Recurrent Events With a Terminal Event

Why a Terminal Event Changes the Recurrence Process

A person may experience the same event repeatedly—for example, several hospitalizations—until a terminal event such as death makes further recurrences impossible to observe. The terminal event is therefore not merely another follow-up time. It ends the period in which a recurrence can occur and be recorded. Right censoring also ends observation, but it does not imply that the person’s recurrence opportunity has ended; it means only that subsequent event information is unavailable.

Keeping these two endings distinct is essential. A recurrence count must stop when the terminal event occurs, while the observed count must also stop when follow-up is censored. The corresponding time at risk stops at whichever occurs first. These definitions establish the event-process bookkeeping needed before specifying a joint model.

Definition: For subject $i$, let $T_i$ be the time of the terminal event and $C_i$ the right-censoring time. Let $S_{ik}$ denote the time of the $k$th recurrence, when that recurrence occurs before the terminal event. The complete-follow-up recurrent-event count up to time $t$ is $$ N_i^R(t)=\sum_k I(S_{ik}\le t,\ S_{ik}<T_i). $$ The count observed by time $t$, allowing for censoring, is $$ N_i^{R,\mathrm{obs}}(t)=\sum_k I(S_{ik}\le t,\ S_{ik}<T_i,\ S_{ik}\le C_i). $$ The at-risk indicator for observing recurrences is $$ Y_i(t)=I(t\le T_i,\ t\le C_i). $$ Thus, the observed recurrent-event process stops at the earlier of $T_i$ and $C_i$. At a tie between a recurrence and the terminal event, this definition excludes the recurrence by requiring $S_{ik}<T_i$; a study with a different event-ordering convention must state it explicitly.

The complete-follow-up count $N_i^R(t)$ describes recurrence history through the terminal event as it would be known under complete follow-up. If censoring occurs first, its later values are not observed. The censoring-adjusted process $N_i^{R,\mathrm{obs}}(t)$, in contrast, counts only recurrences actually observed by time $t$. In particular, it cannot jump after $C_i$, even when $T_i>C_i$.

The terminal-event process can also be written as a counting process. The observed terminal-event indicator by time $t$ is $N_i^{D,\mathrm{obs}}(t)=I(T_i\le t,\ T_i\le C_i)$. If censoring occurs first, this process does not record a terminal event, because no terminal event was observed during follow-up. The terminal event and censoring have different meanings even though both end the observed recurrence history.

At Risk, Observed, and No Longer Observable

With the stated tie convention, $Y_i(t)=1$ while subject $i$ has not yet had the terminal event and has not yet been censored, including at the endpoint itself. Immediately after either endpoint, $Y_i(t)=0$. This convention makes the risk indicator include a terminal event at its event time while excluding recurrences simultaneous with that terminal event. In continuous-time models, exact ties generally have probability zero, but recorded times may be rounded, so an analysis still needs an explicit convention.

For a recurrence intensity, let $\mathcal H_i(t-)$ denote the relevant event and covariate history just before time $t$. A conditional recurrence intensity $\lambda_i^R(t\mid\mathcal H_i(t-))$ describes the instantaneous recurrence rate among subjects still at risk, conditional on that history. Its units are recurrences per unit time. Under a correctly specified conditional intensity and an observation scheme that does not introduce unaccounted informative censoring, the intensity of the observed counting process is

Y_i(t)\lambda_i^R(t\mid\mathcal H_i(t-)).

The factor $Y_i(t)$ is crucial: the formula applies to $N_i^{R,\mathrm{obs}}(t)$, which stops at censoring as well as at the terminal event. It is not a valid observed-process intensity to multiply by this risk indicator while defining the count to continue after censoring. If censoring depends on unobserved recurrence risk, the censoring mechanism must be addressed; simply applying the at-risk formula does not remove informative censoring.

Worked Example: Reading a Subject’s Event Timeline

Suppose a hypothetical subject has terminal-event time $T_i=9$ months and censoring time $C_i=6$ months. The recorded recurrence times are 1.2, 3.5, and 5.8 months. Find the observed recurrence count at 4 months and at 8 months, and identify the end of observed recurrence follow-up.

At 4 months, the recurrences at 1.2 and 3.5 months have occurred, so $N_i^{R,\mathrm{obs}}(4)=2$. At 8 months, there is still no observed recurrence after 5.8 months, and censoring occurred at 6 months. Therefore $N_i^{R,\mathrm{obs}}(8)=3$, not a count that includes any hypothetical recurrence between months 6 and 8.

Observed recurrence follow-up ends at $\min(T_i,C_i)=\min(9,6)=6$ months. Censoring, rather than the terminal event, ended observation. The data do not establish whether any recurrence occurred between months 6 and 9.

From an Intensity to Expected Observed Counts

A useful consequence of the intensity definition is that the expected observed count is determined by the integrated intensity over the period actually at risk. Suppose the observed process has conditional intensity $Y_i(u)\lambda_i^R(u\mid\mathcal H_i(u-))$, and the process is integrable. Its compensated process is

M_i(t)=N_i^{R,\mathrm{obs}}(t) -\int_0^t Y_i(u)\lambda_i^R(u\mid\mathcal H_i(u-))\,du.

Under the stated intensity conditions, $M_i(t)$ is a martingale with mean-zero increments. Taking expectations gives the compensator identity

E\{N_i^{R,\mathrm{obs}}(t)\} = E\left\{\int_0^t Y_i(u)\lambda_i^R(u\mid\mathcal H_i(u-))\,du\right\}.

The right-hand side uses the same censoring- and terminal-event-adjusted risk indicator as the observed count. The equality is an expectation over repeated realizations; it does not say that a particular person’s observed count must equal that person’s integrated intensity.

