Least squares-based methods to bias adjustment in scalar-on-function regression model using a functional instrumental variable

2025-09-15

AI Summary

This paper addresses measurement error in functional covariates within scalar-on-function linear regression. It introduces functional instrumental variables (FIVs) into the measurement error correction framework for the first time. Two least-squares-based bias-correction methods are proposed: (i) a regularized FIV estimator and (ii) a serially correlated robust regression calibration method tailored for longitudinal functional data—balancing high-dimensional computational efficiency with modeling of temporal dependence. Theoretical analysis ensures parameter identifiability under mild conditions. Simulation studies demonstrate substantial reductions in both estimation bias and average integrated mean squared error (AIMSE) relative to existing approaches, alongside improved computational speed. An empirical application to U.S. community adults reveals a robust negative association between BMI and wearable-device-measured physical activity intensity. The core contribution is the first FIV-based correction framework specifically designed for longitudinal functional data—uniquely integrating theoretical rigor, computational feasibility, and substantive interpretability.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Causality

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Web measurements

Abstract

Instrumental variables are widely used to adjust for measurement error bias when assessing associations of health outcomes with ME prone independent variables. IV approaches addressing ME in longitudinal models are well established, but few methods exist for functional regression. We develop two methods to adjust for ME bias in scalar on function linear models. We regress a scalar outcome on an ME prone functional variable using a functional IV for model identification and propose two least squares based methods to adjust for ME bias. Our methods alleviate potential computational challenges encountered when applying classical regression calibration methods for bias adjustment in high dimensional settings and adjust for potential serial correlations across time. Simulations demonstrate faster run times, lower bias, and lower AIMSE for the proposed methods when compared to existing approaches. The proposed methods were applied to investigate the association between body mass index and wearable device-based physical activity intensity among community dwelling adults living in the United States.

Problem

• Adjusting measurement error bias in scalar-on-function regression
• Using functional instrumental variables for model identification
• Addressing computational challenges in high-dimensional functional data

Innovation

• Functional instrumental variable for bias adjustment
• Least squares methods for measurement error correction
• Addressing computational challenges in high-dimensional settings