Neuroimaging data on functional connections in the brain are frequently represented by weighted networks. These networks share the same set of labeled nodes corresponding to a fixed atlas of the brain, while each subject’s network has their own edge weights. This thesis focuses on developing statistical tools for analyzing samples of weighted networks with applications to neuroimaging.

We first propose a method for modeling such brain networks via linear mixed effects models, which takes advantage of the community structure, or functional regions, known to be present in the brain. The model allows for comparing two populations, such as patients and healthy controls, globally, at functional systems level, and at individual edge level, with systems-level inference in particular allowing for a biologically meaningful interpretation. We incorporate correlation between edge weights into the model by allowing for a general variance structure, and show this leads to much more accurate inference. A thorough study comparing schizophrenics to healthy controls illustrates the full potential of our methods, and obtains results consistent with the medical literature on schizophrenia.

While we focus on networks as the main object of analysis, auxillary information about subjects is frequently available. The subject’s age is a particularly important covariance, since studying how the brain changes over time can lead to insights about brain development in children and adolescents and the effects of aging for older subjects. A typical neuroimaging study, however, is cross-sectional rather than longitudinal, meaning we measure subjects of many different ages, but only once. We developed two methods for analyzing such samples of multiple, time-stamped networks. One is a parametric approach utilizing a linear mixed effects model with age included as a covariate; the other one is a nonparametric method which can be viewed as a network version of principal component analysis, where we look for components that explain age-related trends and vary smoothly with age. Both approaches take network community structure into account and allow for concise and interpretable representation of the data by obtaining developmental curves for functional regions of the brain that vary smoothly with age. We apply the methods to fMRI data of subjects who are 8 to 22 years old, and extract developmental curves consistent with the current understanding of brain maturation in neuroscience.

Clustering is of special interest in neuroimaging studies of mental illness, because psychiatrists believe that many psychiatric conditions present in multiple distinct and not yet identified subtypes. Clustering brain connectivity networks of patients with a certain disorder can lead to discovering these subtypes, and ideally identifying the differences in connectivity patterns that distinguish between subtypes. Clustering with a large number of features is challenging in itself, and the network nature of the observations presents additional difficulties. Our goal is to develop a clustering method that respects the network nature of the data, allows for feature selection, and scales well to high dimensions. One general method for clustering and feature selection in high dimensions is sparse K-means, which performs feature selection by minimizing the K-means objective function plus a lasso penalty. Here we develop network-aware sparse K-means, using a network-induced penalty for simultaneously clustering weighted networks and performing feature selection. We also develop a Gaussian mixture model version of the algorithm, particularly useful when features are highly correlated, which is the case in neuroimaging. We illustrate the method on simulated networks and an fMRI dataset of youth.