Dynamic Single‐Index Scalar‐On‐Function Model

Environmental exposures often exhibit temporal variability, prompting extensive research to understand their dynamic impacts on human health. There has been a growing interest in studying time-dependent exposure mixtures beyond a single exposure. However, current analytic methods typically assess each exposure individually or assume an additive relationship. This paper aims to fill the gap in method development for evaluating the joint effects of multiple time-dependent exposures on a scalar outcome. We introduce a dynamic single-index scalar-on-function model to characterize the exposure mixture's time-varying effect through a non-parametric bivariate exposure-time-outcome surface function. Utilizing B-spline tensor product bases to approximate the surface function, we propose a profiling algorithm for model estimation and establish large-sample properties for the resulting single-index estimators. In addition, we introduce a non-parametric hypothesis testing procedure to determine whether the surface function varies over time at each fixed mixture level and a model averaging solution to circumvent the issue of knot selection for spline approximations. The performance of our proposed methods is examined through extensive simulations and further illustrated using real-world applications.