Confounder adjustment in single index function-on-scalar regression model

The function-on-scalar regression model serves as a potent tool for elucidating the connection between functional responses and covariates of interest. Despite its widespread utilization in numerous extensive neuroimaging investigations, prevailing methods often fall short in accounting for the intricate nonlinear relationships and the enigmatic confounding factors stemming from imaging heterogeneity. This heterogeneity may originate from a myriad of sources, such as variations in study environments, populations, designs, protocols, and concealed variables. To address this challenge, this paper develops a single index function-on-scalar regression model to investigate the nonlinear associations between functional responses and covariates of interest while making adjustments for concealed confounding factors arising from potential imaging heterogeneity. Both estimation and inference procedures are established for unknown parameters within our proposed model. In addition, the asymptotic properties of estimated functions and detected confounding factors are also systematically investigated. The finite-sample performance of our proposed method is assessed by using both Monte Carlo simulations and a real data example on the diffusion tensor images from the Alzheimer's Disease Neuroimaging Initiative study.