Gap-Time Recurrent-Event Joint Models

Why Model Time Since the Last Recurrence?

For recurrent events, the time since a person’s last event can matter as much as the time since study entry. A recurrence may change the risk of another event, and a model based only on time since entry does not represent that reset directly. A gap-time model uses the elapsed time since the preceding recurrence as the clock for the next one. When a terminal event occurs, such as death, the person is no longer at risk for further recurrences; the final observed gap is therefore incomplete rather than an event-free period extending indefinitely.

Here the recurrence hazard is also linked to latent biomarker effects: subject-specific random effects in the longitudinal model can be associated with recurrence risk. This allows the event model to use features of the underlying biomarker trajectory while accounting for measurement error. The association is a model-based conditional association, not evidence that changing the biomarker would cause a change in recurrence risk.

A Joint Gap-Time Model

A typical longitudinal component represents the latent biomarker trajectory as \(m_i(t)=x_i^L(t)^\mathsf{T}\beta_L+z_i(t)^\mathsf{T}b_i\), with measurements \(y_{ij}=m_i(t_{ij})+\epsilon_{ij}\). Here \(x_i^L(t)\) and \(z_i(t)\) are known covariate vectors, \(\beta_L\) is a fixed-effect vector, and \(b_i\) is a subject-specific random-effect vector. For example, the components of \(b_i\) might represent deviations in a subject’s underlying biomarker level and slope. The recurrence model uses these latent effects rather than a noisy observed measurement at a particular visit.

Let \(x_{ik}\) denote recurrence covariates, and let \(\lambda_{0k}(s)\) be the baseline hazard for the \(k\)th recurrence as a function of gap time. A flexible event-order-specific specification is:

The baseline may differ across event orders \(k\), allowing the hazard after two prior recurrences to differ from the hazard after none. The vector \(\alpha_k\) describes how the latent biomarker random effects are associated with recurrence \(k\)’s hazard. If a component of \(b_i\) measures biomarker level, its coefficient is interpreted conditionally: a one-unit increase in that latent component multiplies the recurrence hazard by \(\exp(\alpha_k)\), holding the other modeled terms fixed.

Let \(D_i\) be terminal-event time, and let \(C_i\) be censoring time. The recurrence process is observed only up to \(Y_i=\min(D_i,C_i)\). A terminal-event model can be written on calendar time as \(\lambda_{Di}(t\mid b_i)=\lambda_{D0}(t)\exp(w_i^\mathsf{T}\theta+\xi^\mathsf{T}b_i)\), where \(w_i\) contains terminal-event covariates. The gap-time recurrence clock resets after a recurrence, but the terminal-event clock in this specification continues from study entry.

Deriving the Recurrence Contribution

For recurrence \(k\), define the cumulative gap-time hazard \(\Lambda_{ik}(s\mid b_i)=\int_0^s\lambda_{ik}(u\mid b_i)\,du\). Conditional on \(b_i\), \(S_{ik}(s\mid b_i)=\exp\{-\Lambda_{ik}(s\mid b_i)\}\) is the recurrence-component (net) survival, not generally the real-world probability of no recurrence by gap time \(s\) in the presence of a competing terminal event. That real-world probability depends on both event processes. This follows because the survival function satisfies \(dS(s)/ds=-\lambda(s)S(s)\), with \(S(0)=1\); solving this differential equation gives the exponential of the negative cumulative hazard. The density of a recurrence at \(s\) is then \(\lambda_{ik}(s\mid b_i)S_{ik}(s\mid b_i)\).

Thus each completed gap contributes its event density. If the last recurrence occurs at \(T_{iN_i}\), the next gap is only observed for \(R_i=Y_i-T_{iN_i}\) time units. No next recurrence before \(Y_i\) contributes survival through that partial gap. The recurrence likelihood conditional on \(b_i\) is:

When \(N_i=0\), the product is empty and the contribution is survival through the observed first gap. The final survival factor applies whether follow-up ends at a terminal event or at censoring: it says that another recurrence was not observed before the recurrence process ended. It does not assert that the subject would remain recurrence-free after that endpoint.

