Are you Normal? The Problem of Confounded Residual Structures in Hierarchical Linear Models

We encounter hierarchical data structures in a wide range of applications. Regular linear models are extended by random effects to address correlation between observations in the same group. Inference for random effects is sensitive to distributional mis-specifications of the model, making checks for (distributional) assumptions particularly important. The investigation of residual structures is complicated by the presence of different levels and corresponding dependencies. Ignoring these dependencies leads to erroneous conclusions using our familiar tools, such as Q-Q plots or normal tests. We first show the extent of the problem, then we introduce the fraction of confounding as a measure of the level of confounding in a model and finally introduce rotated random effects as a solution to assessing distributional model assumptions. This article has supplementary materials online.分层数据结构广泛存在于各类应用场景中。常规线性模型通过引入随机效应（random effects）以处理同一分组内观测值间的相关性。随机效应的统计推断对模型的分布设定误设十分敏感，因此对（分布类）假设进行检验显得尤为关键。由于存在不同层级及对应的相关性，残差结构的分析变得更为复杂。若忽略此类相关性，使用Q-Q图、正态性检验等常规工具进行分析将得到错误的结论。本文首先阐明该问题的严重程度，随后引入混杂分数作为衡量模型中混杂程度的指标，最终提出旋转随机效应方法以解决模型分布假设的检验问题。本文附带在线补充材料。