Smooth Estimation of the Survival for Interval Censored Data: An Empirical Study

SUMMARY Interval censored data arise naturally in large-scale panel studies where subjects can only be followed periodically and the event of interest can only be observed between some time intervals. To estimate the survival function the nonparametric maximum likelihood estimator (NPMLE) is commonly used, which is a step function having some large jumps. But in many applications the underlying survival function can be reasonably assumed to be smooth, then the NPMLE does not eeciently use this information. Two smooth estimators have been proposed in the literature: one is based on the kernel smoothing of the NPMLE; another is the logspline density model. However, to our knowledge there is no nite sample study yet to assess their performance in the literature. In this paper we rst show by simulation that both smooth estimators improve over the NPMLE for smooth survivals, though further studies are still necessary to ease their use and establish their theoretical properties. We then apply these estimators to compare two survival curves. The test statistics based on the maximum diierence between two survivals (Kolmogorov-Smirnov test) and on the integrated weighted diierence of two survivals (IWDB test) are investigated via bootstrap. From our limited simulation studies the IWDB test seems particularly promising for some stochastically ordered survivals that do not satisfy the proportional hazards model. The methods are illustrated by reanalyzing the Breast Cosmesis Study dataset.