A Robust Monotonic Single-Index Model for Skewed and Heavy-Tailed Data: A Deep Neural Network Approach Applied to Periodontal Studies

2025-05-04

AI Summary

This paper addresses the prevalent skewness, heavy-tailedness, and nonlinear associations with clinical risk factors in periodontal pocket depth data by proposing a robust single-index modal regression model. Methodologically, it introduces a novel two-piece scaled Student-*t* error distribution to enhance robustness against outliers; employs a deep neural network with monotonicity constraints to estimate the single-index link function—balancing modeling flexibility and clinical interpretability; and rigorously establishes model identifiability and universal approximation capability. A corresponding R package, DNNSIM, is developed for implementation. In both simulation studies and real-world periodontal data from HealthPartners, the proposed model demonstrates superior outlier resistance and higher modal estimation accuracy compared to state-of-the-art methods, while preserving stable, clinically meaningful interpretations of key covariate effects.

Technology Category

• Machine Learning: Multimodal Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Periodontal pocket depth is a widely used biomarker for diagnosing risk of periodontal disease. However, pocket depth typically exhibits skewness and heavy-tailedness, and its relationship with clinical risk factors is often nonlinear. Motivated by periodontal studies, this paper develops a robust single-index modal regression framework for analyzing skewed and heavy-tailed data. Our method has the following novel features: (1) a flexible two-piece scale Student-$t$ error distribution that generalizes both normal and two-piece scale normal distributions; (2) a deep neural network with guaranteed monotonicity constraints to estimate the unknown single-index function; and (3) theoretical guarantees, including model identifiability and a universal approximation theorem. Our single-index model combines the flexibility of neural networks and the two-piece scale Student-$t$ distribution, delivering robust mode-based estimation that is resistant to outliers, while retaining clinical interpretability through parametric index coefficients. We demonstrate the performance of our method through simulation studies and an application to periodontal disease data from the HealthPartners Institute of Minnesota. The proposed methodology is implemented in the extsf{R} package href{https://doi.org/10.32614/CRAN.package.DNNSIM}{ extsc{DNNSIM}}.

Problem

• Model skewed and heavy-tailed periodontal pocket depth data
• Estimate nonlinear relationships with clinical risk factors robustly
• Ensure interpretability via monotonic deep neural network constraints

Innovation

• Flexible two-piece scale Student-t error distribution
• Monotonic deep neural network for single-index function
• Robust mode-based estimation resistant to outliers