Using auxiliary time-dependent covariates to recover information in nonparametric testing with censored data

Murray and Tsiatis (1996) described a weighted survival estimate that incorporates prognostic time-dependent covariate information to increase the efficiency of estimation. We propose a test statistic based on the statistic of Pepe and Fleming (1989, 1991) that incorporates these weighted survival estimates. As in Pepe and Fleming, the test is an integrated weighted difference of two estimated survival curves. This test has been shown to be effective at detecting survival differences in crossing hazards settings where the logrank test performs poorly. This method uses stratified longitudinal covariate information to get more precise estimates of the underlying survival curves when there is censored information and this leads to more powerful tests. Another important feature of the test is that it remains valid when informative censoring is captured by the incorporated covariate. In this case, the Pepe-Fleming statistic is known to be biased and should not be used. These methods could be useful in clinical trials with heavy censoring that include collection over time of covariates, such as laboratory measurements, that are prognostic of subsequent survival or capture information related to censoring.