  Statistical analysis of multimodal imaging data is a challenging task, since
the data involves high-dimensionality, strong spatial correlations and complex
data structures. In this paper, we propose rigorous statistical testing
procedures for making inferences on the complex dependence of multimodal
imaging data. Motivated by the analysis of multi-task fMRI data in the Human
Connectome Project (HCP) study, we particularly address three hypothesis
testing problems: (a) testing independence among imaging modalities over brain
regions, (b) testing independence between brain regions within imaging
modalities, and (c) testing independence between brain regions across different
modalities. Considering a general form for all the three tests, we develop a
global testing procedure and a multiple testing procedure controlling the false
discovery rate. We study theoretical properties of the proposed tests and
develop a computationally efficient distributed algorithm. The proposed methods
and theory are general and relevant for many statistical problems of testing
independence structure among the components of high-dimensional random vectors
with arbitrary dependence structures. We also illustrate our proposed methods
via extensive simulations and analysis of five task fMRI contrast maps in the
HCP study.
