Scalar-on-function regression (SOFR) is a widely used tool in the medical and behavioral sciences, which elucidates the association between a scalar response and data collected repeatedly across a continuous domain. However, estimating and interpreting SOFR models pose significant challenges due to the high autocorrelation and dimensionality of functional predictors.

This work presents novel estimation and inference tools for Bayesian SOFR models. Firstly, we propose a locally adaptive and highly scalable Bayesian SOFR model. By combining a B-spline basis expansion with dynamic shrinkage priors on the regression coefficient function, our model achieves more accurate point estimates and precise uncertainty quantification, particularly when capturing both smooth and rapidly-changing features.

Secondly, we provide decision analysis tools for Bayesian SOFR models that extract locally constant summaries based on the posterior predictive distribution. These summaries help identify critical windows—regions in the domain of the functional covariates that predict the scalar response. Leveraging the proposed Bayesian SOFR model and decision analysis tools, we investigate the relationship between prenatal daily PM2.5 exposure and standardized 4th-grade reading test scores in a large cohort of North Carolina students. Our findings indicate that prenatal PM2.5 exposure during early and late pregnancy has the most adverse impact on the testing scores.

Lastly, we extend the proposed Bayesian SOFR model and decision analysis strategy to handle multiple functional covariates, nonlinear relationships, and binary indicator response variables. Using this generalized framework, we explore the effects of prenatal temperature and PM2.5 exposure on birth weight in Michigan. Our analysis reveals that higher temperature and PM2.5 exposure are associated with lower birth weights, with higher temperature exhibiting a stronger effect than PM2.5. We provide an R package (BaiSOFR), that implements the proposed model and decision analysis strategy, along with a vignette illustrating its application on simulated data.