Recurrent Event Models for Repeated Clinical Events

Why Repeated Events Need Their Own Hazard Formulation

A time-to-event model usually focuses on the time to one event per subject. Many clinical endpoints, however, can occur more than once: exacerbations, seizures, infections, or rescue-medication episodes. A subject who has already had an event remains informative, but their later event risk may depend on how many events have occurred and how long it has been since the last one. A recurrent-event model makes those choices explicit while allowing the hazard to depend on drug exposure.

The key modeling decision is the time scale for the hazard after an event. An event-order model can use time since study entry while allowing a different baseline hazard for the first, second, or later event. A gap-time model resets the clock after each event and describes risk as a function of time since the preceding event. These formulations answer different questions; neither is automatically preferable.

Definition: For subject \(i\), let \(N_i(t)\) be the number of recurrent events observed by calendar time \(t\), and let \(T_{ij}\) be the time of the subject’s \(j\)th event. The recurrent-event hazard for event \(j\) is the instantaneous event rate at time \(t\), conditional on the subject being under observation and on the event history up to \(t\). It is written \(\lambda_{ij}(t\mid\mathcal{H}_i(t))\), where \(\mathcal{H}_i(t)\) denotes that history.

A hazard has units of inverse time, such as events per day. It is not itself the probability of an event over a day. For a subject who is at risk over an interval from \(a\) to \(b\), the cumulative hazard is the integral of the hazard over that interval; under the stated conditional hazard model, the probability of at least one next event over the interval is one minus the exponential of the negative cumulative hazard.

Event-Order and Gap-Time Formulations

In an event-order, or total-time, formulation, time \(t\) is measured from a common origin, usually study entry. A common stratified model gives each event order its own baseline hazard. Subject \(i\) is eligible for the \(j\)th-event risk set only after having experienced event \(j-1\); for \(j=1\), the subject is eligible from entry. This sequential risk-set rule is often called a Prentice-Williams-Peterson total-time formulation.

\lambda_{ij}^{\mathrm{TT}}(t\mid\mathcal{H}_i(t)) = Y_{ij}^{\mathrm{TT}}(t)\lambda_{0j}(t) \exp\{\beta C_i(t)\}, \qquad t>T_{i,j-1}.

Here \(Y_{ij}^{\mathrm{TT}}(t)\) is 1 when the subject is under observation and eligible for event \(j\), and 0 otherwise; \(T_{i,0}\) denotes entry time. The baseline hazard \(\lambda_{0j}(t)\) may differ by event order, \(C_i(t)\) is the exposure at calendar time \(t\), and \(\beta\) is the log hazard ratio for a one-unit increase in exposure. The proportional-hazards form assumes that the exposure multiplies the event-order-specific baseline hazard by \(\exp\{\beta C_i(t)\}\).

A gap-time formulation instead measures \(u=t-T_{i,j-1}\), the time since the preceding event. Its baseline hazard is indexed by gap time, and it can also differ by event order:

\lambda_{ij}^{\mathrm{GT}}(u\mid\mathcal{H}_i(t)) = Y_{ij}^{\mathrm{GT}}(u)\lambda_{0j}^{\mathrm{GT}}(u) \exp\{\beta C_i(T_{i,j-1}+u)\}, \qquad u>0.

The exposure is still evaluated at calendar time: at gap time \(u\), the subject’s calendar time is \(T_{i,j-1}+u\). This distinction matters when exposure changes with dosing or when the timing of previous events differs among subjects. The event-order baseline describes how risk changes over time since entry; the gap-time baseline describes how risk changes over time since the previous event.

Formulation	Clock for the baseline hazard	Risk-set feature	Typical question
Event-order, total time	Time since study entry	Separate eligible risk set for each event order	Does risk vary with time on study, and does that pattern differ by event number?
Event-order, gap time	Time since the preceding event	Separate eligible risk set for each event order	How does risk evolve within each interval between events?
Andersen-Gill recurrent-event model	Time since study entry	One recurrent-event process; event order need not have a separate baseline	What is the common event intensity over follow-up?

The Andersen-Gill form is a useful alternative when a common baseline intensity over calendar time is scientifically reasonable. It does not automatically make repeated events independent: the intensity is conditional on observed history, and within-subject dependence still matters for uncertainty estimates and model adequacy. An event-order or gap-time baseline is more flexible about how the event number or interval clock shapes risk, at the cost of more parameters and potentially smaller risk sets for later events.

