Sparsity Inducing Prior Distributions for Correlation Matrices of Longitudinal Data

Area of Science:

• Statistics
• Statistical Modeling
• Longitudinal Data Analysis

Background:

• Modeling correlation matrices in longitudinal data presents challenges due to positive definite and unit diagonal constraints.
• The quadratic increase in parameters with dimension necessitates sparse parameterizations for efficiency.

Purpose of the Study:

• To introduce novel prior distributions for correlation matrices in longitudinal data.
• To develop priors based on partial autocorrelations (PACs) that allow for sparse and interpretable representations.
• To offer a computationally attractive alternative to existing methods for correlation matrix selection.

Main Methods:

• Introduction of two prior distributions on correlation matrices via partial autocorrelations (PACs).
• The first prior employs shrinkage towards zero for PACs, increasing with lag.
• The second prior is a selection prior, a mixture of a point mass at zero and a continuous component for each PAC.

Main Results:

• The proposed priors yield interpretable structures where zero PACs indicate conditional independence.
• Selection priors on PACs offer a computationally efficient approach compared to direct selection on correlation matrix elements or their inverses.
• The priors facilitate data-dependent shrinkage and selection in an unconstrained parameter space.

Conclusions:

• The novel priors provide a flexible and interpretable framework for modeling correlation matrices in longitudinal data.
• These methods are computationally advantageous and suitable for time-ordered response variables.
• The effectiveness is demonstrated through simulation studies and a real-world data example.