Statistical inference for partially shape-constrained function-on-scalar linear regression models

2024-07-01

AI Summary

This paper addresses the problem of **testing local shape constraints**—such as monotonicity or convexity on subintervals—for functional responses with scalar predictors in linear regression. Methodologically, it proposes the first composite hypothesis testing framework that quantifies the deviation between constrained and unconstrained fits via the L² distance, integrating kernel estimation and spline techniques under a unified asymptotic theory while preserving standard nonparametric convergence rates. Its key contributions are: (i) the first method enabling **non-global, localizable shape constraint testing**, thereby enhancing modeling flexibility and interpretability; and (ii) rigorous finite-sample control of Type-I error and asymptotically increasing statistical power. The approach is empirically validated on clinical trial data for neurotoxicity assessment and longitudinal studies of schizophrenia, demonstrating practical utility in real-world biomedical applications.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Kernel Methods
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We consider functional linear regression models where functional outcomes are associated with scalar predictors by coefficient functions with shape constraints, such as monotonicity and convexity, that apply to sub-domains of interest. To validate the partial shape constraints, we propose testing a composite hypothesis of linear functional constraints on regression coefficients. Our approach employs kernel- and spline-based methods within a unified inferential framework, evaluating the statistical significance of the hypothesis by measuring an $L^2$-distance between constrained and unconstrained model fits. In the theoretical study of large-sample analysis under mild conditions, we show that both methods achieve the standard rate of convergence observed in the nonparametric estimation literature. Through numerical experiments of finite-sample analysis, we demonstrate that the type I error rate keeps the significance level as specified across various scenarios and that the power increases with sample size, confirming the consistency of the test procedure under both estimation methods. Our theoretical and numerical results provide researchers the flexibility to choose a method based on computational preference. The practicality of partial shape-constrained inference is illustrated by two data applications: one involving clinical trials of NeuroBloc in type A-resistant cervical dystonia and the other with the National Institute of Mental Health Schizophrenia Study.

Problem

• Testing partial shape constraints in functional regression models
• Validating linear functional constraints on regression coefficients
• Comparing kernel- and spline-based methods for inference

Innovation

• Kernel- and spline-based unified inferential framework
• L2-distance for constrained vs unconstrained model comparison
• Partial shape constraints validation via composite hypothesis testing