Total-Time Clocks For Recurrent Events

Why the Clock Matters

A recurrent-event record contains more than a list of event counts. It also contains information about when events occur, and “when” depends on the time origin. If the scientific question concerns how event occurrence changes with duration since enrollment, a natural choice is to start each participant’s clock at study entry and let it run continuously. An event occurring after a participant has already had several events is still located on that same elapsed-time scale.

This is the total-time clock. It provides a common within-person scale for event occurrence: time zero is that participant’s study entry, and time \(t\) means \(t\) units of elapsed time after entry. A participant’s calendar entry date can differ from another’s, but both are measured from their own entry.

For clarity, this tutorial uses continuous time measured in days in the examples. The same indexing works in other units, such as weeks or years, provided event times and follow-up durations use the same unit. If a participant experiences \(m_i\) events during observed follow-up, their ordered event times are \(T_{i1},\ldots,T_{im_i}\), usually with \(T_{i1}<\cdots<T_{im_i}\). When exact ties are possible, they can be recorded as tied times rather than forced into an artificial order.

Indexing Events and Observation

A convenient summary of a participant’s recurrent-event history is the counting process \(N_i(t)\): the number of that participant’s events recorded by total time \(t\). With observed event times \(T_{ij}\), it can be written as

Here \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise. The formula counts events without changing the time coordinate after any event. For example, if a participant’s events occur at days 18 and 63 after entry, then \(N_i(20)=1\) and \(N_i(70)=2\). Day 63 remains day 63 on the study-entry clock; it is not recoded as day 45 just because 45 days elapsed after the first event.

At total time \(t\), the basic observed risk set consists of participants still under observation immediately before \(t\), subject to whatever eligibility rules the analysis defines. In a simple cohort in which everyone enters at time zero and is observed until censoring at \(C_i\), this is the set of participants whose follow-up has not ended before \(t\). A recurrent event by itself does not mean follow-up has ended. An analysis may impose additional eligibility conditions, but the clock still indexes time since entry.

The distinction between observation and eligibility is important. A participant may be observed at day 60 after entry and therefore available for an analysis at total time 60, even if they have already had events. Whether that participant belongs in a particular model’s risk set can depend on the model’s target and rules. The total-time clock specifies the time coordinate; it does not, by itself, settle every risk-set decision.

Risk Sets on the Total-Time Scale

The total-time risk set is formed by asking who is still observed at a common elapsed duration after entry. For a basic recurrent-event process, a participant with an earlier event can remain under observation and contribute later follow-up. For instance, someone followed to day 80 is still observed at total time 50 even if they experienced an event on day 30. Whether a particular analysis includes that person at day 50 depends on its eligibility definition, not on resetting the clock.

The translation from calendar time to total time also explains why participants entering on different dates can be compared at the same value of \(t\). For participant \(i\), calendar time \(s\) corresponds to total time \(s-E_i\). Thus a calendar date is not itself a common total-time value across participants.

A useful justification for this indexing is direct: subtracting the same participant-specific entry time from every event date preserves event order and elapsed-time differences. If \(S_{i1}<S_{i2}\), then \(S_{i1}-E_i<S_{i2}-E_i\), and \((S_{i2}-E_i)-(S_{i1}-E_i)=S_{i2}-S_{i1}\). Thus the transformation locates events relative to entry without changing the amount of calendar time between them. It does not make participants’ calendar dates equal; rather, it places their follow-up histories on the same kind of within-person scale.

Assumptions, Interpretation, and Checks

Choosing a total-time clock is a definition of the time axis, not a claim that a particular statistical model is correct. Its use relies on the following data and design conditions for the stated interpretation:

• Entry is meaningful and recorded. Study entry supplies the time origin for each participant, and date or time discrepancies are not large enough to alter the intended ordering or elapsed durations.
• Event times use a consistent convention. Recurrent events are recorded on a compatible scale, and the event-order labels correspond to the participant’s actual sequence when events are distinct.
• Follow-up is represented accurately. Censoring time denotes the end of observed follow-up, so events after that time are not treated as observed events.
• The clock matches the question. The analysis is intended to relate event occurrence to duration since entry, rather than to time since the most recent event or a shared calendar date.

In a model that uses this scale, time dependence is indexed by elapsed time since entry. A generic event intensity may be described as a conditional event rate at total time \(t\), given the event and covariate history just before \(t\). Its interpretation is a rate per unit of elapsed follow-up time for the specified history and covariates. This description does not assert that event rates are constant over time or that participants have identical event histories.

Diagnostics begin with the event-time records themselves. Check that each event time is nonnegative, that every observed event time falls within that participant’s follow-up, and that the date-to-duration conversions are consistent. Tabulate event counts and follow-up by elapsed-time intervals; an implausible pile-up at time zero, a sudden drop in follow-up not explained by censoring, or reversed event ordering can signal coding errors. Compare records against source dates when possible, and specify how same-day events and event/censoring ties are handled.

These checks do not establish that censoring is noninformative or validate a model’s functional form. Those are separate issues. They do help catch a particularly consequential clock error: mixing calendar dates for some participants with durations since entry for others. Such a mistake can put records on incompatible axes and create incorrect event ordering or risk-set membership.

Choosing This Clock and Common Pitfalls

The chief alternative clock for recurrent events measures time since the previous event, restarting the time origin after each event. That is a different question and a different indexing choice; it is treated separately in the material on gap-time clocks. Total time is suitable when elapsed duration since study entry is the meaningful time coordinate, for example when participants may experience changes in event occurrence as their follow-up progresses, regardless of how recently an event occurred.

Total time has limitations. If risk is driven primarily by time since the previous event, a total-time scale may not directly represent that mechanism. If entry dates differ and the scientific question concerns calendar-period effects, elapsed time since entry alone does not encode calendar time; calendar time may need separate representation. Also, a clock choice does not resolve dependence among recurrent events, informative censoring, or confounding. Those require appropriate estimands, models, and assumptions.

Common mistake: After a participant’s first event at day 20, a second event 35 days later is recorded at total time 35. That reasoning has silently restarted the clock. The correction is to retain the original entry origin: the second event occurs at day \(20+35=55\) after entry. Under total time, event order is an index attached to event times, not a series of new time origins.

A second mistake is to treat every participant’s calendar date as if it were the same total-time value. The March 15 example shows why that fails: total time is calculated from each individual’s entry date. Always make the origin explicit, calculate durations using a consistent unit, and verify them against follow-up limits.