Bayesian Effect Selection for Additive Quantile Regression with an Application to Air Pollution Thresholds

2026-06-10

AI Summary

This study addresses the challenge of distinguishing between linear and nonlinear effects of covariates in quantile regression, particularly in the context of air pollution modeling. The authors propose a Bayesian effect selection approach based on an additive quantile regression framework, wherein each additive component is orthogonally decomposed into linear and nonlinear parts using Demmler–Reinsch bases. Independent spike-and-slab priors are employed to facilitate effect selection, enabling, for the first time, orthogonal separation and consistent estimation of both types of effects within the quantile regression paradigm. The method establishes a data-driven Bayesian modeling mechanism and demonstrates robustness to likelihood misspecification in simulations, outperforming conventional mixed representations. Application to Madrid NO₂ data reveals distinct driving mechanisms of meteorological and traffic-related spatial structures on high-concentration pollution episodes.

Abstract

Air pollution regulatory limits are typically defined in terms of exceedances of concentration thresholds which are naturally related to conditional quantiles of the pollutant distribution and are therefore of direct relevance for assessing severe pollution events. At the same time, it is important to determine not only whether a covariate affects air pollution but also whether this effect is linear, nonlinear, or both. We address these issues by developing a Bayesian effect selection approach for additive quantile regression. While commonly used mixed model representations (MMRs) of penalized splines allow for flexible nonlinear effects, they do not provide a meaningful separation of linear and nonlinear effect components. We therefore employ a Demmler-Reinsch basis expansion, which yields an orthogonal decomposition of each additive effect into linear and nonlinear parts and show theoretically that both effect components can be estimated consistently. To facilitate data-driven model building, we propose Bayesian effect selection with separate spike and slab priors on the scalar importance parameters associated with the linear and nonlinear components and implement an efficient Gibbs sampler. Through simulation studies, we demonstrate robustness to the misspecification induced by the employed asymmetric Laplace working likelihood and show superior performance relative to the MMR. In a detailed analysis of air pollution data in Madrid, Spain we highlight the added value of flexibly modeling extreme nitrogen dioxide (NO$_2$) concentrations and reveal that threshold-relevant pollution levels are driven differently by climatological variables and traffic-related spatial structure. These findings underline the need for advanced statistical models that support short-term decision-making and help local authorities mitigate, or potentially prevent, exceedances of NO$_2$ concentration limits.

Problem

• quantile regression
• effect selection
• linear vs nonlinear effects
• air pollution thresholds
• additive models

Innovation

• Bayesian effect selection
• additive quantile regression
• Demmler-Reinsch basis
• orthogonal decomposition
• spike-and-slab priors