Estimating power for multilevel models is complex because there are many moving parts, several sources of variation to consider, and unique sample sizes at Level 1 and Level 2. Monte Carlo computer simulation is a flexible tool that has received considerable attention in the literature. However, much of the work to date has focused on very simple models with one predictor at each level and one cross-level interaction effect, and approaches that do not share this limitation require users to specify a large set of population parameters. The goal of this tutorial is to describe a flexible Monte Carlo approach that accommodates a broad class of multilevel regression models with continuous outcomes. Our tutorial makes three important contributions. First, it allows any number of within-cluster effects, between-cluster effects, covariate effects at either level, cross-level interactions, and random coefficients. Moreover, we do not assume orthogonal effects, and predictors can correlate at either level. Second, our approach accommodates models with multiple interaction effects, and it does so with exact expressions for the variances and covariances of product random variables. Finally, our strategy for deriving hypothetical population parameters does not require pilot or comparable data. Instead, we use intuitive variance-explained effect size expressions to reverse-engineer solutions for the regression coefficients and variance components. We describe a new R package mlmpower that computes these solutions and automates the process of generating artificial data sets and summarizing the simulation results. The online supplemental materials provide detailed vignettes that annotate the R scripts and resulting output. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

中文翻译：

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一种简单的蒙特卡罗方法，用于估计多级设计中的功效。

估计多级模型的功效非常复杂，因为有许多移动部件、需要考虑多个变化源，以及第 1 级和第 2 级的独特样本大小。蒙特卡罗计算机模拟是一种灵活的工具，在文献中受到了相当多的关注。然而，迄今为止的大部分工作都集中在非常简单的模型上，每个级别都有一个预测变量和一个跨级别交互效应，而没有这种限制的方法需要用户指定一大组总体参数。本教程的目标是描述一种灵活的蒙特卡罗方法，该方法适用于具有连续结果的广泛的多级回归模型。我们的教程做出了三个重要贡献。首先，它允许任意数量的簇内效应、簇间效应、任一级别的协变量效应、跨级别交互作用和随机系数。此外，我们不假设正交效应，并且预测变量可以在任一水平上相关。其次，我们的方法适应具有多种交互作用的模型，并且它通过乘积随机变量的方差和协方差的精确表达式来实现。最后，我们推导假设总体参数的策略不需要试点数据或可比数据。相反，我们使用直观的方差解释效应大小表达式来对回归系数和方差分量的解决方案进行逆向工程。我们描述了一个新的 R 包 mlmpower，它可以计算这些解决方案，并自动生成人工数据集和总结模拟结果的过程。在线补充材料提供了对 R 脚本和结果输出进行注释的详细插图。（PsycInfo 数据库记录 (c) 2023 APA，保留所有权利）。