Reduced Rank Mixed Effects Models for Spatially Correlated Hierarchical Functional Data

Area of Science:

• Statistics
• Biostatistics
• Functional Data Analysis

Background:

• Hierarchical functional data are common in complex studies with nested structures.
• Modeling within-unit and sub-unit variations and their correlations is challenging.

Purpose of the Study:

• To propose a general framework for functional mixed effects models tailored for hierarchical data.
• To address the complexities of modeling correlated functions and covariance kernels in nested data.

Main Methods:

• Utilized a functional mixed effects model with two sets of principal components for variations.
• Employed penalized splines for mean and principal component functions, with roughness penalties for regularization.
• Developed an EM algorithm for efficient model fitting, leveraging the covariance structure to avoid large matrix operations.

Main Results:

• The proposed dimension reduction using principal components effectively models random function covariance and correlations.
• The EM algorithm provides computational efficiency by avoiding large matrix storage and inversion.
• The methodology demonstrated effectiveness in simulations and a colon carcinogenesis study.

Conclusions:

• The developed functional mixed effects model framework offers a powerful tool for analyzing hierarchical functional data.
• Principal component analysis provides an effective dimension reduction strategy for complex functional data.
• The approach is validated through simulations and real-world application in a biomedical study.