Worked Example: Integrated Intensity During Observed Follow-Up

Consider a hypothetical subject whose recurrence intensity is constant at $0.25$ recurrences per month while at risk. The subject has a terminal event at month 5 and is censored at month 4. Calculate the integrated at-risk intensity through month 6 and interpret it.

The observed recurrence period ends at $\min(5,4)=4$ months. Thus $Y_i(u)=1$ during the first 4 months and $0$ afterward. The integrated intensity is $$ \int_0^6 Y_i(u)\lambda_i^R(u)\,du =\int_0^4 0.25\,du+\int_4^6 0\,du =(0.25)(4)=1.00. $$

The value 1.00 is the compensator through month 6; under the assumed intensity, its expectation corresponds to one observed recurrence over that exposure period. It is not a prediction that exactly one recurrence must be observed. If the subject actually had two recurrences by month 4, the observed count would be 2 while the integrated intensity remained 1.00.

This derivation also shows why a count and its exposure must be paired. A subject who is no longer at risk contributes no further recurrence intensity after the terminal event or censoring. Treating time after either endpoint as recurrence exposure would make the integrated rate inconsistent with the observed process.

Worked Comparison Across Subjects

Worked Example: Counting Events and Person-Time

Consider three hypothetical subjects followed from time 0. Subject A has $T_A=4$ months, $C_A=5$ months, and observed recurrences at months 1 and 3. Subject B has $T_B=6$ months, $C_B=2$ months, and an observed recurrence at month 0.5. Subject C has $T_C=3$ months, $C_C=5$ months, and an observed recurrence at month 2. Find each observed follow-up duration, the total observed recurrence count, and the crude event rate per person-month.

For each subject, the observed duration is the smaller of the terminal and censoring times. A contributes $\min(4,5)=4$ months and 2 recurrences. B contributes $\min(6,2)=2$ months and 1 recurrence. C contributes $\min(3,5)=3$ months and 1 recurrence. Total observed time is $4+2+3=9$ person-months, and total observed recurrences are $2+1+1=4$.

The crude rate is $$ \frac{4\ \text{recurrences}}{9\ \text{person-months}} =0.4444\ \text{recurrences per person-month}. $$

This arithmetic summarizes the observed data; it does not by itself account for differences in recurrence intensity over time, subject characteristics, or the way terminal events alter who remains at risk. In particular, B contributes only 2 months of observed exposure even though the hypothetical terminal-event time is 6 months.

Diagnostics, Alternatives, and Limits

A useful data check is to draw each subject’s timeline with recurrence times, the terminal-event time, and the censoring time marked separately. The observed recurrent-event count must be nondecreasing, its jumps must occur only at recorded recurrences before the terminal event and no later than censoring, and it must remain constant after the observed period ends. The risk indicator must be zero after $\min(T_i,C_i)$. Investigate any recurrence recorded after a terminal event or after censoring; it may reflect a coding error, an incorrect endpoint, or a mismatch between the event definition and the data.

The definitions also clarify what the data cannot answer. Once a terminal event makes recurrence impossible, there is no post-terminal recurrence history to recover. Once censoring occurs, later recurrence history is unobserved, not known to be zero. A model for recurrence after censoring or for the dependence between recurrence and terminal-event processes requires additional assumptions; the event-count definitions alone do not supply them.

Different summaries answer different questions. Counting all recurrences while at risk retains repeated events. A first-event analysis discards later recurrences and instead describes time to the first event. A composite endpoint that combines recurrence and terminal events treats distinct outcomes as one endpoint, changing the scientific question. None is automatically preferable: the choice depends on whether the target is repeated-event burden, first-event timing, or occurrence of a combined outcome.

Approach	What it summarizes	Key limitation
Observed recurrent-event count	Repeated events while recurrence observation is possible	Follow-up duration and terminal-event occurrence affect opportunity to observe events
Time to first recurrence	Timing of the first recurrence	Does not use later recurrences
Composite endpoint	Occurrence of any event included in the composite	Can combine outcomes with different meanings

Common pitfall: “Censoring occurred at month 4, so the subject had no recurrences after month 4 through the terminal event at month 7.” This confuses absence of observation with absence of events. The observed count stops at censoring; what happened afterward is unknown. By contrast, after a terminal event that truly ends recurrence opportunity, further recurrence is not merely unobserved under this event definition: the recurrence process has ended.

Key takeaway: Define the recurrence count, terminal-event time, censoring time, and at-risk indicator together. The observed count and its intensity must both stop at the earlier of terminal event and censoring.

Check Your Understanding

Answer each question using the event-process definitions above. Brief solutions follow the questions.

A subject has $T_i=8$, $C_i=5$, and recurrences at months 1, 4, and 6 in a hypothetical complete record. How many recurrences are observed?
Why does an observed recurrence count not include a recurrence after censoring, even if the terminal event occurs later?
If the at-risk recurrence intensity is $0.10$ per week for 6 observed weeks and zero afterward, what is the integrated at-risk intensity?
A recurrence is recorded at the same time as a terminal event. What must be specified before deciding whether it counts?
What does a first-event analysis omit when subjects can have recurrent events?

Solutions. (1) Two: the events at months 1 and 4 occur by censoring; the month-6 event is not observed. (2) Censoring ends observation, so later recurrence history is unavailable rather than known to be zero. (3) $0.10\times6=0.60$ expected-count units over the observed exposure; it is not a guarantee of 0.60 actual events. (4) The event-ordering convention, including whether a recurrence simultaneous with the terminal event is counted. Under the definition here, it is excluded. (5) It omits all recurrence events after the first.