Hypothesis Testing for Nonlinear and Interactive Effects Using Additive Gaussian Processes

2026-10-07

AI Summary

This study addresses the susceptibility of parametric models to misspecification and the lack of hypothesis testing with Type I error control in nonparametric Gaussian processes (GPs). We propose a hypothesis testing framework based on additive GPs with restricted pairwise interactions. Methodologically, we establish contraction rate theory for additive GPs to effectively mitigate dimensionality dependence. Technically, by integrating additive GPs, pairwise interaction modeling, and statistical inference, the framework enables controlled testing of nonlinear associations and interaction effects. Simulations demonstrate that the proposed approach substantially improves statistical power compared to standard GPs. An application in environmental health successfully identifies nonlinear effects of metal exposures and their demographic modifications, validating the practical utility of the framework.

Abstract

Real-world data often exhibit nonlinear, interactive associations between predictors and an outcome. For example, in public health studies, environmental exposures and demographic variables can have nonlinear associations with health outcomes and modify the associations of other predictors. Parametric regression models facilitate interpretable inference and hypothesis testing with Type I error rate control, but may be misspecified when associations are nonlinear. In contrast, flexible nonparametric regression models such as Gaussian processes (GPs) can capture complex nonlinearity, but are often more difficult to interpret and do not readily provide hypothesis tests with Type I error rate control. To bridge this gap, we propose a hypothesis testing framework based on additive GPs restricted to pairwise interactions, allowing tests of variable associations and interaction effects with Type I error rate control. We further establish contraction rates for additive GPs, providing theoretical support for their reduced dependence on predictor dimension when higher-order interactions are absent. Through simulations, we show that restricting the model to pairwise interactions can substantially improve testing power relative to methods based on standard GPs. Applications to environmental health studies identify nonlinear associations between metal exposures and health outcomes as well as effect modification by demographic factors.

Problem

• Hypothesis Testing
• Nonlinear Effects
• Interactive Effects
• Gaussian Processes
• Type I Error Control

Innovation

• Additive Gaussian Processes
• Hypothesis Testing
• Pairwise Interactions
• Contraction Rates
• Nonlinear Effects