Nonparametric Estimation With Recurrent Event Data

The problem of nonparametric estimation for the distribution function governing the time to occurrence of a recurrent event in the presence of censoring is considered. We derive Nelson–Aalen and Kaplan–Meier-type estimators for the distribution function, and establish their respective finite-sample and asymptotic properties. We allow for random observation periods for each subject under study and explicitly account for the informative sum-quota nature of the data accrual scheme. These allowances complicate technical matters considerably and, in particular, invalidate the direct use of martingale methods. Consistency and weak convergence of our estimators are obtained by extending an approach due to Sellke, who considered a single renewal process (i.e., recurrent events on a single subject) observed over an infinite time period. A useful feature of the present analysis is that strong parallels are drawn to the usual “single-event” setting, providing a natural route toward developing extensions that involve covariates (e.g., weighted log-rank tests, Cox-type regression, and frailty models). Another advantage is that we obtain explicit, closed-form expressions for the asymptotic variances for these estimators. This enables, for instance, the characterization of the efficiency loss that results from employing only the first, possibly right-censored, observation per subject. An interesting feature of these results is the prominent role of the renewal function. Finally, we discuss the case of correlated interoccurrence times, propose an estimator in the case where the within-unit interoccurrence times follow a gamma frailty model, and compare the performance of our estimators to an estimator recently proposed by Wang and Chang.