A general theory on stochastic curtailment for censored survival data

Stochastic curtailment is a valuable tool in monitoring long-term medical s tudies. Under this approach, one calculates the conditional power, which is the probability of rejecting the null hypothesis at the scheduled end of t he study given the existing data at the interim analysis, along with certai n speculation about the future data. The conditional power may be used to a id the decision to terminate a study prematurely or to extend a study beyon d its originally planned duration. This article provides a formal and syste matic investigation into the use of stochastic curtailment in the context o f censored survival data. To enhance generality, we introduce a broad class of statistics that includes two-sample weighted log-rank statistics, as we ll as the partial likelihood score statistic for testing treatment differen ce with covariate adjustment under the proportional hazards model. We estab lish the weak convergence under both the null hypothesis and contiguous alt ernatives for this class of statistics when calculated repeatedly over the calendar time (i.e., time of interim analysis). Further, we derive the cond itional distributions of these statistics calculated at the end of the stud y given all the data collected up to the interim look or given the statisti cs calculated at the interim look, and provide analytic expressions for the corresponding conditional powers. These results enable us to address sever al subtle issues involved in the definition and implementation of condition al power for censored survival data, especially when there is staggered pat ient entry with a potential time trend in the survival distribution, when t he Gehan-type weight function is used, or when treatment is not independent of covariates. For randomized clinical trials, we show that very simple fo rmulas can be used to calculate the conditional powers of the unweighted lo g-rank test (with or without covariate adjustment) under both the null and alternative hypotheses. Simulation studies demonstrate that the conditional powers for survival studies can be accurately evaluated through the propos ed formulas even when the sample size is small. An illustration with data t aken from a colon cancer study is provided.