Recurrent-Event Path Likelihoods

Why the Entire Recurrent-Event Path Matters

For a single event, the likelihood reflects both whether the event occurred and how long the subject remained event-free. Recurrent-event data require the same accounting over and over: every observed event contributes a term for its occurrence, and every interval without an event contributes a term for remaining event-free. Follow-up continues after each event, so the likelihood includes the whole observed history rather than stopping at the first jump.

This tutorial constructs that likelihood for one fully observed recurrent-event process. It does not require events to have the same conditional rate throughout follow-up. In particular, the intensity after an event may depend on the event history up to that time. The central bookkeeping rule is simple: evaluate the intensity just before each event, then subtract its cumulative value over the entire follow-up interval.

Definition: Let $N(t)$ be a simple counting process observed continuously on $[0,\tau]$, with $N(0)=0$. Its observed event times are $0<T_1<\cdots<T_m\leq\tau$, where $m=N(\tau)$. Given the observed history just before time $t$, write its conditional intensity as $\lambda(t\mid\mathcal{H}_{t-};\theta)$. The recurrent-event path likelihood, up to a factor that does not depend on $\theta$, is $$ L(\theta)=\left\{\prod_{j=1}^{m}\lambda(T_j\mid\mathcal{H}_{T_j-};\theta)\right\} \exp\left\{-\int_0^\tau \lambda(u\mid\mathcal{H}_{u-};\theta)\,du\right\}. $$ An empty product is 1, so a subject with no events contributes only the exponential term.

Here $\mathcal{H}_{t-}$ denotes the event history strictly before $t$. The left-limit convention matters: the intensity at $T_j$ describes the rate immediately before the event at $T_j$, not a rate updated using that event. The intensity has units of events per unit time; its integral is dimensionless, as required for the exponential term. The product of intensities is a density for the observed event times with respect to event-time measure, not a dimensionless probability by itself.

Assumptions Behind the Path Likelihood

The expression applies when the conditional intensity specifies the event process on the observation interval and the observed path is complete. More precisely, it uses the following conditions:

Simple events: events occur one at a time, with no simultaneous jumps. Thus each observed jump increases $N(t)$ by one.
Continuous-time intensity: conditional on the past, the probability of an event in a short interval of length $dt$ is approximately $\lambda(t\mid\mathcal{H}_{t-};\theta)dt$, with the probability of two or more events in that interval negligible relative to $dt$.
Predictability and non-explosion: the intensity is determined by information available just before $t$, and its cumulative value is finite over the observed interval, so the process does not have infinitely many events in finite time.
Complete observation and specified follow-up: event times are recorded through $\tau$. If the end of observation is random and its mechanism contains information about event risk, that mechanism must be modeled rather than silently ignored.
Correct conditional model: the proposed intensity, including its dependence on event history, is the model being used to describe the conditional event rate.

The likelihood is conditional on the starting history and any fixed or conditioned-on covariates. It is not automatically a likelihood for the observation process, treatment assignment, or other random quantities that may also appear in a study. If a subject is only at risk during selected intervals, an at-risk indicator $Y(t)$ can be included: the intensity is $Y(t)\lambda(t\mid\mathcal{H}_{t-};\theta)$, and the integrated term must cover the time for which $Y(t)=1$. That extension requires correctly specified, observed risk intervals.

Why Events and Event-Free Time Both Enter

The likelihood can be justified by considering what the conditional intensity says over a short interval. Given the history at time $t$, the conditional probability of no event during $[t,t+dt)$ is approximately $1-\lambda(t\mid\mathcal{H}_{t-};\theta)dt$. Multiplying these no-event contributions across event-free time yields the exponential of minus the integrated intensity. At an observed event time, the event-time density contributes the intensity evaluated at that time.

A more explicit justification uses the conditional survival of the next waiting time. Starting at time $a$ with a fixed past history, suppose the path has no new event up to time $s$. Along that no-event history, write the intensity as $\lambda_a(u;\theta)$. Its conditional survival function $S_a(s)$ satisfies

\frac{d}{ds}S_a(s)=-\lambda_a(s;\theta)S_a(s),\qquad S_a(a)=1.

