The use of hierarchical data (also called multilevel data or clustered data) is common in behavioural and psychological research when data of lower-level units (e.g., students, clients, repeated measures) are nested within clusters or higher-level units (e.g., classes, hospitals, individuals). Over the past 25 years we have seen great advances in methods for computing the sample sizes needed to obtain the desired statistical properties for such data in experimental evaluations. The present research provides closed-form and iterative formulas for sample size determination that can be used to ensure the desired width of confidence intervals for hierarchical data. Formulas are provided for a four-level hierarchical linear model that assumes slope variances and inclusion of covariates under both balanced and unbalanced designs. In addition, we address several mathematical properties relating to sample size determination for hierarchical data via the standard errors of experimental effect estimates. These include the relative impact of several indices (e.g., random intercept or slope variance at each level) on standard errors, asymptotic standard errors, minimum required values at the highest level, and generalized expressions of standard errors for designs with any-level randomization under any number of levels. In particular, information on the minimum required values will help researchers to minimize the risk of conducting experiments that are statistically unlikely to show the presence of an experimental effect.

中文翻译：

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基于置信区间的样本大小确定公式和分层数据的一些数学属性。

当较低级别单元（例如，学生、客户、重复测量）的数据嵌套在群集或更高级别单元（例如，班级、医院、个人）。在过去的 25 年中，我们已经看到计算样本量的方法取得了巨大进步，这些样本量在实验评估中为此类数据获得所需的统计特性。本研究提供了用于确定样本大小的封闭形式和迭代公式，可用于确保分层数据的置信区间的所需宽度。为四级分层线性模型提供了公式，该模型假定在平衡和不平衡设计下均存在斜率方差和包含协变量。此外，我们通过实验效果估计的标准误差来解决与分层数据的样本大小确定相关的几个数学属性。其中包括几个指标（例如，每个级别的随机截距或斜率方差）对标准误差、渐近标准误差、最高级别的最低要求值以及在任意数量的级别。特别是，有关最低要求值的信息将帮助研究人员最大限度地降低进行实验的风险，这些实验在统计上不太可能显示实验效果的存在。其中包括几个指标（例如，每个级别的随机截距或斜率方差）对标准误差、渐近标准误差、最高级别的最低要求值以及在任意数量的级别。特别是，有关最低要求值的信息将帮助研究人员最大限度地降低进行实验的风险，这些实验在统计上不太可能显示实验效果的存在。其中包括几个指标（例如，每个级别的随机截距或斜率方差）对标准误差、渐近标准误差、最高级别的最低要求值以及在任意数量的级别。特别是，有关最低要求值的信息将帮助研究人员最大限度地降低进行实验的风险，这些实验在统计上不太可能显示实验效果的存在。