+I(9.0\le6)=1+1+0=2\).
For B, \(T_{B1}=4\), \(K_B=1\), and \(C_B=7\) months. For C, \(K_C=0\) and \(C_C=10\) months; there is no event time to enter, but the 10 months of observed follow-up remain part of the record. These values describe observed events and observation time, not the subjects’ unobserved event histories after follow-up ends.
A time-to-first-event summary would retain only 2.0 months for A and 4 months for B, while classifying C as having no event during observed follow-up. It would discard A’s events at 5.5 and 9.0 months. That reduction is appropriate only if the analysis question concerns first events; it is not a faithful representation of all recurrences.
Choosing a Data Layout
The underlying information is the same whether stored in one row per subject or several rows per subject. A subject-level layout is convenient for checking total follow-up and event counts. An event-level layout places each recurrent event on its own row and makes event order explicit. An interval-style layout can record successive periods between events and the final period ending at censoring. These are storage choices, not different event histories; the definitions of start and end points must be consistent.
For interval-style records, a common convention is to have an event occur at the end of an interval, while a final interval ends at censoring without an event. The interval endpoint convention must be documented so an event is not counted twice or omitted at a boundary. Later model choices can impose additional requirements on these records; an interval layout alone does not determine a model or its risk sets.
Worked Example: Translating One Subject’s Record Into Event Rows
A hypothetical subject is followed for 8 weeks and has events at weeks 1.5 and 6.0. Use an event-level layout with one row per event and a separate row marking the final event-free observation period. For this illustration, each event is placed at the end of its preceding interval.
| Subject | Event order | Interval start (weeks) | Interval end (weeks) | Event at end? |
|---|---|---|---|---|
| R | 1 | 0 | 1.5 | Yes |
| R | 2 | 1.5 | 6.0 | Yes |
| R | — | 6.0 | 8.0 | No; censored at week 8 |
The event rows show two recurrences in order; the last row preserves the remaining 2 weeks of observed follow-up without inventing another event. The event count is \(K_R=2\), and the cumulative count by week 5 is \(N_R(5)=1\), since only the event at week 1.5 has occurred by then. The interval starts and ends are a rearrangement of the event times and follow-up end, not additional observed events.
An event-level table should retain a subject identifier, event time, and event order, and it should be possible to recover each subject’s end of follow-up. A subject-level table should retain all event times or a structure that preserves their sequence; a single event count cannot recover when the events occurred. Whichever format is used, a subject with no recurrence must remain represented, rather than disappearing because there is no event row.
Censoring and Terminal Events
Censoring ends observation without establishing that a recurrent event occurred at that time. A terminal event, such as death, is different: it is an observed event that ends the possibility of subsequent recurrence as defined in the study. The record should identify which type of endpoint occurred. Treating death as ordinary loss to follow-up obscures an observed outcome; treating loss to follow-up as a terminal event invents one.
Let \(D_i\) denote a terminal-event time when one occurs. If recurrent events are observed only before the terminal event or censoring, the end of the recurrence observation window is \(\min(C_i,D_i)\), with the endpoint type recorded separately. The event protocol must specify whether a recurrent event and terminal event recorded at the same time are both included and, if needed, how their ordering is resolved. Without that information, a tie should be flagged as ambiguous rather than settled by an arbitrary ordering.
Worked Example: Distinguishing Censoring From a Terminal Event
Consider two hypothetical subjects with study administrative end at month 8. Subject M has recurrent events at months 1.2 and 3.0, then dies at month 5. Subject N has an event at month 1.5 and is lost to follow-up at month 4.2, with no terminal event recorded. Assume event ascertainment is complete through each endpoint and no recurrence is recorded at the endpoint itself.
For M, \(C_M=8\), \(D_M=5\), and the recurrence follow-up ends at \(\min(8,5)=5\) months because death is terminal. The observed recurrent-event count is \(K_M=2\); death is separately recorded as the terminal event at month 5. It is incorrect to label month 5 as censoring, because death was observed.
For N, \(C_N=4.2\) months and there is no recorded \(D_N\). The observed recurrence follow-up ends at month 4.2, with \(K_N=1\). Loss to follow-up does not establish that N had a second event, nor does it establish that no second event occurred afterward. The data record supports neither claim beyond month 4.2.
The distinction is essential even before an analysis method is chosen: the end time alone is not enough. The endpoint reason determines whether the record says observation stopped without an observed terminal outcome or a terminal outcome occurred.
Justification, Checks, and Common Errors
The count representation follows directly from the ordered event record. For any observed time \(t\), each event time contributes one indicator: it contributes 1 exactly when it has occurred by \(t\), and 0 otherwise. Summing those indicators therefore gives the number of observed events by \(t\). At the follow-up endpoint, every recorded event time is included, so the sum equals the number of recorded events, \(K_i\). This identity depends on retaining the event times and applying one consistent boundary rule.
Before analysis, check that each event belongs to a known subject, event times are in the stated units, event order is consistent with time, and no recorded event falls outside the subject’s observed window unless a protocol explains the boundary. Reconcile the event-level rows against subject-level event counts. Check that zero-event subjects and their follow-up durations are retained, and that every follow-up endpoint has a reason. Investigate duplicate times: they may indicate genuine simultaneous events, rounding, duplicate records, or a data-processing error.
- Keeping only the first event. This is appropriate for a first-event question, but it discards later recurrences and cannot represent the full event history.
- Calling the last event time the follow-up end. Follow-up may continue after the last recurrence. Record censoring or terminal-event time separately.
- Equating “no event recorded” with “no event occurred.” A subject may have no event during observed follow-up, or may have an event after censoring. The data distinguish neither history beyond the observed endpoint.
- Dropping subjects with no event rows. This erases their observed time without recurrence. Keep a subject-level record or an explicit follow-up row.
- Sorting events without checking ties and definitions. A timestamp does not resolve event order when times are tied or rounded. Verify the event definition and recording precision.
The main alternative is a time-to-first-event representation. It is simpler when the scientific outcome is specifically the first occurrence, but it cannot answer questions that depend on subsequent events. A total event count preserves how many events were observed but loses their timing and order. The full ordered event-time record with follow-up retains both, while requiring careful handling of event definitions, endpoint status, and recording precision. No layout removes the need for assumptions appropriate to the eventual analysis; in particular, the record alone does not justify treating censoring as non-informative.
Check Your Understanding
Answer each question using the definitions above. Solutions follow in the same box.
- A subject has events at days 3 and 8 and is censored at day 12. What are \(K_i\) and \(N_i(6)\), and what information would a first-event summary discard?
- A subject has no recorded events and is followed through day 20. Should the subject be omitted from an event-level data set? Explain.
- A subject has one recurrence at month 2 and dies at month 7. The administrative study end is month 10. What is the recurrence follow-up endpoint, and how should it be described?
- Two events for one subject are both recorded at day 5, but the source data report dates only to the nearest day. What can be concluded about their order?
Solutions. (1) \(K_i=2\), and \(N_i(6)=I(3\le6)+I(8\le6)=1+0=1\). A first-event summary keeps day 3 but discards the event at day 8. (2) No. The subject contributes 20 days of observed follow-up with zero recorded events; an event-only table must be supplemented by a subject-level follow-up record or an explicit no-event row. (3) The endpoint is month 7, because \(\min(10,7)=7\). Record death as the observed terminal event, not as censoring. (4) The recorded dates do not establish which event occurred first. Check finer-resolution source data or retain the tie as unresolved; do not invent an ordering.