Linear Models with Nested Random Effects

Symmetry is a crucial concept in various fields of mathematics, offering a systematic approach to understanding structural properties and simplifying complex problems. This study focuses on linear mixed models, emphasizing the role of symmetry in the design of experiments and the structure of variance–covariance matrices. Building on foundational works, this paper introduces the concept of nested models and employs Wishart matrices for variance component estimation, enhancing the efficiency of complex model analysis. The methodology is particularly applicable when variance–covariance matrices conform to a commutative Jordan algebra (CJA). The practical significance of this modeling approach is demonstrated through numerical applications using both simulated and real-world data.