If \(\delta_{Di}=1\) indicates an observed terminal event, its conditional contribution is \(\lambda_{Di}(Y_i\mid b_i)\exp\{-\int_0^{Y_i}\lambda_{Di}(t\mid b_i)\,dt\}\). If \(\delta_{Di}=0\), the contribution is terminal-event survival alone, \(\exp\{-\int_0^{Y_i}\lambda_{Di}(t\mid b_i)\,dt\}\). The joint contribution also includes the longitudinal measurement density and the random-effects density. In schematic form, the observed-data likelihood contribution is the integral over \(b_i\) of their product with \(L_{Ri}(b_i)\) and the appropriate terminal contribution. This integration is essential: the same latent effects connect the longitudinal and event information.

When a Terminal Event Ends Recurrence Follow-Up

Suppose the same subject’s recurrences occur at calendar months 3 and 8, followed by a terminal event at month 11. The first two completed gaps are 3 and 5 months, and the final partial gap is 3 months. With the preceding hazards and a third-gap hazard of \(0.080\) per month, the recurrence contribution is:

The final factor is survival over the three observed months after recurrence 2. There is no third recurrence-event density, because the terminal event occurred before a third recurrence. If the terminal hazard is a constant \(0.030\) per month, the terminal contribution is \(0.030e^{-0.030(11)}=0.02157\) per month. The product of these two conditional event-process contributions is approximately \(0.00003652\ \text{per month}^3\), before including longitudinal measurements or integrating over random effects. It is a likelihood density, not a probability for the exact event times.

This example also emphasizes the different time scales: the final recurrence survival is evaluated over a three-month gap, while terminal-event survival is evaluated over eleven months from study entry. Treating the terminal event as ordinary censoring for recurrence is appropriate for the recurrence component’s risk interval, but its terminal-event contribution must still be included when the terminal event is observed.

Diagnostics, Alternatives, and Limitations

A gap-time model is useful when recurrence risk plausibly depends on time since the most recent event, such as when an event is followed by recovery or a renewed period of elevated risk. Its fit should be assessed rather than inferred from the clock choice. Examine whether predicted gap-time survival or recurrence patterns match observed patterns across event orders and relevant covariate groups. Check the adequacy of each baseline hazard, the proportional-hazards form of the latent-effect association, and the longitudinal and random-effects distributions. Sparse later recurrences can make event-order-specific baselines and associations unstable. Convergence problems, extreme random-effect estimates, or large changes under reasonable baseline specifications are warnings to investigate.

The main alternative clocks encode different questions. In the Andersen–Gill approach, the hazard is indexed by time since study entry while a subject is at risk, and recurrent events are represented through a counting-process intensity. An event-order-specific calendar-time model can allow different hazards by recurrence number but still measures time from study entry. A gap-time model instead measures elapsed time from the preceding event. These approaches are not interchangeable: each encodes a different risk-time scale and should be chosen to match the scientific process and observation scheme.

The latent-effect link allows the longitudinal and recurrence components to be associated through shared subject-specific trajectory features. It does not make the association causal, and it relies on the chosen random-effects and hazard specifications. A single gap-time scale may be inadequate if risk depends on both time since the last recurrence and calendar time, or if intervention and other time-varying covariates alter risk. The model also depends on the assumption that recurrence risk ends at the terminal event; after death, for example, further recurrences are not merely unobserved but impossible. Modeling the terminal process jointly is important when terminal-event occurrence is associated with latent effects or recurrence history.

Common pitfall: A tempting calculation is to multiply event densities for every observed gap and stop at the final recurrence, even when the subject is followed afterward. That omits the information that no next recurrence occurred during the remaining at-risk time. The correction is to include final-gap survival up to \(Y_i\); if a terminal event ends follow-up, use only the interval from the last recurrence to that event for the recurrence survival factor, and include the terminal-event contribution separately.