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Generalized Heterogeneous Functional Model with Applications to Large-scale Mobile Health Data

Xiaojing Sun, Bingxin Zhao, Fei Xue

TL;DR

This work tackles the challenge of time-varying, subgroup-specific effects of physical activity on diseases in large-scale mobile health data by introducing the Generalized Heterogeneous Functional Model (GHFM). GHFM jointly identifies subgroup structure and estimates subject-specific functional effects $\beta_{ij}(t)$ within a generalized scalar-on-function regression, employing a pairwise fusion penalty and a roughness penalty, with a scalable pre-clustering strategy and a hypothesis test for subgroup heterogeneity. The authors establish theoretical consistency for both coefficient estimation and subgroup identification and demonstrate strong performance in simulations versus homogeneous and competing mixture methods. In a large UK Biobank dementia study, GHFM detects three distinct subgroups with divergent time-varying effects of activity, improving predictive accuracy and offering interpretable, subgroup-specific guidance for interventions. The approach extends to Gaussian and binary outcomes, is scalable to very large datasets, and provides a practical framework for personalized, time-dependent health risk assessment.

Abstract

Physical activity is crucial for human health. With the increasing availability of large-scale mobile health data, strong associations have been found between physical activity and various diseases. However, accurately capturing this complex relationship is challenging, possibly because it varies across different subgroups of subjects, especially in large-scale datasets. To fill this gap, we propose a generalized heterogeneous functional method which simultaneously estimates functional effects and identifies subgroups within the generalized functional regression framework. The proposed method captures subgroup-specific functional relationships between physical activity and diseases, providing a more nuanced understanding of these associations. Additionally, we develop a pre-clustering method that enhances computational efficiency for large-scale data through a finer partition of subjects compared to true subgroups. We further introduce a testing procedure to assess whether the different subgroups exhibit distinct functional effects. In the real data application, we examine the impact of physical activity on the risk of dementia using the UK Biobank dataset, which includes over 96,433 participants. Our proposed method outperforms existing methods in future-day prediction accuracy, identifying three distinct subgroups, with detailed scientific interpretations for each subgroup. We also demonstrate the theoretical consistency of our methods. Codes implementing the proposed method are available at: https://github.com/xiaojing777/GHFM.

Generalized Heterogeneous Functional Model with Applications to Large-scale Mobile Health Data

TL;DR

This work tackles the challenge of time-varying, subgroup-specific effects of physical activity on diseases in large-scale mobile health data by introducing the Generalized Heterogeneous Functional Model (GHFM). GHFM jointly identifies subgroup structure and estimates subject-specific functional effects within a generalized scalar-on-function regression, employing a pairwise fusion penalty and a roughness penalty, with a scalable pre-clustering strategy and a hypothesis test for subgroup heterogeneity. The authors establish theoretical consistency for both coefficient estimation and subgroup identification and demonstrate strong performance in simulations versus homogeneous and competing mixture methods. In a large UK Biobank dementia study, GHFM detects three distinct subgroups with divergent time-varying effects of activity, improving predictive accuracy and offering interpretable, subgroup-specific guidance for interventions. The approach extends to Gaussian and binary outcomes, is scalable to very large datasets, and provides a practical framework for personalized, time-dependent health risk assessment.

Abstract

Physical activity is crucial for human health. With the increasing availability of large-scale mobile health data, strong associations have been found between physical activity and various diseases. However, accurately capturing this complex relationship is challenging, possibly because it varies across different subgroups of subjects, especially in large-scale datasets. To fill this gap, we propose a generalized heterogeneous functional method which simultaneously estimates functional effects and identifies subgroups within the generalized functional regression framework. The proposed method captures subgroup-specific functional relationships between physical activity and diseases, providing a more nuanced understanding of these associations. Additionally, we develop a pre-clustering method that enhances computational efficiency for large-scale data through a finer partition of subjects compared to true subgroups. We further introduce a testing procedure to assess whether the different subgroups exhibit distinct functional effects. In the real data application, we examine the impact of physical activity on the risk of dementia using the UK Biobank dataset, which includes over 96,433 participants. Our proposed method outperforms existing methods in future-day prediction accuracy, identifying three distinct subgroups, with detailed scientific interpretations for each subgroup. We also demonstrate the theoretical consistency of our methods. Codes implementing the proposed method are available at: https://github.com/xiaojing777/GHFM.
Paper Structure (17 sections, 3 theorems, 25 equations, 4 figures, 4 tables)

This paper contains 17 sections, 3 theorems, 25 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Generalized Heterogeneous Functional Method (GHFM)
  4. Heterogeneous Functional Gaussian Method (HFGM)
  5. Heterogeneous Functional Logistic Method (HFLM)
  6. Pre-clustering
  7. Generalized Test for Functional Subgroup Effects
  8. Theory
  9. Consistency of GHFM
  10. Theory for Pre-clustering
  11. Simulation
  12. Setting 1: Random B-spline structure.
  13. Setting 2: Structured polynomial forms.
  14. Real Data Application
  15. The UKB Data
  16. ...and 2 more sections

Key Result

Theorem 1

When Conditions (C1) to (C6) are satisfied, for the loss function of the HFGM in Equation (loss func: gauss beta t), with probability tending to one, there exists a local minimizer $(\hat{\alpha}, \hat{{\beta}}) = \arg \min_{\alpha\in\mathbb{R}, \beta \in \mathcal{S}_{d M}} Q_{n}^G({\alpha, \beta})$

Figures (4)

  • Figure 1: An example for relationship between true subgroups and pre-clustering groups when there are two true subgroups and five pre-clustering groups.
  • Figure 2: The flowchart of the proposed method.
  • Figure 3: Comparison of raw and smoothed physical activity signals for four subjects, based on minute-level data from 0:00 to 23:59. The smoothed curves are obtained using 50 B-spline basis functions. Each panel represents one subject, with distinct color schemes for raw and smoothed curves.
  • Figure 4: Estimated coefficient functions for Dementia data.

Theorems & Definitions (3)

  • Theorem 1
  • Proposition 2
  • Theorem 3