In this paper, we demonstrate a purely Bayesian approach for estimating within-group and between-group effect sizes for learning outcomes encountered in educational research, taking naturally into account the multilevel structure of the data, as well as heterogeneous residual variances among time points and conditions. We provide a detailed implementation using the brms package in R serving as a wrapper for the probabilistic programming language Stan. We recommend that for a pooled design, one computes an effect size $d_s$ similar to a Cohen's $d$, and for a paired design, one should compute two possibly different quantities $d_s$ and $d_z$ to correct for correlations in within-group designs and allow for comparability across different studies. All these effect sizes are based on ideas coming from Hedge's total effect size $δ_t$ introduced in 2007. Ultimately, these estimates allow us to study the differential effectiveness of educational interventions with respect to classes.

翻译：本文展示了一种纯粹贝叶斯方法，用于估计教育研究中学习成果的组内和组间效应量，自然考虑了数据的多层次结构以及不同时间点和条件下的异质残差方差。我们提供了使用R语言中brms包（作为概率编程语言Stan的封装）的详细实现。我们建议，对于合并设计，应计算类似于Cohen's d的效应量$d_s$；对于配对设计，应计算两个可能不同的量$d_s$和$d_z$，以校正组内设计中的相关性并确保不同研究间的可比性。所有这些效应量均基于Hedge于2007年提出的总效应量$δ_t$的概念。最终，这些估计使我们能够研究教育干预措施在班级层面的差异化效果。