Matrix-Response Generalized Linear Mixed Model with Applications to Longitudinal Brain Images

TL;DR

The paper addresses the challenge of modeling longitudinal brain networks by introducing a matrix-response generalized linear mixed model (MR-GLMM) that handles matrix-valued, time-indexed responses. It imposes a low-rank structure on the population connectivity $\boldsymbol{\Theta} = \boldsymbol{U}\boldsymbol{V}^{\top}$ and exact sparsity on the covariate slope tensor $\mathcal{B}$, estimating parameters via a scalable Monte Carlo EM algorithm with Metropolis-within-Gibbs in the E-step and proximal-gradient updates with hard-thresholding in the M-step. Through extensive simulations, MR-GLMM outperforms element-wise GLMMs in both edge estimation and covariate-effect identification, and it is validated on real DTI and fMRI datasets where age, sex, APOE4, and task status modulate connectivity patterns. The framework provides interpretable, edge-level biomarkers of brain network changes over time and offers a flexible, scalable tool for longitudinal neuroimaging analysis with potential for multi-modal extensions.

Abstract

Longitudinal brain imaging data facilitate the monitoring of structural and functional alterations in individual brains across time, offering essential understanding of dynamic neurobiological mechanisms. Such data improve sensitivity for detecting early biomarkers of disease progression and enhance the evaluation of intervention effects. While recent matrix-response regression models can relate static brain networks to external predictors, there remain few statistical methods for longitudinal brain networks, especially those derived from high-dimensional imaging data. We introduce a matrix-response generalized linear mixed model that accommodates longitudinal brain networks and identifies edges whose connectivity is influenced by external predictors. An efficient Monte Carlo Expectation-Maximization algorithm is developed for parameter estimation. Extensive simulations demonstrate effective identification of covariate-related network components and accurate parameter estimation. We further demonstrate the usage of the proposed method through applications to diffusion tensor imaging (DTI) and functional MRI (fMRI) datasets.

Figures (8)

• Figure 1: The heatmap of $\widehat{\boldsymbol{\Theta}}$ in Model 1, where rows and columns are arranged according to the K-means clustering. The black dashed lines represent the groups of functional modules identified in the clustering process with module names on the right table.
• Figure 2: Estimated $\widehat{\mathcal{B}}_2$ and $\widehat{\mathcal{B}}_3$ in Model 1. Red and blue edges indicate positive and negative entries in $\widehat{\mathcal{B}}_2$ and $\widehat{\mathcal{B}}_3$, respectively.
• Figure 3: The heatmap of $\widehat{\boldsymbol{\Theta}}$ in Model 2, where rows and columns are arranged according to the K-means clustering. The black dashed lines represent the groups of functional modules identified in the clustering process with module names on the right table.
• Figure 4: Estimated $\widehat{\mathcal{B}}_2$ and $\widehat{\mathcal{B}}_3$ in Model 2. Red and blue edges indicate positive and negative entries in $\widehat{\mathcal{B}}_2$ and $\widehat{\mathcal{B}}_3$, respectively.
• Figure 5: The heatmap of $\widehat{\boldsymbol{\Theta}}$ in Model 3, where rows and columns are arranged according to the K-means clustering. The black dashed lines represent the groups of functional modules identified in the clustering process with module names on the right table.