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

Although homoscedasticity is often assumed in linear regression, real data may show variance patterns or residual structures that violate this assumption. We propose VarGuid, a variance-guided framework for two related settings: Covariate-dependent conditional variance under a global linear mean model, and residual nonlinear mean structure that can mimic heteroscedasticity. The framework has two deliberately separated components. The first uses an iteratively reweighted regression (IRR) algorithm to estimate a sparse global linear mean-variance model and support coefficient interpretation. The second uses a biconvex artificial-grouping algorithm for conditional prediction, keeping the fitted linear backbone fixed while adding group-specific local intercept corrections. We establish predictive-risk guarantees for the global estimator, and simulations and empirical studies show improved out-of-sample accuracy. VarGuid is illustrated in two applications: Health-related quality of life in low- and middle-income countries, and high-dimensional genomic prediction of lymph node evaluation in breast cancer.