From a Hazard to a Prediction

For a specified event order and a subject who has not yet had that next event, let \(\lambda(v)\) denote the conditional hazard along the future interval. The probability \(S(b)\) of no next event through time \(b\), starting at \(a\), satisfies \(dS(v)/dv=-\lambda(v)S(v)\) and \(S(a)=1\). Dividing by \(S(v)\) and integrating gives the cumulative-hazard relation:

\frac{dS(v)}{S(v)}=-\lambda(v)\,dv \quad\Longrightarrow\quad S(b)=\exp\left\{-\int_a^b\lambda(v)\,dv\right\}, \qquad P(\text{at least one next event by }b)=1-S(b).

This calculation predicts the next event conditional on the subject’s history and on remaining observed and eligible. It does not by itself imply a Poisson count model. In particular, a count distribution such as Poisson requires additional assumptions about the event process; it is not a consequence of knowing only a first-event probability.

Worked Example: First-Event Risk Under Constant Exposure

Suppose, hypothetically, the first-event baseline hazard is constant at \(0.04\) events per day. Let exposure remain at \(C=2\) concentration units for three days, with \(\beta=0.50\) per concentration unit. Assume the subject is event-free and observed throughout, and that this proportional-hazards model applies over the interval.

The hazard is constant because both baseline hazard and exposure are constant:

\lambda=0.04\exp(0.50\times2) =0.04e^1 \approx0.1087\ \text{per day}.

Using the unrounded rate, the integrated hazard is \(0.04\exp(1)\times3\approx0.3262\). Therefore, the probability of at least one first event by day 3 is:

1-\exp\{-0.04\exp(1)\times3\} \approx1-\exp(-0.326194) \approx0.2783.

Under the specified conditional hazard, the predicted first-event probability over these three days is about 27.83%. This is not a probability that the hazard itself is 27.83%; it is the interval event probability obtained by integrating the hazard.

Event Order Can Change the Predicted Recurrence Pattern

An event-order model can assign different hazards to successive events even when exposure is unchanged. The next example shows why the probability of two events by a fixed time is not obtained by simply doubling the first-event probability.

Worked Example: Probability of a Second Event in an Event-Order Model

Suppose, hypothetically, that the first-event hazard is constant at \(\lambda_1=0.10\) per day and the second-event hazard, after the first event, is constant at \(\lambda_2=0.20\) per day. Both are total-time hazards. Assume complete observation through day 3, no terminal event, and no exposure change. These assumptions make the waiting time to the first event exponential with rate \(\lambda_1\), and the subsequent waiting time exponential with rate \(\lambda_2\).

The time to the second event is the sum of those two waiting times. For distinct rates, its probability of exceeding \(t\) is:

P(T_{i2}>t) = \frac{\lambda_2e^{-\lambda_1t}-\lambda_1e^{-\lambda_2t}} {\lambda_2-\lambda_1}.

At \(t=3\) days, substitution gives:

P(T_{i2}>3) = \frac{0.20e^{-0.10(3)}-0.10e^{-0.20(3)}}{0.20-0.10} = \frac{0.20(0.7408)-0.10(0.5488)}{0.10} \approx0.9328.

Thus \(P(T_{i2}\leq3)=1-0.9328=0.0672\), or about 6.72%. This result uses the sequential event-order hazards: the second-event clock cannot begin until the first event occurs. A constant-rate Poisson shortcut would not represent these order-specific hazards.

Gap-Time Prediction with Changing Exposure

For a gap-time prediction, the integration starts at the previous event and proceeds on the gap-time clock. The exposure profile remains indexed by calendar time, so the model must map each future gap time back to the corresponding calendar time.

Worked Example: Next-Event Risk After a Previous Event

Consider a hypothetical gap-time model with a constant baseline hazard of \(0.10\) per day and \(\beta=\log(2)\) per concentration unit. Exposure is \(C(t)=1\) from study entry until day 2, then \(C(t)=0\). A subject’s first event occurs at calendar day 1. Predict the probability of a second event by day 3, conditional on the event at day 1 and continued observation.

The second-event gap begins at day 1. During the first day of that gap, calendar time runs from day 1 to day 2 and \(C=1\), so the hazard is \(0.10\exp\{\log(2)\}=0.20\) per day. During the next day, from calendar day 2 to day 3, exposure is zero and the hazard is \(0.10\) per day. The cumulative hazard is therefore:

\Lambda = 0.10\exp\{\log(2)\}(2-1)+0.10(3-2) =0.20+0.10 =0.30.