Solving this differential equation gives $S_a(s)=\exp\{-\int_a^s\lambda_a(u;\theta)\,du\}$. The density that the next event occurs at $s$ is its hazard times its survival, so it is $\lambda_a(s;\theta)\exp\{-\int_a^s\lambda_a(u;\theta)\,du\}$. Apply this argument from the start to the first event, then after each event to the next, and finally from the last event to $\tau$, where the subject has no further observed event. Multiplying those conditional contributions gives the stated product of event intensities and the single exponential of the integrated intensity over $[0,\tau]$. This also shows why a terminal event-free interval remains in the likelihood.

Worked Example: A Constant Recurrent-Event Rate

Suppose one hypothetical subject is observed continuously for 10 months and has events at months 2, 5.5, and 9. Assume a homogeneous Poisson process with constant intensity $\lambda(t;\theta)=\theta$, where $\theta>0$ is measured in events per month. This model assumes that the rate is constant and does not change after an event; it also assumes a simple process with complete observation to month 10.

There are three event contributions, each equal to $\theta$, and the integrated intensity is $10\theta$. Therefore,

L(\theta)=\theta^3\exp(-10\theta).

The log-likelihood is $\ell(\theta)=3\log(\theta)-10\theta$. Differentiating gives $d\ell/d\theta=3/\theta-10$. Setting this equal to zero yields $\hat{\theta}=3/10=0.3000$ events per month. The estimate is the observed event count divided by the observed time. Under this model, the data do not show whether the rate changes with time since the last event; that pattern has been excluded by assumption.

When the Rate Resets After Each Event

A recurrent-event model may instead describe each waiting time since the most recent event. Let $G_j=T_j-T_{j-1}$ be the $j$th completed gap, with $T_0=0$, and let $G_{m+1}=\tau-T_m$ be the final, right-censored gap. If the conditional intensity during a gap depends only on its elapsed time $u$, call it $h(u;\theta)$. The path likelihood then has one factor $h(G_j;\theta)$ for each completed gap and survival through the final gap. The cumulative-intensity form is still the general rule; this gap notation simply displays its interval-by-interval contributions.

Worked Example: Weibull Gap-Time Intensity

Consider a hypothetical recurrent process observed for 10 months, with events at months 2, 5, and 9. Suppose each waiting time follows a Weibull model with shape 2 and scale 4 months. Its gap-time hazard is $h(u)=2u/16=u/8$ per month, and its cumulative hazard over a gap of length $g$ is $H(g)=g^2/16$. This model assumes that the conditional hazard resets to a function of elapsed time since the latest event and that successive gaps follow the specified conditional law.

The completed gaps are 2, 3, and 4 months; the final event-free gap is 1 month. The hazard contributions at the three events are $2/8$, $3/8$, and $4/8$, respectively. The total cumulative hazard includes all four gaps:

\sum_{j=1}^{4}H(G_j) =\frac{2^2+3^2+4^2+1^2}{16} =\frac{30}{16} =1.875.

Consequently, the path likelihood is

L=\left(\frac{2}{8}\right)\left(\frac{3}{8}\right)\left(\frac{4}{8}\right)e^{-1.875} =0.046875\,e^{-1.875} \approx 0.00719.

The final one-month gap contributes no event-intensity factor, because no event occurred in it, but it does contribute $H(1)=1/16$ to the cumulative hazard. Omitting that term would treat the observed follow-up as ending at the last event rather than at month 10.

When the Rate Depends on the Event Count

Another possibility is that the event history changes the future rate. For example, an intensity may increase with the number of previous events. In such a model, the intensity must be recalculated at each jump using the history just before that jump, and the integrated intensity must be calculated over each stretch of time on which that history remains unchanged.

Worked Example: An Intensity Increasing With Previous Events

Suppose a hypothetical subject is observed for 5 months, with events at months 1, 3, and 4.5. Let the intensity be $\lambda(t)=\alpha+\beta N(t-)$, with $\alpha=0.2$ events per month and $\beta=0.1$ events per month per previous event. Assume complete follow-up and that this specified history-dependent intensity is the model for the process.

