Gap-Time Clocks For Recurrent Events

Why Use Time Since the Previous Event?

A recurrent-event record has two natural notions of elapsed time. One is time since study entry; the other is time since the most recent recurrence. These clocks answer different questions. If the clinical question concerns how risk changes as a patient spends longer without another event, time since the previous event is a direct analysis clock. At each recurrence, elapsed gap time returns to zero, even though the patient’s calendar follow-up continues.

The reset is a change in the time scale used to describe the event process. It does not erase earlier events, restart study follow-up, or imply that the patient’s condition has returned to its original state. In particular, a gap-time analysis can still condition on event order and the history of prior events.

Definition: For subject $i$, let $T_{ij}$ be the calendar time of the $j$th recurrent event, with $T_{i0}=0$ denoting study entry. The $j$th gap time is $W_{ij}=T_{ij}-T_{i,j-1}$. While subject $i$ is between events $j-1$ and $j$, the gap-time clock is $u=t-T_{i,j-1}$, where $t$ is calendar time. It starts at zero at the beginning of that interval and increases until the next event or the end of observation.

The first gap runs from study entry to the first event. Each subsequent gap runs from one event to the next. If the record ends before the next recurrence, the final interval is not a completed gap: it is an observed, event-free portion of a gap, ending at censoring or another event that prevents further observation.

From Calendar Event Times to Gap Times

Suppose events occur at ordered calendar times $T_{i1}<T_{i2}<\cdots<T_{im}$. Subtract consecutive times to obtain the completed gaps. The event order is preserved, but the origin for measuring elapsed time changes at each event. The identities of the events and any recorded covariates or prior-event history remain attached to the subject; only the clock’s zero point changes.

Worked Example: Converting Event Times to Gaps

Consider a hypothetical participant observed from day 0 through day 30, with recurrent events on days 6, 14, and 25. There is no censoring before day 30.

The first gap is $W_{i1}=6-0=6$ days. The second is $W_{i2}=14-6=8$ days. The third is $W_{i3}=25-14=11$ days. After the event on day 25, the gap-time clock resets to zero. It then reaches $30-25=5$ days at the end of observation, without another event.

The three completed gaps are therefore 6, 8, and 11 days, followed by 5 days of observed event-free time in the next gap. Their total is $6+8+11+5=30$ days, matching the calendar follow-up. The final 5 days are not a fourth event time or a completed inter-event duration.

More generally, if observation ends at calendar time $C_i$, and the last observed event is $T_{im}$, then the final gap has observed duration $C_i-T_{im}$ if no further recurrent event occurs before observation ends. For a subject with no observed events, the only observed interval is the first gap, from 0 through $C_i$. These calculations describe the data; they do not, by themselves, establish that censoring is statistically noninformative.

Worked Example: A Right-Censored Final Gap

In a hypothetical study, a participant is followed through day 23 and has recurrent events on days 4 and 13. The study record then ends, with no later event observed.

The first completed gap is $4-0=4$ days. The second completed gap is $13-4=9$ days. The last observed interval begins at day 13 and ends at day 23, so its gap-time duration is $23-13=10$ days. This is a right-censored gap: the participant was observed event-free for 10 days in that interval, but the time to the next event is unknown beyond day 23.

The observed calendar follow-up is $4+9+10=23$ days. It would be incorrect to record a third event at day 23 or to treat 10 days as the participant’s full time to the next event. The correct information is that the next gap exceeded 10 days, subject to the study’s observation and censoring assumptions.

What a Gap-Time Hazard Describes

A gap-time hazard describes the instantaneous rate of a next recurrence at a specified elapsed time since the previous event, among subjects who have reached that point in the interval without another recurrence and remain observed. It is conditional on the information used to define the model, which may include event order and prior history. A gap-time scale does not require successive gaps to be independent or identically distributed.

Definition: Let $H_{i,j-1}$ denote the observed history available at the start of gap $j$. The conditional gap-time survival function $S_{ij}(u\mid H_{i,j-1})$ is the probability that the next recurrent event occurs later than elapsed time $u$, conditional on that history. When this survival function is differentiable, its hazard is $$ \lambda_{ij}(u\mid H_{i,j-1}) = \lim_{\Delta u\to 0^+} \frac{ P\!\left(u\le W_{ij}<u+\Delta u\mid W_{ij}\ge u,\ H_{i,j-1}\right) }{\Delta u}. $$ The hazard has units of events per unit of elapsed gap time.

To connect this definition to calendar time, the gap beginning at event $j-1$ has $u=t-T_{i,j-1}$. A point $u$ days into that gap corresponds to calendar time $T_{i,j-1}+u$. The gap-time scale therefore re-expresses when risk is evaluated; it does not alter the underlying event time.

Why the Clock Resets: A Short Derivation

The conditional survival function for the $j$th gap can be written as $S_{ij}(u\mid H_{i,j-1})=P(W_{ij}>u\mid H_{i,j-1})$. For a continuous gap-time distribution, the conditional event probability over a short interval after $u$, given survival to $u$, is $1-S_{ij}(u+\Delta u\mid H_{i,j-1})/S_{ij}(u\mid H_{i,j-1})$. Dividing by $\Delta u$ and taking the limit gives the hazard:

\begin{aligned} \lambda_{ij}(u\mid H_{i,j-1}) &= -\frac{d}{du}\log S_{ij}(u\mid H_{i,j-1})\\ &= -\frac{S'_{ij}(u\mid H_{i,j-1})} {S_{ij}(u\mid H_{i,j-1})}. \end{aligned}

This derivation assumes a differentiable conditional survival function and positive survival at the elapsed time being considered. It shows why the time argument is $u$, not the absolute calendar time $t$: for the $j$th gap, the interval begins at $T_{i,j-1}$, and its elapsed time is $t-T_{i,j-1}$. At the event, a new interval begins and its elapsed time is zero. If event times have a discrete or mixed distribution, a hazard may instead need an appropriate discrete-time or jump-process definition; the clock-reset calculation still applies.

