Event-Order-Specific Recurrent Hazards

Why Event Order Can Matter

A recurrent-event model need not assume that every recurrence has the same intensity. The first event may occur under a different hazard pattern from the second or later events: prior events can mark a changed risk state, and the association between a person’s latent longitudinal trajectory and event risk may itself vary with recurrence order. A model that lets these effects differ can represent such patterns directly, while still linking them to the shared random effects in a joint model.

The focus here is the event-order-specific intensity, not a change in the time scale. As in Andersen–Gill intensity models, calendar time can be the analysis time and the individual’s event history determines whether an event of a given order is currently possible. The essential extension is to let model parameters depend on that order.

Model Specification and Assumptions

Let \(b_i\) denote the random effects shared with the individual’s longitudinal model, and let \(X_i(t)\) be a predictable vector of covariates. “Predictable” means that the covariate value used at time \(t\) is determined by information available just before \(t\). One flexible specification uses a common baseline hazard and order-specific terms:

Here \(\mathcal{F}_{i,t-}\) denotes the individual’s event and covariate history just before \(t\); \(\lambda_0(t)\) is a baseline hazard; \(\beta_k\) describes covariate associations for event order \(k\); \(\gamma_k\) is an order-specific log-rate multiplier; and \(\alpha_k\) describes how the shared random effects are associated with that order’s intensity. The intensity is zero when \(Y_i(t)=0\). For identifiability with a common baseline, one may set \(\gamma_1=0\), treating order 1 as the reference. A simpler model can keep \(\beta_k\) or \(\alpha_k\) constant across orders; the specification should reflect which differences are scientifically and statistically supportable.

An alternative is to use an order-specific baseline \(\lambda_{0k}(t)\). That permits the baseline time pattern to differ freely by event order, rather than representing a constant order shift through \(\gamma_k\). Do not include an unrestricted order-specific baseline and an unrestricted order intercept for the same purpose: their contributions may be redundant. The choice between these forms is substantive as well as computational.

The model is interpreted conditional on \(b_i\) and the history. A joint model supplies a distribution for \(b_i\), usually through the longitudinal submodel, and combines the longitudinal and event information when estimating parameters. Standard regularity requirements include a well-defined intensity, a suitable predictable risk process, and finite cumulative hazard over the observed period. A typical likelihood calculation additionally assumes that event times are observed accurately, recurrent events are simple (not simultaneous for one individual), and censoring or loss to follow-up is noninformative conditional on the modeled history and random effects. If observation or censoring depends on unmodeled risk, the resulting event likelihood may not describe the observed data process adequately.

Interpreting Event-Order Effects and Shared Effects

With a common baseline, compare event orders \(k\) and \(r\) at the same time, covariate values, and random effects, when both orders are at risk. Their conditional intensity ratio is

Thus an order contrast need not be a single rate ratio for all individuals: if \(\alpha_k\ne\alpha_r\), it depends on \(b_i\). This is the direct way event-order effects interact with shared longitudinal random effects. The same latent feature of a person’s trajectory can be associated more strongly with one recurrence order than another. The interpretation remains conditional and associational; it does not establish that changing event order or a longitudinal feature would cause a change in event risk.

Conditional Likelihood and Its Justification

For a specified random effect \(b_i\), the event-process likelihood combines the intensity at each observed event with the probability of remaining event-free over the intervening risk time. If the individual has recurrent event times \(t_{i1},\ldots,t_{in_i}\) in the observation period, the event contribution is

where \(C_i\) is the end of the observed risk period and the intensity includes \(Y_i(u)\). The integral therefore accumulates hazard only while the individual is at risk. The event-order index changes after each event, so the integrand uses the appropriate order-specific parameters on each segment.

The expression follows from the defining small-interval behavior of an intensity. Conditional on the history and \(b_i\), the probability of an event in a short interval of width \(du\) is approximately \(\lambda_i(u)du\), while the probability of no event over that interval is approximately \(1-\lambda_i(u)du\). Multiplying the no-event probabilities over a partition gives, in the limit, the survival contribution \(\exp\{-\int\lambda_i(u)du\}\); multiplying by the event intensity at each observed event supplies the event-density factors. This is the conditional event likelihood. In a joint model it is combined with the longitudinal likelihood and integrated over the random-effects distribution; it is not generally valid to replace \(b_i\) by an estimate and treat that value as known.

Allowing the Random-Effect Association to Vary by Order

An order-specific association parameter can produce materially different contrasts for people with different latent trajectories, even if their measured covariates are the same. The following calculation isolates that feature.

Estimation Choices, Diagnostics, and Limitations

The main alternative is a common-parameter recurrent-event intensity, such as the Andersen–Gill form, which applies one set of regression and shared-effect association coefficients across event orders. It is more parsimonious and may be preferable when data are sparse or when there is no strong reason to expect order variation. At the other extreme, separate order-specific baselines and regression effects provide more flexibility but require enough observed events of each order for stable estimation. A common baseline with order multipliers is a compromise when order shifts are plausible but separate time patterns are not supported.

Model assessment should examine how many individuals reach each order, the distribution of risk exposure by order, and whether estimates are dominated by a few subjects. Compare fitted and observed event counts or rates across recurrence orders and time intervals, and inspect residual or calibration summaries suitable for recurrent-event intensities. Unstable estimates, very wide uncertainty intervals, or large changes when a high-order term is removed can signal sparse information. Also check whether the assumed functional form for the shared random effects is plausible and whether the event process or censoring shows patterns unexplained by the model.

A common mistake is to interpret \(\exp(\gamma_2)\) as the second-to-first event intensity ratio whenever the model includes order-specific random-effect associations. That is correct only under additional restrictions: with a common baseline and equal covariate effects, the conditional ratio is \(\exp\{(\gamma_2-\gamma_1)+(\alpha_2-\alpha_1)^\top b_i\}\). If \(\alpha_2\ne\alpha_1\), the ratio also depends on the shared random effects. Another mistake is to describe a conditional intensity ratio as a population-average rate ratio. Averaging over a random-effects distribution generally does not preserve the same ratio, especially when risk-set membership and event order are related to those effects.

Order-specific models cannot by themselves explain why recurrence order changes risk or establish a causal mechanism. They also do not remove bias from misspecified event histories, informative observation, or an unsuitable random-effects distribution. Finally, allowing every parameter to vary by order may overfit, particularly at high event orders. Parsimony, scientific interpretation, and the amount of information available for each order should guide model complexity.