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 misspecifications 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.

Journal of Computation and Graphical Statistics 24(4), p. 1191–1209