11.4 Recurrent events

When the event can recur, each subject contributes a sequence of times, and the data are kept in the counting-process form of the section on checking proportional hazards, one record \((t_{\text {start}}, t_{\text {stop}}]\) per interval between events. A simple descriptive summary is the mean cumulative function \(M(t)\), the expected number of events per subject by time \(t\), estimated by the Nelson–Aalen sum \(\hat M(t) = \sum _{t_j \le t}d_j/n_j\), where \(n_j\) now counts subjects under observation at \(t_j\) whatever their number of previous events. Three regression models are common.

• The Andersen–Gill model treats each subject’s events as a counting process with intensity \(h_0(t)e^{\beta ^{\mathsf T}x}\), the same for every event. The subject stays in the risk set after each event. Robust standard errors clustered on the subject allow for the dependence between a subject’s events.
• The Prentice–Williams–Peterson (PWP) model stratifies by event number, so the \(m\)th event has its own baseline hazard and only subjects who have had \(m - 1\) events are at risk of it; it suits events that change the subject, where a second infection is not like the first.
• The Wei–Lin–Weissfeld marginal model fits a separate Cox model for the time from the start to each event number, with robust standard errors.

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