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Group-Sparse Smoothing for Longitudinal Models with Time-Varying Coefficients

Yu Lu, Tianni Zhang, Yuyao Wang, Mengfei Ran

TL;DR

Time-varying effect selection, TV-Select, a unified framework for structural identification that simultaneously selects relevant variables and determines whether their effects are constant or time-varying is proposed.

Abstract

Longitudinal data analysis is fundamental for understanding dynamic processes in biomedical and social sciences. Although varying coefficient models (VCMs) provide a flexible framework by allowing covariate effects to evolve over time, fitting all effects as time-varying may lead to overfitting, efficiency loss, and reduced interpretability when some effects are actually constant. In contrast, standard linear mixed models (LMMs) may suffer substantial bias when temporal heterogeneity is ignored. To address this issue, we propose time-varying effect selection, TV-Select, a unified framework for structural identification that simultaneously selects relevant variables and determines whether their effects are constant or time-varying. The proposed method decomposes each coefficient function into a time-invariant mean component and a centered time-varying deviation, where the latter is approximated by B-splines. We then construct a doubly penalized objective function that combines a group Lasso penalty for structural sparsity with a roughness penalty for smoothness control. An efficient block coordinate descent algorithm is developed for computation. Under regular semiparametric conditions, we establish selection consistency and oracle-type asymptotic properties, including asymptotic normality for the constant-effect component after correct structure recovery. Simulation studies and a real-data application show that TV-Select achieves more accurate structural recovery, smoother functional estimation, and better predictive performance than competing methods.

Group-Sparse Smoothing for Longitudinal Models with Time-Varying Coefficients

TL;DR

Time-varying effect selection, TV-Select, a unified framework for structural identification that simultaneously selects relevant variables and determines whether their effects are constant or time-varying is proposed.

Abstract

Longitudinal data analysis is fundamental for understanding dynamic processes in biomedical and social sciences. Although varying coefficient models (VCMs) provide a flexible framework by allowing covariate effects to evolve over time, fitting all effects as time-varying may lead to overfitting, efficiency loss, and reduced interpretability when some effects are actually constant. In contrast, standard linear mixed models (LMMs) may suffer substantial bias when temporal heterogeneity is ignored. To address this issue, we propose time-varying effect selection, TV-Select, a unified framework for structural identification that simultaneously selects relevant variables and determines whether their effects are constant or time-varying. The proposed method decomposes each coefficient function into a time-invariant mean component and a centered time-varying deviation, where the latter is approximated by B-splines. We then construct a doubly penalized objective function that combines a group Lasso penalty for structural sparsity with a roughness penalty for smoothness control. An efficient block coordinate descent algorithm is developed for computation. Under regular semiparametric conditions, we establish selection consistency and oracle-type asymptotic properties, including asymptotic normality for the constant-effect component after correct structure recovery. Simulation studies and a real-data application show that TV-Select achieves more accurate structural recovery, smoother functional estimation, and better predictive performance than competing methods.
Paper Structure (23 sections, 6 theorems, 108 equations, 8 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 6 theorems, 108 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Longitudinal Model
  4. Spline Approximation
  5. Objective Function
  6. Algorithm
  7. Block updates
  8. Algorithm
  9. Asymptotic Properties
  10. Assumptions
  11. Results
  12. Numerical Study
  13. Data Generation
  14. Simulation Scenarios
  15. Competing Methods
  16. ...and 8 more sections

Key Result

Theorem 1

Under Assumptions ass:A1--ass:A5, with probability at least $1-c_1 p^{-c_2}$, the estimator $(\hat{\beta}_0,\hat{\mu},\{\hat{\boldsymbol{\theta}}_k\})$ satisfies for constants $C,c_1,c_2>0$ independent of $(N,p)$. Consequently, for each $k\in\mathcal{S}_{\mathrm{vary}}$, the estimated function $\hat{g}_k(t)=\tilde{\boldsymbol{B}}(t)^\top\hat{\boldsymbol{\theta}}_k$ obeys In particular, choosing

Figures (8)

  • Figure 1: ClassAcc acrosss different configurations.
  • Figure 2: TPR acrosss different configurations.
  • Figure 3: FPR acrosss different configurations.
  • Figure 4: Stability acrosss different configurations.
  • Figure 5: ISE acrosss different configurations.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 1: Estimation error and rate
  • Theorem 2: Selection consistency for time-varying effects
  • Corollary 1: Consistent classification of $\mathcal{S}_{\mathrm{zero}}$ vs $\mathcal{S}_{\mathrm{const}}$
  • Theorem 3: Oracle asymptotic normality for constant effects
  • Lemma 1: Spline approximation error
  • proof
  • Lemma 2: Uniform control of noise--design correlations
  • proof