Typically, trials investigate the impact of either an individual-level intervention on participant outcomes, or the impact of a cluster-level intervention on participant outcomes. Factorial designs consider two (or more) treatments for each of two (or more) different factors. In factorial trial designs, trial units (individuals or clusters) are each randomised to a level of each of the treatments; these designs allow assessment of the interactions between different interventions. Recently, there has been growing interest in the design of trials that jointly assess the impact of individual- and cluster-level interventions (i.e. multi-level interventions); requiring the development of methodology that accommodates randomisation at multiple levels. While recent work has developed sample size methodology for variants combining standard cluster randomisation and individual randomisation, that work does not apply to longitudinal cluster randomised trial designs such as the stepped wedge design or cluster randomised crossover design. Here we present dedicated sample size methodology for "split-plot factorial longitudinal cluster randomised trials" with continuous outcomes: allowing for joint assessment of individual-level and cluster-level interventions that allows for the impact of the cluster-level intervention to be assessed using any longitudinal cluster randomised trial design. We show how the power to detect given effects of the individual-level intervention, the cluster-level intervention, and the interaction between the two depends on standard results for individually-randomised trials and longitudinal cluster randomised trials. We apply these results to the SharES trial, which considered the effects of a patient- and clinician-level interventions for patients with breast cancer on patient knowledge about the risks and benefits of treatment.

翻译：通常，试验研究个体层面干预对参与者结局的影响，或整群层面干预对参与者结局的影响。析因设计考虑两个（或更多）因子的各两个（或更多）处理水平。在析因试验设计中，试验单元（个体或整群）分别被随机分配到各处理的某个水平；此类设计可评估不同干预之间的交互作用。近年来，联合评估个体层面与整群层面干预（即多水平干预）的试验设计日益受到关注，这需要发展能够容纳多水平随机化的方法论。尽管近期研究已开发出适用于结合标准整群随机化和个体随机化的变体样本量方法，但该方法不适用于阶梯楔形设计或整群随机交叉设计等纵向整群随机试验设计。本文针对具有连续结局的"裂区析因纵向整群随机试验"提出专门的样本量方法：允许联合评估个体层面和整群层面干预，且整群层面干预的影响可采用任意纵向整群随机试验设计进行评估。我们证明了个体层面干预、整群层面干预及其交互作用的特定效应检验效能如何依赖于个体随机试验与纵向整群随机试验的标准结果。我们将这些结果应用于SharES试验——该试验评估了针对乳腺癌患者的患者层面和临床医生层面干预对患者关于治疗风险与获益认知的影响。