Smooth Estimation and Inference with Interval Censored Data

In many biostatistical applications one is concerned with estimating the distribution of a survival time T. This time T is interval censored if its current status is observed at several monitoring times so that one observes an interval containing T. Construction of confidence intervals for smooth functionals of the survival function and smooth estimators with confidence limits for the survival function itself are open problems. In this paper we provide solutions to these problems, assuming that the observed monitoring times are independent of T. Our proposed smooth estimator is a standard univariate kernal regression smoother applied to the pooled sample of N dependent current status observations. The proposed kernal smoother converges at the optimal rate to a normal distribution identical to that of a kernal smoother applied to N independent and identically distributed current status observations with the same set of monitoring times. We show that existing bandwidth selection techniques and confidence limit procedures for standard nonparametric regressions yield the correct answers despite the dependence in the pooled sample. Our results give rise to the practical insight that for large n, each additional monitoring time for subjects already monitored actually carries as much or more information for estimation of the survival function than a monitoring time for a new subject. We provide a simple method for constructing confidence intervals for smooth functionals of the survival function. A simulation demonstrates the excellent performance of these confidence intervals and we apply the method to analyze the HIV seroconversion distribution for a cohort of hemophiliacs