Worked Example: Interpreting the Same Elapsed Gap at Different Calendar Times

Imagine a hypothetical participant with events on days 5 and 17. The second gap begins at day 5 and ends at day 17, so its duration is 12 days. At calendar day 11, the participant is 6 days into that gap: $11-5=6$. At calendar day 17, immediately before the second event, the gap-time clock has reached 12 days; immediately after the event, the next gap begins at zero.

Suppose a specified conditional gap-time hazard at elapsed time 6 days is $0.04$ per day for the relevant history and event interval. Its interpretation is a rate at the point 6 days after the preceding event, conditional on remaining event-free to that point and on the stated history. It is not a 4% probability of an event on day 11, and it does not say that the participant’s overall risk is the same on every calendar day. A probability over a finite interval requires integrating the hazard, with the appropriate conditional survival information.

Assumptions, Data Checks, and Failure Modes

Defining gap times requires correctly ordered event dates and a clear observation origin. If exact event times are unavailable, the analyst must account for their recorded time resolution rather than silently treating rounded dates as exact. If two events share a recorded time, their ordering or whether they are distinct events must be resolved using the study’s event definitions and recording protocol.

The clock itself is a descriptive transformation and does not require independent gaps. Inference about a gap-time distribution or hazard does require assumptions appropriate to the chosen analysis. In particular, right censoring must be handled using a method whose censoring assumptions are credible, potentially conditional on measured history. If the chance observation ends depends on an unobserved impending recurrence after conditioning on the information used in the analysis, treating the observed censored gap as if it were an ordinary completed gap can bias conclusions.

Useful checks include verifying that all event times fall within observed follow-up, that gaps are nonnegative and sum with the final observed interval to the calendar time covered, and that the event order agrees with source records. Review subjects with zero events separately: they contribute observed first-gap time but have no reset. For each subject, confirm that censoring is assigned to the interval in which observation actually ends. Implausibly short or long gaps can indicate date-entry errors, duplicate events, or a mismatch between the recurrence definition and the recorded data.

Common pitfall: “After an event, follow-up starts over at day zero, so the participant’s earlier history no longer matters.” The clock’s elapsed-time value resets, but the participant’s calendar follow-up and event history do not. A later gap can have a different distribution because it is a later event order or follows a different clinical history. The correction is to reset only the gap-time origin and retain relevant history and observation information.

Gap Time and Total Time Answer Different Questions

A total-time clock measures elapsed time since study entry throughout follow-up. A gap-time clock measures elapsed time since the most recent event, resetting that value at every recurrence. The distinction is substantive: a gap-time description concerns how recurrence risk relates to the length of the current event-free interval, whereas total time concerns how it relates to time since entry. Neither clock is automatically preferable. The choice should match the scientific question and the time pattern believed relevant.

Feature	Gap-time clock	Total-time clock
Origin for the current event	Most recent recurrent event (or study entry for the first gap)	Study entry
Clock value after a recurrence	Returns to zero for the next interval	Continues increasing
Time-scale question	How does recurrence relate to time event-free since the last event?	How does recurrence relate to time since study entry?
History implication	Resetting elapsed time does not remove event-order or prior-history information	Elapsed time alone likewise does not summarize all prior history

The gap-time scale is especially interpretable when elapsed time since the latest event is central to the question. It can be less informative if the scientific mechanism is tied mainly to duration since entry, or if the reset obscures an important cumulative-time pattern. In either case, a clock choice is not a claim that other time scales or histories have no role.

Worked Example: Checking Whether Two Clocks Are Being Confused

A hypothetical participant has events at days 7 and 19 and is observed through day 28. The total-time values at the two events are 7 and 19 days. The gap times are 7 days for the first event and $19-7=12$ days for the second. After the second event, the participant has 9 days of observed time in the next gap, since $28-19=9$.

A report stating “the second recurrence occurred 19 days after the first” confuses the clocks: 19 is the second event’s calendar time, while the interval between events is 12 days. A report stating “the participant was followed for 12 days” is also wrong; total follow-up was 28 days. Correctly labeling all three quantities—calendar event time, completed gap duration, and total observed follow-up—prevents this mix-up.

Check Your Understanding

Answer each question using the distinction between calendar time and elapsed gap time. Solutions follow.

A hypothetical participant enters at day 0 and has events at days 3, 10, and 16. Give the completed gaps and the gap-time value immediately after the third event.
A participant has an event on day 8 and is censored on day 20 without another event. What is the observed duration of the last gap, and is it completed?
In one sentence, interpret a conditional gap-time hazard of $0.02$ per day at elapsed time 5 days.
Does resetting the gap-time clock imply that event gaps are independent or identically distributed? Explain.
A participant’s second event occurs on day 15, and the first occurred on day 4. State the second event’s total time and second gap time.

Solutions. (1) The completed gaps are $3-0=3$, $10-3=7$, and $16-10=6$ days; immediately after the third event, the next gap-time clock is zero. (2) The observed final-gap duration is $20-8=12$ days. It is right-censored, not completed, because no next event was observed. (3) Conditional on the specified history and remaining event-free to 5 days after the prior event, the instantaneous rate of the next recurrence at that elapsed time is $0.02$ per day; this is not a 2% finite-interval probability. (4) No. Resetting specifies the time origin for each interval, not the dependence structure; gaps may depend on event order, prior history, or other factors. (5) The second event’s total time is 15 days since entry, while its gap time is $15-4=11$ days since the first event.