Smooth estimation of the survival function for interval censored data

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Abstract
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 in some time intervals. To estimate the survival function the non‐parametric maximum likelihood estimator (NPMLE) is commonly used, which is a step function having some large jumps. However, in many applications the underlying survival function can be reasonably assumed to be smooth, and then the NPMLE does not efficiently 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 finite sample study yet to assess their performance in the literature. In this paper we first show by simulation that both smooth estimators improve over the NPMLE for smooth survival functions. We then apply these estimators to compare two survival curves. The test statistics based on the maximum difference between two survival functions (Kolmogorov–Smirnov test) and on the integrated weighted difference of two survival functions (IWDB test) are investigated via the bootstrap. From our simulation studies the IWDB test seems particularly promising for some stochastically ordered survival functions that do not satisfy the proportional hazards model. The methods are illustrated by reanalysing the Breast Cosmesis Study data set. Copyright © 2000 John Wiley & Sons, Ltd.

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