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Adaptive Penalized Doubly Robust Regression for Longitudinal Data

Yuyao Wang, Yu Lu, Tianni Zhang, Mengfei Ran

TL;DR

An iterative reweighting algorithm is developed and establish estimation and prediction error bounds, support recovery consistency, and oracle-type asymptotic normality in a doubly adaptive robust regression framework for longitudinal linear mixed effects models.

Abstract

Longitudinal data often involve heterogeneity, sparse signals, and contamination from response outliers or high-leverage observations especially in biomedical science. Existing methods usually address only part of this problem, either emphasizing penalized mixed effects modeling without robustness or robust mixed effects estimation without high-dimensional variable selection. We propose a doubly adaptive robust regression (DAR-R) framework for longitudinal linear mixed effects models. It combines a robust pilot fit, doubly adaptive observation weights for residual outliers and leverage points, and folded concave penalization for fixed effect selection, together with weighted updates of random effects and variance components. We develop an iterative reweighting algorithm and establish estimation and prediction error bounds, support recovery consistency, and oracle-type asymptotic normality. Simulations show that DAR-R improves estimation accuracy, false-positive control, and covariance estimation under both vertical outliers and bad leverage contamination. In the TADPOLE/ADNI Alzheimer's disease application, DAR-R achieves accurate and stable prediction of ADAS13 while selecting clinically meaningful predictors with strong resampling stability.

Adaptive Penalized Doubly Robust Regression for Longitudinal Data

TL;DR

An iterative reweighting algorithm is developed and establish estimation and prediction error bounds, support recovery consistency, and oracle-type asymptotic normality in a doubly adaptive robust regression framework for longitudinal linear mixed effects models.

Abstract

Longitudinal data often involve heterogeneity, sparse signals, and contamination from response outliers or high-leverage observations especially in biomedical science. Existing methods usually address only part of this problem, either emphasizing penalized mixed effects modeling without robustness or robust mixed effects estimation without high-dimensional variable selection. We propose a doubly adaptive robust regression (DAR-R) framework for longitudinal linear mixed effects models. It combines a robust pilot fit, doubly adaptive observation weights for residual outliers and leverage points, and folded concave penalization for fixed effect selection, together with weighted updates of random effects and variance components. We develop an iterative reweighting algorithm and establish estimation and prediction error bounds, support recovery consistency, and oracle-type asymptotic normality. Simulations show that DAR-R improves estimation accuracy, false-positive control, and covariance estimation under both vertical outliers and bad leverage contamination. In the TADPOLE/ADNI Alzheimer's disease application, DAR-R achieves accurate and stable prediction of ADAS13 while selecting clinically meaningful predictors with strong resampling stability.
Paper Structure (25 sections, 10 theorems, 106 equations, 5 figures, 8 tables, 1 algorithm)

This paper contains 25 sections, 10 theorems, 106 equations, 5 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Model Specification
  4. Doubly Adaptive Objective
  5. Computation
  6. EM Algorithm
  7. E-step: Updating Weights and Random Effects
  8. M-step: Updating Fixed Effects ($\beta$)
  9. Algorithm
  10. Coordinate Descent for the Penalized Fixed-Effect Update
  11. Theoretical Properties
  12. Notation and Assumptions
  13. Theoretical Results
  14. Simulation Study
  15. Data Generation
  16. ...and 10 more sections

Key Result

Theorem 1

Under Assumption ass_a1--ass_a7, assume $\lambda\ge 2\|\nabla \mathcal{L}_N(\beta^\star)\|_\infty$. Then for all sufficiently large $N$, there exist constants $C_1,C_2>0$ such that where $\tilde{W}=\mathrm{blkdiag}(\tilde{W}_1,\ldots,\tilde{W}_n)$.

Figures (5)

  • Figure 1: Boxplots of $\mathrm{MSE}(S)$ under scenarios S1--S3. All values are $\log_{10}$-transformed).
  • Figure 2: Boxplots of $\mathrm{MSE}(S^c)$ under scenarios S1--S3. All values are $\log_{10}$-transformed.
  • Figure 3: Boxplots of MSPE under scenarios S1--S3. All values are $\log_{10}$-transformed.
  • Figure 4: Boxplots of covariance estimation error for the random-effect. All values are $\log_{10}$-transformed.
  • Figure 5: Boxplots of the application-study prediction error (MSPE) across repeated resampling runs for all competing methods on the TADPOLE data.

Theorems & Definitions (26)

  • Theorem 1: Non-asymptotic bounds
  • Remark 1
  • Theorem 2: Consistency
  • Remark 2
  • Theorem 3: Support Recovery
  • Remark 3
  • Theorem 4: Oracle Asymptotic Normality
  • Remark 4
  • Proposition 1: Breakdown point
  • Remark 5
  • ...and 16 more