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Frequency of Recurrent Events at Failure Time

Recurrent events arise in many longitudinal medical studies where time to a terminal event or failure is the primary endpoint. With incomplete follow-up data, the analysis of recurrent events is a challenge owing to their association with the failure. One specific quantity of interest rarely addressed in the statistical literature is the recurrence frequency at the failure time; an example is hospitalization frequency, which is often used as a rough measure of lifetime medical cost. In this article we show that a marginal model (e.g., the log-linear model) of the recurrence frequency, although desirable, is typically not identifiable. For this reason, we advocate modeling the recurrent events and the failure time jointly, and propose an approach to forming semiparametric joint models from prespecified marginal ones. We suggest two conceptually simple and nested regression models aiming at the recurrence frequency as a mark of the failure and at the process of recurrent events. We formulate monotone estimating functions and propose novel interval-estimation procedures to accommodate non-smooth estimating functions. The resulting estimators are consistent and asymptotically normal. Simulation studies and the application to an AIDS clinical trial exhibit that these proposals are easy to implement and reliable for practical use. Finally, we generalize our proposals to marked recurrent events, and also devise a global inference procedure for recurrent events of multiple types.