Modeling Longitudinal Data with Multivariate GLMMs: A Skew Normal Approach

Music City Center 

In recent decades, multivariate Generalized Linear Mixed Models (mGLMMs) have become essential for analyzing complex data structures, particularly in longitudinal studies. These models provide a flexible framework for handling correlated responses across subjects and time points, integrating both fixed and random effects. However, the assumption of multivariate normal distribution is often violated. We proposed mGLMM for modeling data from a multivariate skew normal distribution (mGLMM-SKN) that incorporates skewness. Parameter estimation was performed using the Expectation-Maximization (EM) algorithm, iteratively updating fixed effects, random effects, covariance matrices, and skewness parameters. The model's goodness-of-fit was assessed through residual analysis. We demonstrated the new approach using both simulation and real datasets with R statistical software. The simulation scenarios included N=1000, j= 5 time points, and k=8 outcomes. We applied the proposed method to examine the relationship between the exposures (APOE ε4 allele status and baseline Aβ+ status) and 8 outcomes from the motivating ADNI study, while adjusting for age, sex, and education. By incorporating skewness effects, we enable a more flexible analysis that accommodates the asymmetry, and heavy tails often present in real-world datasets.