In this thesis, we aim to address two topical issues at the forefront of task-based functional magnetic resonance imaging (fMRI). The first of these is a growing apprehension within the field about the reproducibility of findings that make up the neuroimaging literature. To confront this, we assess how the choice of software package for analyzing fMRI data can impact the final group-level results of a neuroimaging study. We reanalyze data from three published task-fMRI studies within the three most widely-used neuroimaging software packages -- AFNI, FSL, and SPM -- and then apply a range of comparison methods to gauge the scale of variability across the results. While qualitatively we find similarities, our quantitative assessment methods discover considerable differences between the final statistical images obtained with each package. Ultimately, we conclude that exceedingly weak effects may not generalize across fMRI analysis software. In the second part of this work we shift our attention to the analytical methods applied for fMRI inference. Here, we seek to overcome limitations with the traditional statistical approach, where for sufficiently large data sizes current methods determine universal activation across the brain, rendering the results as uninterpretable. We extend on a method proposed by (Sommerfeld et al., 2018; SSS) to develop spatial Confidence Sets (CSs) on clusters found in thresholded raw blood-oxygen-level-dependent (BOLD) effect size maps. The CSs give statements on the locations where raw effect sizes exceed, and fall short of, a purposeful non-zero threshold. We propose several theoretical and practical implementation advancements to the original method formulated in SSS, delivering a procedure with superior performance in sample sizes as low as N = 60. We validate the method with 3D Monte Carlo simulations that resemble fMRI data. We then compute CSs for the Human Connectome Project (HCP) working memory task contrast images, illustrating the brain regions that show a reliable %BOLD for a given %BOLD threshold. In the final part of this thesis, we develop the CSs to operate on standardized Cohen's d effect size images. We derive the statistical properties of the Cohen's d estimator to motivate three algorithms for computing Cohen's d CSs, including a novel method based on normalizing the distribution of Cohen's d. With intensive 3D Monte Carlo simulations, we find that two of these methods can be effectively applied to fMRI data. We compute Cohen's d CSs on the HCP data, and by comparing the CSs with results obtained from a standard testing procedure, exemplify the improved localization of effects that can be gained by using the Confidence Sets.
3D Monte Carlo simulations on the control of the aggregate structures of cubic magnetic particles in terms of an external electric field
