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

• Activity Number: 292
• Type: Topic Contributed
• Date/Time: Tuesday, August 6, 2013 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Computing
• Author(s): Adam Loy*+ and Heike Hofmann
• Companies: Iowa State University and Iowa State University
• Keywords: residuals ; hierarchical linear models ; statistical graphics ; diagnostics

Abstract

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 it fraction of confounding as a measure of the level of confounding in a model and finally introduce minimally confounded residuals as a solution to assessing distributional model assumptions.