Multivariate causal effects: a Bayesian causal regression factor model

The impact of wildfire smoke on air quality is a growing concern, contributing to air pollution through a complex mixture of chemical species with important implications for public health. Although previous studies have focused mainly on its association with total particulate matter (pm$_{2.5}$), the causal relationship between wildfire smoke and the chemical composition of pm$_{2.5}$ remains largely unexplored. To fill this gap, we propose a Bayesian causal regression factor model that estimates the multivariate causal effects of wildfire smoke on the concentration of 27 chemical species in pm$_{2.5}$ across the United States. Our approach introduces two key innovations: (i) a causal inference framework for multivariate potential outcomes, and (ii) a novel Bayesian factor model that employs a probit stick-breaking process as prior for treatment-specific factor scores. By focusing on factor scores, our method addresses the missing data challenge common to causal inference and enables a flexible, data-driven characterization of the latent factor structure, which is crucial to capturing the complex correlation between multivariate outcomes. Through Monte Carlo simulations, we show the accuracy of the model in estimating the causal effects in multivariate outcomes and characterizing the treatment-specific latent structure. Finally, we apply our method to US air quality data, estimating the causal effect of wildfire smoke on 27 chemical species in pm$_{2.5}$, providing a deeper understanding of their interdependencies.