The Augmented Hat‐Matrix of Hierarchical Generalised Linear Models and Its Use in Leverage Diagnostics

Abstract

Defining a 'hat-matrix' for a model is essential in many model diagnostic procedures, as it acts as an orthogonal projector from the observation space to the model space. In this paper, we introduce a unique Hat-matrix for the class of hierarchical generalised linear models (HGLMs), which includes, as a special case, the subclass of generalised linear mixed models (GLMMs). We provide a practical discussion on interpreting the hat matrix values in HGLMs across various settings, aimed at assisting practitioners in model diagnostics. Additionally, we propose two new empirical thresholds to identify high-leverage observations and clusters. We demonstrate the advantages of using these empirical thresholds over the traditional approach with a simulation study. Lastly, we present an application to real data to illustrate the effectiveness of our proposed methodology in real-world scenarios.