Immediately before the three events, the event counts are 0, 1, and 2. Thus the event-intensity product is

(0.2+0.1\cdot0)(0.2+0.1\cdot1)(0.2+0.1\cdot2) =(0.2)(0.3)(0.4)=0.024.

For the integrated intensity, the count is 0 from month 0 to 1, 1 from month 1 to 3, 2 from month 3 to 4.5, and 3 from month 4.5 to 5. The integrated intensity is therefore

$$ 0.2(1)+0.3(2)+0.4(1.5)+0.5(0.5) =0.2+0.6+0.6+0.25 =1.65. $$

The likelihood is $L=0.024e^{-1.65}\approx0.00461$. At month 4.5 the event contribution is 0.4, not 0.5: the event itself has not yet occurred when its intensity is evaluated. The rate becomes 0.5 only after that event, for the remaining half-month of follow-up.

Log-Likelihood, Comparisons, and Diagnostics

Taking logarithms converts the product into a sum:

\ell(\theta)=\sum_{j=1}^{m}\log\{\lambda(T_j\mid\mathcal{H}_{T_j-};\theta)\} -\int_0^\tau\lambda(u\mid\mathcal{H}_{u-};\theta)\,du.

This form is often easier to compute and optimize. It also makes the roles of the data visible: each event contributes a log-intensity, while all observed time contributes a negative cumulative-intensity term. If the intensity is zero at an observed event, the likelihood is zero and the log-likelihood is minus infinity; such a model cannot assign positive density to the observed path.

A recurrent-event path likelihood differs from approaches that discard some of the path. A first-event likelihood stops at the first event and cannot use later recurrent events. A count-only likelihood retains $N(\tau)$ but not the event times; under a homogeneous Poisson model the count contains the relevant rate information, but under a time-varying or history-dependent intensity, event timing can be informative. A gap-time likelihood is not a competing formula: it is a way to express the same path likelihood when the intensity is specified through elapsed time since the previous event.

Useful checks include confirming that every observed event has exactly one intensity contribution, that intensities are evaluated using the pre-event history, and that the integrated intensity covers all observed at-risk time, including the final event-free interval. Plotting observed event times and cumulative event counts can reveal data errors such as duplicated times or gaps in recording. Comparing observed event counts with those predicted by the fitted intensity can flag poor fit, though such comparisons do not by themselves establish a particular alternative model.

A path likelihood is only as reliable as the observation and intensity assumptions behind it. Unrecorded events, incorrectly recorded follow-up, informative loss to follow-up, or an intensity that omits important history can invalidate the likelihood interpretation. The expression also does not by itself account for dependence between subjects or additional event types; those require an appropriate joint model. In applications, numerical integration must be accurate enough to represent the cumulative-intensity term, especially when the intensity changes between events or depends on elapsed time.

Common pitfall: Counting only the observed jumps and maximizing a product of their intensities omits the probability of event-free time. For a constant rate, this would use $\theta^m$ rather than $\theta^m e^{-\theta\tau}$, and would increase without bound as $\theta$ grows. The correction is to include $-\int_0^\tau\lambda(u)\,du$ in the log-likelihood, over the full observed at-risk interval.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A constant-rate recurrent process has 4 events during 8 months of complete observation. Write its likelihood as a function of the rate $\theta$, and find the maximum-likelihood estimate.
An event occurs at month 2, and follow-up ends at month 5. Under an intensity that resets after each event, which part of the likelihood represents months 2 through 5 if no further event occurs?
For $\lambda(t)=0.1+0.2N(t-)$, an event occurs when $N(t-)=2$. What intensity contribution belongs to that event?
Why can event times matter even when the total event count is known?

Solutions. (1) $L(\theta)=\theta^4e^{-8\theta}$. The log-likelihood derivative is $4/\theta-8$, so $\hat{\theta}=4/8=0.5000$ events per month. (2) The interval contributes survival through the final gap: $\exp\{-\int_2^5\lambda(u\mid\mathcal{H}_{u-})\,du\}$, with no event-intensity factor for that interval. (3) The pre-event intensity is $0.1+0.2(2)=0.5000$ events per unit time. (4) With time-varying or history-dependent intensity, different event timings can produce different event-intensity factors and different integrated intensities, even when $N(\tau)$ is the same.