Approximate U-Statistics for State Waiting Times Under Right Censoring

We develop two different adaptations of a U-statistic based on the waiting times in a given transient state in a multistate system when the state entry and/or the exit times are subject to right censoring. In the first version, the inverse probability of censoring weights calculated based on the state exit times are used along with m-tuples of fully observed waiting times, m being the degree of the kernel of the U-statistic. In the second version, an approximate statistic is defined as a multiple integral with respect to a product of Satten–Datta estimators of a state waiting time survival function. We provide a simulation study to investigate the finite sample behavior of the statistics. We demonstrate that the second version is more efficient since it utilizes additional data where the exit times may be censored. The asymptotic normality of our estimators is also studied through simulation. We further extend our approximate U-statistics to that of a K-sample U-statistic of waiting times under right censoring. Another extension considers waiting time data that are clustered. We apply our U-statistics to test whether initial functional status has a significant impact on the waiting time of an intermediate state of functional recovery of a spinal cord injured patient.