Hypothesis testing for network data in functional neuroimaging

Abstract

set of signals (usually time series) at each of a collection of pixels (in two dimensions) or voxels (in three dimensions). Building from such data, various forms of higher-level data representations are employed in neuroimaging. Traditionally, two-and three-dimensional images have, naturally, been the norm, but increasingly in recent years there has emerged a substantial interest in network-based representations.1.1. Motivation. Let G = (V, E) denote a graph, based on d = |V | vertices. In this setting, the vertices v ∈ V correspond to regions of interest (ROIs) in the brain, often pre-defined through considerations of the underlying neurobiology (e.g., the putamen or the cuneus). Edges {u, v} ∈ E between vertices u and v are used to denote a measure of association between the corresponding ROIs. Depending on the imaging modality used, the notion of 'association' may vary. For example, in diffusion tensor imaging (DTI), associations are taken to be representative of structural connectivity between brain regions. On the other hand, in functional magnetic resonance imaging (fMRI), associations are instead thought to represent functional connectivity, in the sense that the two regions of the brain participate together in the achievement of some higher-order function, often in the context of performing some task (e.g., counting from 1 to 10).With neuroimaging now a standard tool in clinical neuroscience, and with the advent of several major neuroscience research initiatives -perhaps most prominent being the recently announced Brain Research Accelerated by Innovative Neurotechnologies (BRAIN) initiative -we are quickly moving towards a time in which we will have available databases composed of large collections of secondary data in the form of network-based data objects. Faced with databases in which networks are a fundamental unit of data, it will be necessary to have in place the statistical tools to answer such questions as, "What is the 'average' of a collection of networks?" and "Do these networks differ, on average, from a given nominal network?," as well as "Do two collections of networks differ on average?" and "What factors (e.g., age, gender, etc.) appear to contribute to differences in networks?", or finally, say, "Has there been a change in the networks for a given subpopulation from yesterday to today?" In order to answer these and similar questions, we require network-based analogues of classical tools for statistical estimation and hypothesis testing.While these classical tools are among the most fundamental and ubiquitous in use in practice, their extension to network-based datasets, however, is not immediate and, in fact, can be expected to be highly non-trivial. The main challenge in such an extension is due to the simple fact that networks 2 are not Euclidean objects (for which classical methods were developed)rather, they are combinatorial objects, defined simply through their sets of vertices and edges. Nevertheless, our work here in this paper demonstrates that networks can be associated with ce...

Bayesian Inference and Testing of Group Differences in Brain Networks

“…By modeling the collection of group-dependent pmfs for the network-valued random variable via a flexible mixture of low-rank factorizations with group-specific mixing probabilities, we develop a simple global test for assessing evidence of group differences in the entire distribution of the network-valued random variable, rather than focusing inference only on changes in selected functionals. Differently from Ginestet et al (2014), our procedure additionally incorporates local testing for changes in edge probabilities across groups, in line with Scott et al (2015) -which in turn do not consider global tests. By explicitly borrowing strength within the network via matrix factorizations, we substantially improve power in our multiple local tests compared to standard FDR control procedures.…”

“…Instead of controlling FDR thresholds, Scott et al (2015) gain power in multiple testing by using auxiliary data -such as spatial proximity -to inform the posterior probability that specific pairs of nodes interact differently across groups or with respect to a baseline. Ginestet et al (2014) focus instead on assessing evidence of global changes in the brain structure by testing for group differences in the expected Laplacians. Scott et al (2015) and Ginestet et al (2014) substantially improve state of the art in local and global hypothesis testing for network data, respectively, but are characterized by a similar key issue, motivating our methodology.…”

“…Ginestet et al (2014) focus instead on assessing evidence of global changes in the brain structure by testing for group differences in the expected Laplacians. Scott et al (2015) and Ginestet et al (2014) substantially improve state of the art in local and global hypothesis testing for network data, respectively, but are characterized by a similar key issue, motivating our methodology. Specifically, previous procedures test for changes across groups in marginal (Scott et al, 2015) or expected (Ginestet et al, 2014) structures associated with the network-valued random variable, and hence cannot detect variations in the probabilistic generative mechanism that go beyond their focus.…”

“…Scott et al (2015) and Ginestet et al (2014) substantially improve state of the art in local and global hypothesis testing for network data, respectively, but are characterized by a similar key issue, motivating our methodology. Specifically, previous procedures test for changes across groups in marginal (Scott et al, 2015) or expected (Ginestet et al, 2014) structures associated with the network-valued random variable, and hence cannot detect variations in the probabilistic generative mechanism that go beyond their focus. Similarly to much simpler settings, substantially different joint probability mass functions (pmf) for a network-valued random variable can have equal expectation or induce the same marginal distributions -characterized by the edge probabilities.…”

“…The system of hypotheses (2.1)-(2.2) assesses evidence of global changes in the entire probability mass function, rather than on selected functionals or summary statistics, and hence is more general than Ginestet et al (2014) and joint tests on network measures.…”