Sample size calculation based on normal approximations is often associated with the loss of statistical power for a single-arm trial with a time-to-event endpoint. Recently, Wu (2015) derived the exact variance for the one-sample log-rank test under the alternative and showed that a single-arm one-stage study based on exact variance often has power above the nominal level while the type I error rate is controlled. We extend this approach to a single-arm two-stage design by using exact variances of the one-sample log-rank test for the first stage and the two stages combined. The empirical power of the proposed two-stage optimal designs is often not guaranteed under a two-stage design setting, which could be due to the asymptotic bi-variate normal distribution used to estimate the joint distribution of the test statistics. We adjust the nominal power level in the design search to guarantee the simulated power of the identified optimal design being above the nominal level. The sample size and the study time savings of the proposed two-stage designs are substantial as compared to the one-stage design.

基于正态近似值的样本量计算通常与具有事件时间终点的单臂试验的统计功效损失相关。最近，Wu (2015) 推导出了替代方案下单样本对数秩检验的精确方差，并表明基于精确方差的单臂单阶段研究通常具有高于名义水平的功效，而 I 类错误率被控制。我们通过使用第一阶段和两个阶段组合的单样本对数秩检验的精确方差将这种方法扩展到单臂两阶段设计。在两阶段设计设置下，通常不能保证所提出的两阶段优化设计的经验功效，这可能是由于用于估计检验统计量的联合分布的渐近双变量正态分布。我们在设计搜索中调整标称功效水平，以保证识别出的最佳设计的模拟功效高于标称水平。与单阶段设计相比，拟议的两阶段设计的样本量和研究时间节省是巨大的。