Multiple Recurrent Event Types With Cause-Specific Intensities

Why Model Recurrent Event Types Separately?

A participant may experience more than one kind of recurrent event, such as repeated episodes of two symptoms or repeated visits for different clinical reasons. These event types can have different baseline rates and can relate differently to a longitudinal marker. Combining them into one count process can hide those differences. A type-specific intensity model preserves them while allowing the event processes to share information about a subject’s latent longitudinal trajectory.

The key modeling choices are to define when a subject is at risk for each type, specify an intensity for each type, and state how the longitudinal process enters those intensities. The shared random effects induce dependence between event types even when their type-specific intensity formulas are distinct.

At-Risk Processes and Type-Specific Intensities

Each event type has its own at-risk process because eligibility or observation can differ by type. For a recurrent event, experiencing an event does not by itself remove a subject from risk for future events of that type. The subject may remain at risk immediately, or may re-enter after a clinically specified recovery period. If a subject is not eligible for a type during an interval, the corresponding \(Y_{ik}(t)\) is zero there. At-risk rules should be defined from the study design and applied consistently, rather than inferred from the event model.

Let \(m_i(t;b_i)\) denote the subject-specific longitudinal trajectory implied by the longitudinal submodel and its random effects. A useful type-specific intensity specification is

Here \(\lambda_{0k}(t)\) is the baseline intensity for type \(k\), \(X_i(t)\) contains covariates, \(\gamma_k\) is a type-specific covariate coefficient vector, and \(\alpha_k\) is the association between the current longitudinal trajectory and the type-\(k\) intensity. In this proportional-intensity model, increasing \(m_i(t;b_i)\) by one unit multiplies the type-\(k\) intensity by \(\exp(\alpha_k)\), holding other terms fixed. The units of \(\alpha_k\) are the inverse units of the marker, so the exponential is dimensionless.

The same \(b_i\) appears in \(m_i(t;b_i)\) for every event type. This is what links the event processes through the longitudinal trajectory. The parameters \(\alpha_k\) need not be equal: a higher marker can be associated with a higher intensity for one type and a lower intensity for another. If the scientific model requires a direct shared random-effect term in addition to the trajectory value, that term must be specified explicitly; it should not be assumed to arise automatically from using \(m_i(t;b_i)\).

Shared Longitudinal Effects and Interpretation

For example, suppose the longitudinal submodel has trajectory \(m_i(t;b_i)=\beta_0+\beta_1t+b_{0i}+b_{1i}t\), where \(b_{0i}\) and \(b_{1i}\) are subject-specific random intercept and slope. Substituting this trajectory into each event intensity makes both event types depend on the subject’s latent level and trajectory over time. Conditional on the random effects, each type has its own baseline and association coefficient; marginally, subjects with high latent trajectories can have systematically different rates across both types.

The coefficients describe conditional intensity associations, not causal effects. For instance, \(\exp(\alpha_k)\) is the multiplicative change in type-\(k\) intensity for a one-unit higher current marker, conditional on covariates, the random effects, and the modeled history. It is not generally the ratio of marginal event rates, because integrating over random effects changes the rate relationship and the distribution of random effects among subjects still under observation.

Conditional Likelihood: Why the Intensities Matter

Assume the event processes are simple (no two events occur at precisely the same instant), have locally finite event counts, and have the specified conditional intensities. Given the random effects, the event-process likelihood for subject \(i\), up to factors that do not depend on the model parameters, is

where \(t_{ikj}\) are the observed type-\(k\) event times and \(C_i\) is the end of observation. The event-time factors reward intensity at observed events; the exponential term accounts for the intensity accumulated over all at-risk time, including intervals without events. If \(Y_{ik}(u)=0\), the type contributes no accumulated intensity at that time.

A joint model also includes the longitudinal measurement contribution and the random-effects distribution. Under conditional independence of the longitudinal measurements and event process given \(b_i\), covariates, and the modeled history, the subject’s observed-data likelihood has the form

where \(L_{Li}(b_i)\) is the longitudinal-data likelihood and \(p(b_i)\) is the random-effects density. This factorization is an assumption of the joint model, not a consequence of having written two intensity equations. If the measurement schedule itself is informative after conditioning on the modeled quantities, its observation process may also need to be modeled.

Justification and Modeling Checks

The likelihood form follows by dividing follow-up into short intervals. Conditional on the history and random effects, an interval of width \(dt\) has probability approximately \(\lambda_{ik}(t)dt\) of a type-\(k\) event and probability approximately \(1-\sum_k\lambda_{ik}(t)dt\) of no event of any type, when the subject is at risk and simultaneous events have probability of smaller order than \(dt\). Multiplying these probabilities over intervals yields a factor of the intensity at each observed event and an exponential of minus the integrated total intensity. In the limit as the intervals shrink, this gives the displayed likelihood. Type-specific \(Y_{ik}(t)\) ensures the integrated intensity counts only eligible time.

The event types need not be marginally independent. Sharing random effects generally induces association between their event rates across subjects. The likelihood expression is conditional on the random effects and the modeled history; any additional dependence not captured by those quantities must be represented through suitable history terms or a richer joint event-process model. A simple sum of type-specific intensities describes the instantaneous rate of any event only when the event types form a mutually exclusive marked event process at each instant and the intensities use compatible risk definitions.

Useful checks include verifying the risk indicators against the event and eligibility records, plotting type-specific cumulative event counts against fitted expected counts, and examining residuals or time-rescaled inter-event gaps by type. Systematic residual patterns over time may indicate an inadequate baseline intensity; residual patterns by event order can point to event-order dependence; and remaining association between event types may signal omitted shared structure. Fit and convergence should also be checked across reasonable baseline and random-effects specifications.

Alternatives, Limitations, and Common Pitfalls

Separate type-specific intensities are useful when the event definitions and risk periods are meaningful, and when the data contain enough events to estimate the type-specific parameters. They can be unstable for rare types or highly correlated covariates. They also rely on the longitudinal and event models being adequately specified, on the random-effects distribution being plausible, and on the conditional-independence assumptions used in the joint likelihood. A shared random effect is a structured way to capture dependence, not a guarantee that all residual dependence has been removed.

Common mistake: A student codes \(Y_{ik}(t)=0\) forever after the first type-\(k\) event because the subject is no longer at risk for a first event. That reasoning silently changes a recurrent-event model into a first-event model. For recurrent events, the subject generally returns to the type-specific risk set after an event unless the protocol specifies a recovery period or permanent ineligibility. The correct risk indicator follows the event type’s actual eligibility rule.