Abstract：
Conditional power (CP) serves as a widely utilized approach for futility monitoring in group sequential designs. However, adopting the CP methods may lead to inadequate control of the type II error rate at the desired level. In this study, we introduce a flexible beta spending function tailored to regulate the type II error rate while employing CP based on a predetermined standardized effect size for futility monitoring (a so-called CP-beta spending function). This function delineates the expenditure of type II error rate across the entirety of the trial. Unlike other existing beta spending functions, the CP-beta spending function seamlessly incorporates beta spending concept into the CP framework, facilitating precise stagewise control of the type II error rate during futility monitoring. In addition, the stopping boundaries derived from the CP-beta spending function can be calculated via integration akin to other traditional beta spending function methods. Furthermore, the proposed CP-beta spending function accommodates various thresholds on the CP-scale at different stages of the trial, ensuring its adaptability across different information time scenarios. These attributes render the CP-beta spending function competitive among other forms of beta spending functions, making it applicable to any trials in group sequential designs with straightforward implementation. Both simulation study and example from an acute ischemic stroke trial demonstrate that the proposed method accurately captures expected power, even when the initially determined sample size does not consider futility stopping, and exhibits a good performance in maintaining overall type I error rates for evident futility.
摘要：
条件功率（CP）是组顺序设计中徒劳监视的广泛使用的方法。然而,采用CP方法可能导致II型错误率在所需水平上的控制不足。在这项研究中,我们引入了一种灵活的beta支出函数,该函数可以调整II型错误率,同时基于预定的标准化效应大小采用CP进行无效监测(所谓的CP-beta支出函数).该功能描述了整个试验过程中第二类错误率的支出。与其他现有的beta支出函数不同，CP-β支出功能将β支出概念无缝地纳入CP框架，有助于在徒劳监测期间精确分阶段控制II型错误率。此外,从CP-β支出函数导出的停止边界可以通过类似于其他传统β支出函数方法的积分来计算。此外，拟议的CP-β支出函数在试验的不同阶段适应CP量表上的各种阈值，确保其在不同信息时间场景下的适应性。这些属性使CP-β支出函数在其他形式的β支出函数中具有竞争力，使其适用于任何试验组顺序设计与直接实施。仿真研究和来自急性缺血性卒中试验的示例都表明，所提出的方法准确地捕获了预期功率，即使最初确定的样本量不认为停止是徒劳的，并在保持整体I型错误率方面表现出良好的性能，以明显无效。
