Dissertation

Statistical Inference in fMri Using Random Field Theory and Resampling Methods

Davenport, Samuel

2021

Overview

This thesis provides a set of tools for analysing random images with a specific focus on applications in functional Magnetic Resonance Imaging (fMRI). To do so we employ Random Field Theory (RFT), a set of theoretical parametric results that can be used to analyse multidimensional random processes (known as random fields), and resampling methods, which draw samples from the data (with or without replacement). We extend the voxelwise inference framework of Worsley et al so that it provides accurate control of the familywise error rate in neuroimaging. We drop the standard RFT assumptions of high smoothness and stationarity and develop a quick parametric framework that provides powerful and valid inference even when the underlying data is non-Gaussian. We validate this using a massive resting state analysis, involving brain imaging data from 7000 subjects from the UK Biobank. We further use RFT techniques to derive an asymptotic distribution for the extent of a cluster above a threshold u in a non-stationary Gaussian random field as u → ∞. To do so we define the notion of horizontal-window (HW) conditioning and take advantage of recent advances (Cheng and Schwartzman (2015a)) on the HW-distribution of the height of a peak in a non-stationary Gaussian random field. Our results extend those of Nosko (1969) in which the asymptotic cluster size distribution is derived under the assumption of stationarity. In order to infer upon random fields whose mean is non-zero we derive asymptotic confidence regions for the location of a peak of the true signal given multiple realizations of random fields. These results are valid under non-stationarity and are derived using the theory of extremum estimators. Under the assumption of stationarity we improve upon these asymptotic results using a Monte Carlo approach that provides confidence regions for peaks of the mean which have better coverage in the finite sample. A second quantity of interest when considering fields whose mean is non-zero is the height of the true signal at the location of a peak in the observed random field. These peaks are typically subject to the winner's curse, which causes inflated effect sizes at peak locations (Vul et al., 2009). We develop a resampling based procedure that obtains low bias estimates of the true signal at the location of the peak. We validate this using task data from over 8000 subjects from the UK Biobank, setting aside 4000 subjects to compute a ground truth, and dividing the remaining subjects into small samples on which to test the results.

• Publisher: ProQuest Dissertations & Theses
• Subject: Biobanks