Background: Stepped wedge trials are longitudinal randomised evaluations, usually cluster-randomised, in which the experimental intervention is introduced in a staggered fashion. Incomplete stepped wedge designs focus the effort of data collection on particular periods in particular sequences. Methods: We suppose there is a cost for every period in every cluster where we collect data, and that there are a fixed number of individuals, m, with data available in each period in each cluster. If we are willing to pay the cost of data collection in that cluster-period then we collect the data on all m individuals, and if we are not willing to pay the cost then we collect no data in that cluster-period. We consider the problem of designing a trial to minimise the total number of cluster-periods of data collection needed to achieve given precision for the treatment effect estimator, or equivalently, to maximise precision for a given number of cluster-periods of data collection. Results: We present the solution for two-period trials, which has two distinct forms, depending on the correlation between two cluster-period means from the same cluster in different periods. We also present a conjecture on the form of the solution for multi-period trials, informed by results from a greedy search of the design space. Conclusions: A real-life stepped wedge design problem will involve trading off the costs of various design elements subject also to constraints on the scale of data collection. Nevertheless, the solutions to the problem considered here add significantly to our understanding of the optimal design of incomplete stepped wedge trials.

翻译：背景：阶梯楔形试验是一种纵向随机化评估（通常采用整群随机化），其中实验性干预以交错方式引入。不完全阶梯楔形设计将数据收集工作集中于特定序列的特定时期。方法：我们假设在每个收集数据的整群-时期组合中均存在成本，且每个整群每个时期有固定数量的个体m可供采集数据。若愿意承担该整群-时期的数据采集成本，则收集全部m个个体的数据；若不愿承担成本，则不收集该整群-时期的数据。我们考虑在给定处理效应估计精度要求下，最小化所需数据采集的整群-时期总数；或等价地，在给定数据采集整群-时期数量下最大化估计精度的问题。结果：我们给出两时期试验的解，该解根据同一整群不同时期两个整群-时期均值间的相关性呈现两种不同形式。同时，基于贪心搜索设计空间的结果，我们提出多时期试验解形式的猜想。结论：现实中的阶梯楔形设计问题需要在考虑数据采集规模约束的同时，权衡多种设计要素的成本。尽管如此，本文所探讨的解法显著加深了我们对不完全阶梯楔形试验最优设计的理解。