ABSTRACT
This chapter introduces inferential methods for scalar-on-function regression (SoFR), which is regression with a scalar outcome and a combination of scalar and functional covariates. This is also referred to in the literature as signal regression or Functional Linear Model (FLM) with a scalar outcome and functional predictors. Regression models with functional predictors avoid reducing the functional predictors to single-number summaries or treating observed data within each individual function as a separate, unrelated observation. The general idea is to (1) where necessary, project observed functions on a functional principal component basis to account for noise, irregular observation grids, and/or missing data; (2) use rich-basis spline expansions for functional coefficients and induce smoothing using penalties on the spline coefficients; (3) identify the mixed effects models that correspond to the specific functional regression; and (4) use existing mixed effects model software. Methods using unpenalized basis expansions are implemented in traditional software, while estimation and inference for SoFR using penalized splines is conducted using the refund and mgcv R packages.
