Exploring the Relationship Between Random Effects Variance Components and Model Hierarchical Complexity

Posted: Nov 28, 2017

• Levi Scott
• Maria Wilson
• Michael Miller

Abstract

Hierarchical models, also known as multilevel or mixed-effects models, have revolutionized statistical practice across numerous scientific domains by enabling researchers to account for structured dependencies in data. These models partition variance into fixed and random components, with the latter capturing heterogeneity across grouping structures. Despite their widespread adoption and theoretical foundations, a fundamental gap persists in our understanding of how variance components systematically relate to the complexity of hierarchical structures. The prevailing literature has largely treated variance estimation as a technical challenge to be solved through computational methods, while neglecting the deeper mathematical relationships between model architecture and variance partitioning. This paper addresses this critical gap by developing a comprehensive framework for characterizing and quantifying hierarchical complexity and examining its systematic relationship with random effects variance components. We move beyond conventional approaches that focus primarily on computational efficiency or specific application domains, instead pursuing a principled investigation of the mathematical properties that govern variance decomposition in complex hierarchical systems. Our work is motivated by the observation that practitioners often encounter counterintuitive results when working with elaborate hierarchical structures, particularly when variance estimates appear inconsistent with theoretical expectations or exhibit unexpected sensitivity to model specification. We formulate three primary research questions that guide our investigation: First, how can hierarchical complexity be formally quantified in a manner that captures the multidimensional nature of nesting structures, cross-classifications, and random effect specifications? Second, what mathematical relationships exist between complexity metrics and variance component estimates across different hierarchical configurations? Third, do these relationships exhibit consistent patterns that can inform model selection and diagnostic procedures in practical applications?