Permutation tests are a powerful and flexible approach to inference via resampling. As computational methods become more ubiquitous in the statistics curriculum, use of permutation tests has become more tractable. At the heart of the permutation approach is the exchangeability assumption, which determines the appropriate null sampling distribution. We explore the exchangeability assumption in the context of permutation tests for multiple linear regression models. Various permutation schemes for the multiple linear regression setting have been previously proposed and assessed in the literature. As has been demonstrated previously, in most settings, the choice of how to permute a multiple linear regression model does not materially change inferential conclusions. Regardless, we believe that (1) understanding exchangeability in the multiple linear regression setting and also (2) how it relates to the null hypothesis of interest is valuable. We also briefly explore model settings beyond multiple linear regression (e.g., settings where clustering or hierarchical relationships exist) as a motivation for the benefit and flexibility of permutation tests. We close with pedagogical recommendations for instructors who want to bring multiple linear regression permutation inference into their classroom as a way to deepen student understanding of resampling-based inference.

翻译：置换检验是一种通过重采样进行推断的强大而灵活的方法。随着计算方法在统计学课程中日益普及，置换检验的应用变得更加可行。置换方法的核心在于可交换性假设，该假设决定了适当的零抽样分布。本文探讨了多重线性回归模型置换检验背景下的可交换性假设。文献中先前已提出并评估了适用于多重线性回归场景的各种置换方案。正如先前研究所证明的，在大多数情况下，多重线性回归模型的置换选择方式不会实质性地改变推断结论。尽管如此，我们认为：（1）理解多重线性回归场景中的可交换性，以及（2）其与目标零假设的关联，具有重要价值。我们还简要探讨了超越多重线性回归的模型场景（例如存在聚类或层次关系的场景），以展示置换检验的优势与灵活性。最后，我们为希望将多重线性回归置换推断引入课堂教学的教师提供教学建议，以此深化学生对基于重采样的推断方法的理解。