Neuroimaging data are a vital part of advancing our understanding of brain health and the etiology of neuropsychiatric disorders. But the inherent complexity and dimensionality of these data necessitate new statistical methods to produce replicable and interpretable results. In this work, we contribute to a growing body of methodological research in neuroimaging, with a focus on two main areas: feature extraction and hypothesis testing. Our first method is motivated by the need to mitigate the influence of nuisance variables (e.g., confounders) in observational studies of Alzheimer's disease. Specifically, it can be challenging to distinguish patterns in brain structure related to disease progression from those that may be attributable to healthy aging. To address this problem, we propose a new method involving penalized matrix decomposition, which identifies spatial patterns in the brain that are highly predictive of disease status but are uncorrelated with age. This is a step towards improving the utility of observational data in developing more interpretable neuroimaging-based biomarkers of neuropsychiatric disorders. Our second contribution is motivated by a growing interest in integrating information across multiple neuroimaging modalities (e.g., measures of brain structure and function). We describe a permutation-based hypothesis test of global intermodal associations using participant-level data. We then describe a method for testing and spatially localizing intermodal associations, where we leverage spatial information to enhance statistical power and interpretability. Collectively, these methods broaden the scope of questions that can be investigated using neuroimaging data.