The conditional probability of a second event by day 3 is:

1-\exp(-\Lambda) = 1-\exp(-0.30) \approx0.2592.

The predicted conditional probability is about 25.92%. The clock reset at day 1, but exposure did not reset: its calendar-time change at day 2 was carried into the gap-time integral.

For comparison, if the second-event total-time baseline were instead \(0.05\) per day with the same exposure effect, its cumulative hazard from day 1 to day 3 would be \(0.05(2)(1)+0.05(1)=0.15\), giving \(1-\exp(-0.15)\approx0.1393\). The difference is not an intrinsic consequence of one clock being superior; the two baseline hazards encode different assumptions and would need to be estimated or justified for the application.

Assumptions, Diagnostics, and Common Errors

A recurrent-event hazard model needs a well-defined event and observation process. Each event time must be recorded or handled using an appropriate event-time likelihood; subjects must enter the risk set only when they are eligible and observed; and the exposure used at time \(t\) must be available based on information up to that time. If events are terminal, or if a competing event permanently prevents recurrence, follow-up for the recurrent endpoint ends at that event and the risk process must represent that fact. The relevant competing-outcome framework is distinct from treating a nonterminal event as recurrent.

The proportional-hazards exposure term assumes that a one-unit exposure difference has the same multiplicative effect on the hazard at all relevant times and event orders unless the model explicitly allows interactions or order-specific effects. A common \(\beta\) also assumes the exposure association is shared across event orders. These assumptions are not guaranteed by choosing an event-order or gap-time baseline. Event timing, exposure histories, and fitted residual or predictive patterns should be inspected for systematic mismatch, such as an unmodeled increase in risk immediately after an event, a time-varying exposure association, or poor fit for later event orders with sparse data.

Repeated outcomes from the same subject are not separate independent subjects. Inference should account for within-subject clustering when the analysis method’s variance assumptions require it. A simple conditional hazard specification also does not automatically account for persistent differences in underlying event propensity across subjects. If such heterogeneity is substantial, predictions based only on a common covariate profile may be poorly calibrated for individuals. A model’s assumptions should therefore be checked against the purpose of prediction and the structure of the observed event histories.

Common mistake: “The event happened once, so the risk clock must restart at zero and the exposure clock restarts too.” In a gap-time model, the baseline clock does reset, but calendar-time exposure does not. For event \(j\), evaluate exposure at \(T_{i,j-1}+u\), not at gap time \(u\) as though the exposure profile began again after the previous event.

A second frequent error is to use a different event-order baseline while interpreting the fitted exposure coefficient as if every other aspect of the model were unchanged. The event-order and gap-time choices alter the baseline clock and can alter risk sets and predictions. They should be selected according to the clinical question and the data structure, not as interchangeable labels. Finally, a model that predicts the next event conditional on history is not automatically a model for the total number of future events; count predictions require integrating the sequential recurrence process and accounting for when each new event changes the risk history.

Check Your Understanding

Answer each question using the stated model assumptions. Short solutions follow the questions.

In an event-order total-time model, a subject has a first event on day 4. What time clock determines the second-event baseline hazard?
A gap-time model has a baseline hazard of \(0.08\) per day and a constant exposure effect \(\exp(\beta C)=1.5\) for five days after a previous event. Calculate the conditional probability of a next event during those five days.
Why is a second-event probability not generally equal to the first-event probability in a model with event-order-specific baselines?
Exposure changes at calendar day 6. A subject’s preceding event occurred at day 5. At gap time \(u=2\), which calendar-time exposure belongs in the hazard?

Solutions. (1) Time since study entry remains the baseline clock; the subject becomes eligible for the second-event risk set after day 4. (2) The hazard is \(0.08(1.5)=0.12\) per day, so the cumulative hazard is \(0.12(5)=0.60\), and the probability is \(1-\exp(-0.60)\approx0.4512\), or 45.12%. (3) The second-event hazard applies only after the first event and may have a different baseline, so the probability depends on both the waiting time to event one and the subsequent hazard for event two. (4) The relevant calendar time is \(5+2=7\), so use exposure at day 7, not exposure at gap time 2 or at the preceding event.