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Bayesian Scalar-on-Tensor Quantile Regression for Longitudinal Data on Alzheimer's Disease

Rongke Lyu, Marina Vannucci, Suprateek Kundu

Abstract

As a general and robust alternative to traditional mean regression models, quantile regression avoids the assumption of normally distributed errors, making it a versatile choice when modeling outcomes such as cognitive scores that typically have skewed distributions. Motivated by an application to Alzheimer's disease data where the aim is to explore how brain-behavior associations change over time, we propose a novel Bayesian tensor quantile regression for high-dimensional longitudinal imaging data. The proposed approach distinguishes between effects that are consistent across visits and patterns unique to each visit, contributing to the overall longitudinal trajectory. A low-rank decomposition is employed on the tensor coefficients which reduces dimensionality and preserves spatial configurations of the imaging voxels. We incorporate multiway shrinkage priors to model the visit-invariant tensor coefficients and variable selection priors on the tensor margins of the visit-specific effects. For posterior inference, we develop a computationally efficient Markov chain Monte Carlo sampling algorithm. Simulation studies reveal significant improvements in parameter estimation, feature selection, and prediction performance when compared with existing approaches. In the analysis of the Alzheimer's disease data, the flexibility of our modeling approach brings new insights as it provides a fuller picture of the relationship between the imaging voxels and the quantile distributions of the cognitive scores.

Bayesian Scalar-on-Tensor Quantile Regression for Longitudinal Data on Alzheimer's Disease

Abstract

As a general and robust alternative to traditional mean regression models, quantile regression avoids the assumption of normally distributed errors, making it a versatile choice when modeling outcomes such as cognitive scores that typically have skewed distributions. Motivated by an application to Alzheimer's disease data where the aim is to explore how brain-behavior associations change over time, we propose a novel Bayesian tensor quantile regression for high-dimensional longitudinal imaging data. The proposed approach distinguishes between effects that are consistent across visits and patterns unique to each visit, contributing to the overall longitudinal trajectory. A low-rank decomposition is employed on the tensor coefficients which reduces dimensionality and preserves spatial configurations of the imaging voxels. We incorporate multiway shrinkage priors to model the visit-invariant tensor coefficients and variable selection priors on the tensor margins of the visit-specific effects. For posterior inference, we develop a computationally efficient Markov chain Monte Carlo sampling algorithm. Simulation studies reveal significant improvements in parameter estimation, feature selection, and prediction performance when compared with existing approaches. In the analysis of the Alzheimer's disease data, the flexibility of our modeling approach brings new insights as it provides a fuller picture of the relationship between the imaging voxels and the quantile distributions of the cognitive scores.
Paper Structure (20 sections, 9 equations, 5 figures, 9 tables)

This paper contains 20 sections, 9 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Bayesian Tensor Quantile Regression for Longitudinal Data
  4. Prior model
  5. MCMC Algorithm
  6. Posterior Inference
  7. Simulation Study
  8. Data Generation
  9. Parameter Settings
  10. Performance Metrics
  11. Results
  12. Computation Times and MCMC Convergence
  13. Sensitivity Analysis
  14. Performance Under Model Mis-specification
  15. ADNI-1 Data Analysis
  16. ...and 5 more sections

Figures (5)

  • Figure 1: Simulated signals for scenarios 1-4, as described in the text. Each column represents one scenario, with rows 1-3 representing the longitudinal visits.
  • Figure 2: Simulated signals for 3D setup. From left to right: visit 1( with magnitude 1), visit 2( with magnitude 1), and visit 3( with magnitude 1.5).
  • Figure 3: Estimated signals for one replicate of scenarios 1-4, as described in the text. Each column represents one scenario, with rows 1-3 representing the longitudinal visits.
  • Figure 4: From left to right: Top panel: The distribution of observed ADAS scores measured at baseline, 6-month visit, and 12-month visit for AD cohort; Bottom panel: The distribution of observed ADAS scores measured at baseline, 6-month visit, and 12-month visit for MCI cohort. The test scores have skewed distributions. The quantiles are labeled on the x-axis.
  • Figure 5: Proportion of significant voxels in selected ROIs in the DKT atlas at the 0.5 quantile, for some select ROIs having the highest number of significant voxels. The remaining ROIs are depicted in gray and correspond to regions with limited proportion of significant voxels.