Notation and frameworks

For a single recurrent event process starting at ${\displaystyle t=0}$, let ${\displaystyle 0\leq T_{1}<T_{2}<\dots }$ denote the event times, where ${\displaystyle T_{k}}$ is the time of the ${\displaystyle k}$th event. The associated counting process ${\displaystyle \{N(t),0\leq t\}}$ records the cumulative number of events generated by the process; specifically, ${\textstyle N(t)=\sum _{k=1}^{\infty }I(T_{k}\leq t)}$ is the number of events occurring over the time interval ${\displaystyle [0,t]}$.

Models for recurrent events can be specified by considering the probability distribution for the number of recurrences in short intervals ${\displaystyle [t,t+\Delta t)}$, given the history of event occurrence before time ${\displaystyle t}$. The intensity function describes the instantaneous probability of an event occurring at time ${\displaystyle t}$, conditional on the process history, and describes the process mathematically. Define the process history as ${\displaystyle H(t)=\{N(s):0\leq s<t\}}$, then the intensity is formally defined as

$${\displaystyle \lambda (t|H(t))=\lim _{\Delta t\downarrow 0}{\frac {P(N(t+\Delta t)-N(t)=1)}{\Delta t}}.}$$

When a heterogenous group of individuals or processes is considered, the assumption of a common event intensity is no longer plausible. Greater generality can be achieved by incorporating fixed or time-varying covariates in the intensity function.

Description of recurrent event data

As a counterpart of the Kaplan–Meier curve, which is used to describe the time to a terminal event, recurrent event data can be described using the mean cumulative function, which is the average number of cumulative events experienced by an individual in the study at each point in time since the start of follow-up.

Poisson model

The Poisson model is a popular model for recurrent event data, which models the number of recurrences that have occurred. Poisson regression assumes that the number of recurrences has a Poisson distribution with a fixed rate of recurrence over time. The logarithm of the expected number of recurrences is modeled by a linear combination of explanatory variables.

Marginal means/rates model

The marginal means/rates model considers all recurrent events of the same subject as a single counting process and does not require time-varying covariates to reflect the past history of the process, which makes it a more flexible model.^[2] Instead, the full history of the counting process may influence the mean function of recurrent events.

Multi-state model

In multi-state models, the recurrent event processes of individuals are described by different states. The different states may describe the recurrence number, or whether the subject is at risk of recurrence. A change of state is called a transition (or an event) and is central in this framework, which is fully characterized through estimation of transition probabilities between states and transition intensities that are defined as instantaneous hazards of progression to one state, conditional on occupying another state.^[2]

Extended Cox proportional hazards (PH) models

Extensions of the Cox proportional hazard models are popular models in social sciences and medical science to assess associations between variables and risk of recurrence, or to predict recurrent event outcomes. Many extensions of survival models based on the Cox proportional hazards approach have been proposed to handle recurrent event data. These models can be characterized by four model components:^[3]

• Risk intervals
• Baseline hazard
• Risk set
• Correction for within-subject correlation

Well-known examples of Cox-based recurrent event models are the Andersen and Gill model,^[4] the Prentice, Williams and Petersen model^[5] and the Wei-Lin–Weissfeld model^[6]

Robust variance

It is possible to use robust 'sandwich' estimators for the variance of regression coefficients. Robust variance estimators are based on a jacknife estimate, which anticipates correlation within subjects and provides robust standard errors.

Frailty models

In frailty models, a random effect is included in the recurrent event model which describes the individual excess risk that can not be explained by the included covariates. The frailty term induces dependence among the recurrence times within subjects.