Abstract：
BACKGROUND: Group sequential designs incorporating the option to stop for futility at the time point of an interim analysis can save time and resources. Thereby, the choice of the futility boundary importantly impacts the design\'s resulting performance characteristics, including the power and probability to correctly or wrongly stop for futility. Several authors contributed to the topic of selecting good futility boundaries. For binary endpoints, Simon\'s designs (Control Clin Trials 10:1-10, 1989) are commonly used two-stage designs for single-arm phase II studies incorporating futility stopping. However, Simon\'s optimal design frequently yields an undesirably high probability of falsely declaring futility after the first stage, and in Simon\'s minimax design often a high proportion of the planned sample size is already evaluated at the interim analysis leaving only limited benefit in case of an early stop.
METHODS: This work focuses on the optimality criteria introduced by Schüler et al. (BMC Med Res Methodol 17:119, 2017) and extends their approach to binary endpoints in single-arm phase II studies. An algorithm for deriving optimized futility boundaries is introduced, and the performance of study designs implementing this concept of optimal futility boundaries is compared to the common Simon\'s minimax and optimal designs, as well as modified versions of these designs by Kim et al. (Oncotarget 10:4255-61, 2019).
RESULTS: The introduced optimized futility boundaries aim to maximize the probability of correctly stopping for futility in case of small or opposite effects while also setting constraints on the time point of the interim analysis, the power loss, and the probability of stopping the study wrongly, i.e. stopping the study even though the treatment effect shows promise. Overall, the operating characteristics, such as maximum sample size and expected sample size, are comparable to those of the classical and modified Simon\'s designs and sometimes better. Unlike Simon\'s designs, which have binding stopping rules, the optimized futility boundaries proposed here are not adjusted to exhaust the full targeted nominal significance level and are thus still valid for non-binding applications.
CONCLUSIONS: The choice of the futility boundary and the time point of the interim analysis have a major impact on the properties of the study design. Therefore, they should be thoroughly investigated at the planning stage. The introduced method of selecting optimal futility boundaries provides a more flexible alternative to Simon\'s designs with non-binding stopping rules. The probability of wrongly stopping for futility is minimized and the optimized futility boundaries don\'t exhibit the unfavorable properties of an undesirably high probability of falsely declaring futility or a high proportion of the planned sample evaluated at the interim time point.
METHODS: This work focuses on the optimality criteria introduced by Schüler et al. (BMC Med Res Methodol 17:119, 2017) and extends their approach to binary endpoints in single-arm phase II studies. An algorithm for deriving optimized futility boundaries is introduced, and the performance of study designs implementing this concept of optimal futility boundaries is compared to the common Simon\'s minimax and optimal designs, as well as modified versions of these designs by Kim et al. (Oncotarget 10:4255-61, 2019).
RESULTS: The introduced optimized futility boundaries aim to maximize the probability of correctly stopping for futility in case of small or opposite effects while also setting constraints on the time point of the interim analysis, the power loss, and the probability of stopping the study wrongly, i.e. stopping the study even though the treatment effect shows promise. Overall, the operating characteristics, such as maximum sample size and expected sample size, are comparable to those of the classical and modified Simon\'s designs and sometimes better. Unlike Simon\'s designs, which have binding stopping rules, the optimized futility boundaries proposed here are not adjusted to exhaust the full targeted nominal significance level and are thus still valid for non-binding applications.
CONCLUSIONS: The choice of the futility boundary and the time point of the interim analysis have a major impact on the properties of the study design. Therefore, they should be thoroughly investigated at the planning stage. The introduced method of selecting optimal futility boundaries provides a more flexible alternative to Simon\'s designs with non-binding stopping rules. The probability of wrongly stopping for futility is minimized and the optimized futility boundaries don\'t exhibit the unfavorable properties of an undesirably high probability of falsely declaring futility or a high proportion of the planned sample evaluated at the interim time point.
摘要：
背景：组顺序设计结合了在中期分析的时间点停止徒劳的选项，可以节省时间和资源。因此，无用边界的选择对设计产生的性能特性有重要影响，包括正确或错误地停止徒劳的权力和可能性。几位作者为选择有益的徒劳界限这一主题做出了贡献。对于二进制端点，西蒙的设计（对照试验10：1-10，1989年）是常用的两阶段设计，用于单臂第二阶段研究，包括徒劳停止。然而,西蒙的最优设计经常在第一阶段后错误地宣布无用的可能性很高，在西蒙的minimax设计中，通常在中期分析中已经评估了计划样本大小的高比例，在早期停止的情况下仅留下有限的好处。
方法：这项工作的重点是Schüler等人引入的最优性标准。（BMCMedResMethodol17：119，2017），并将其方法扩展到单臂II期研究中的二元终点。介绍了一种推导优化无用边界的算法，和研究设计实现这个概念的最佳无用边界的性能进行比较，常见的西蒙的minimax和最优设计，以及Kim等人对这些设计的修改版本。（Oncotarget10：4255-61，2019年）。
结果：引入的优化的无用边界旨在最大限度地提高在小的或相反的影响的情况下正确停止无用的概率，同时也设置对中期分析的时间点的约束，功率损耗,以及错误停止研究的可能性，即停止研究，即使治疗效果显示出希望。总的来说,操作特性，如最大样本量和预期样本量，与经典和修改后的西蒙设计相当，有时更好。不像西蒙的设计,有约束力的停止规则，此处提出的优化的无用性边界未进行调整以耗尽全部目标标称显著性水平，因此对于非约束性应用仍然有效。
结论：无效边界的选择和中期分析的时间点对研究设计的性质有重大影响。因此,他们应该在规划阶段进行彻底调查。引入的选择最佳无效边界的方法为Simon的设计提供了更灵活的替代方案，该方案具有无约束力的停止规则。错误停止无效的可能性被最小化，并且优化的无效边界没有表现出错误宣布无效的不期望的高概率或在临时时间点评估的计划样本的高比例的不利特性。
方法：这项工作的重点是Schüler等人引入的最优性标准。（BMCMedResMethodol17：119，2017），并将其方法扩展到单臂II期研究中的二元终点。介绍了一种推导优化无用边界的算法，和研究设计实现这个概念的最佳无用边界的性能进行比较，常见的西蒙的minimax和最优设计，以及Kim等人对这些设计的修改版本。（Oncotarget10：4255-61，2019年）。
结果：引入的优化的无用边界旨在最大限度地提高在小的或相反的影响的情况下正确停止无用的概率，同时也设置对中期分析的时间点的约束，功率损耗,以及错误停止研究的可能性，即停止研究，即使治疗效果显示出希望。总的来说,操作特性，如最大样本量和预期样本量，与经典和修改后的西蒙设计相当，有时更好。不像西蒙的设计,有约束力的停止规则，此处提出的优化的无用性边界未进行调整以耗尽全部目标标称显著性水平，因此对于非约束性应用仍然有效。
结论：无效边界的选择和中期分析的时间点对研究设计的性质有重大影响。因此,他们应该在规划阶段进行彻底调查。引入的选择最佳无效边界的方法为Simon的设计提供了更灵活的替代方案，该方案具有无约束力的停止规则。错误停止无效的可能性被最小化，并且优化的无效边界没有表现出错误宣布无效的不期望的高概率或在临时时间点评估的计划样本的高比例的不利特性。
