Hypothesis Testing For Network Data in Functional Neuroimaging

In recent years, it has become common practice in neuroscience to use networks to summarize relational information in a set of measure- ments, typically assumed to be reective of either functional or structural relationships between regions of interest in the brain. One of the most basic tasks of interest in the analysis of such data is the testing of hypotheses, in answer to questions such as \'Is there a dierence between the networks of these two groups of subjects?" In the classical setting, where the unit of interest is a scalar or a vector, such questions are answered through the use of familiar two-sample testing strategies. Networks, however, are not Euclidean objects, and hence classical methods do not directly apply. We address this challenge by drawing on concepts and techniques from geom- etry, and high-dimensional statistical inference. Our work is based on a precise geometric characterization of the space of graph Laplacian matri- ces and a nonparametric notion of averaging due to Fr echet. We motivate and illustrate our resulting methodologies for testing in the context of net- works derived from functional neuroimaging data on human subjects from the 1000 Functional Connectomes Project. In particular, we show that this global test is more statistical powerful, than a mass-univariate approach. AMS 2000 subject classications: Fr echet mean, fMRI, Graph Lapla- cian, Hypothesis Testing, Matrix manifold, Network data, Neuroscience.