Permutation-based inference for function-on-scalar regression with an application in PET brain imaging

The density of various proteins throughout the human brain can be studied through the use of positron emission tomography (PET) imaging. We report here on data from a study of serotonin transporter (5-HTT) binding. While PET imaging data analysis is most commonly performed on data that are aggregated into several discrete a priori regions of interest, in this study, primary interest is on measures of 5-HTT binding potential that are made at many locations along a continuous anatomically defined tract, one that was chosen to follow serotonergic axons. Our goal is to characterise the binding patterns along this tract and also to determine how such patterns differ between control subjects and depressed patients. Due to the nature of our data, we utilise function-on-scalar regression modelling to make optimal use of our data. Inference on both main effects (position along the tract; diagnostic group) and their interactions are made using permutation testing strategies that do not require distributional assumptions. Also, to investigate the question of homogeneity we implement a permutation testing strategy, which adapts a 'block bootstrapping' approach from time series analysis to the functional data setting.