Nonparametric inference on median residual life function

A simple approach to the estimation of the median residual lifetime is proposed for a single group by inverting a function of the Kaplan-Meier estimators. A test statistic is proposed to compare two median residual lifetimes at any fixed time point. The test statistic does not involve estimation of the underlying probability density function of failure times under censoring. Extensive simulation studies are performed to validate the proposed test statistic in terms of type I error probabilities and powers at various time points. One of the oldest data sets from the National Surgical Adjuvant Breast and Bowel Project (NSABP), which has more than a quarter century of follow-up, is used to illustrate the method. The analysis results indicate that, without systematic post-operative therapy, a significant difference in median residual lifetimes between node-negative and node-positive breast cancer patients persists for about 10 years after surgery. The new estimates of the median residual lifetime could serve as a baseline for physicians to explain any incremental effects of post-operative treatments in terms of delaying breast cancer recurrence or prolonging remaining lifetimes of breast cancer patients.