Variance-Guided Regression for Heteroscedastic Data With a Grouping-Based Extension for Nonlinear Prediction

Area of Science:

• Statistics
• Biostatistics
• Machine Learning

Background:

• Homoscedasticity is a common assumption in linear regression, but real-world data often exhibit heteroscedasticity or complex residual structures.
• Violations of homoscedasticity can lead to biased estimates and reduced predictive accuracy in standard regression models.

Purpose of the Study:

• To introduce VarGuid, a novel variance-guided framework designed to handle covariate-dependent conditional variance and nonlinear mean structures.
• To provide a robust statistical approach that improves upon traditional linear regression methods when assumptions are violated.

Main Methods:

• The VarGuid framework employs two distinct components: an iteratively reweighted regression (IRR) algorithm for estimating a sparse global mean-variance model, and a biconvex artificial-grouping algorithm for conditional prediction.
• The first component focuses on coefficient interpretation and robust mean-variance estimation, while the second refines predictions using group-specific local intercept corrections on a fixed linear backbone.

Main Results:

• The study establishes predictive-risk guarantees for the global estimator within the VarGuid framework.
• Simulations and empirical studies demonstrate that VarGuid achieves improved out-of-sample accuracy compared to existing methods.
• The framework effectively addresses both covariate-dependent variance and nonlinear mean structures that can be mistaken for heteroscedasticity.

Conclusions:

• VarGuid offers a powerful and flexible approach for analyzing data with complex variance patterns and mean structures.
• The framework's ability to provide both interpretable global models and accurate conditional predictions makes it valuable for various applications.
• Applications in health-related quality of life and genomic prediction highlight the practical utility and broad applicability of VarGuid.