Statistical Tools for Samples of Weighted Networks with Applications to Neuroimaging

Abstract

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 develops 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 known community structure, or functional regions of the brain. The model allows for comparing two populations, such as patients and healthy controls, at both the system and the edge levels, with systems-level inference in particular allowing for a biologically meaningful interpretation. We incorporate correlation between edge weights by allowing a general variance structure, and show this leads to more accurate inference. An analysis comparing schizophrenics to healthy controls illustrates the full potential of our methods, and obtains results consistent with the medical literature.

While brain networks are the main focus of analysis, auxiliary information about subjects is frequently available. The subject's age is a particularly important covariate, since studying how the brain changes over time can lead to insights about brain development and aging. A typical neuroimaging study, however, is cross-sectional rather than longitudinal, with subjects of many different ages measured only once. We develop two models for such samples of multiple, time-stamped networks. One is a parametric linear mixed effects model with age included as a covariate; the other is a nonparametric method which can be viewed as a network version of principal component analysis, with components varying smoothly with age. Both approaches take network community structure into account and allow for concise and interpretable representation of the data by obtaining smooth developmental curves for functional regions of the brain. We apply the methods to fMRI dataset from the Philadelphia Neurodevelopmental Cohort study, whose subjects 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 many psychiatric conditions have distinct but not yet identified subtypes. Clustering brain connectivity networks of patients can lead to discovering these subtypes, and ideally to identifying the connectivity patterns that distinguish between subtypes. Clustering with a large number of features is challenging in itself, and the network nature of observations presents additional difficulties. We develop a clustering method that respects the network nature of the data, allows for feature selection, and scales well to high dimensions. We start from a general method for clustering and feature selection in high dimensions called sparse K-means, and develop a network-aware sparse K-means algorithm, using a network-induced penalty for simultaneously clustering weighted networks and performing feature selection. We also develop a Gaussian mixture model version, particularly useful when features are highly correlated, which is the case in neuroimaging. We illustrate the method on simulated networks and the Philadelphia Neurodevelopmental Cohort dataset.