Bayesian monotone single-index quantile regression model with bounded response and misaligned functional covariates

Existing research in mental health has established that rising depressive symptoms in adolescents are associated with parental history of depression and other behavioral risk factors. Our goal is to investigate how these scalar variables, together with multiple functional covariates capturing neural responses to rewards, relate to future adolescent depression. Departing from prior studies that typically relied on simple linear regression to model all covariates, we propose a novel Bayesian quantile regression framework. This approach constructs a single-index summary of both scalar and functional covariates, coupled with a monotone link function that flexibly captures unknown nonlinear relationships and interactions. Our method addresses several limitations of existing approaches. It offers a clinically interpretable index akin to that of linear models, ensures that the estimated quantile remains within the response bounds, and jointly incorporates the registration of functional covariates within the quantile regression analysis. Our simulation studies demonstrate that our method outperforms existing unrestricted single-index-based methods, particularly when there are both scalar and preregistered functional covariates. Furthermore, we showcase the practical utility of our framework using data from a large-scale adolescent depression study, yielding a new, statistically principled summary of neural reward processing with direct relevance to future depression risk.