We consider group-level statistical inference for networks, where outcomes are multivariate edge variables constrained in an adjacency matrix. The graph notation is used to represent a network, where nodes are identical biological units (e.g. brain regions) shared across subjects and edge-variables indicate the strengths of interactive relationships between nodes. Edge-variables vary across subjects and may be associated with covariates of interest. The statistical inference for multivariate edge-variables is challenging because both localized inference on individual edges and the joint inference of a combinatorial of edges (network-level) are desired. Different from conventional multivariate variables (e.g. omics data), the inference of a combinatorial of edges is closely linked with network topology and graph combinatorics. We propose a novel objective function with 𝓁 norm regularization to robustly capture subgraphs/subnetworks from the whole brain connectome and thus reveal the latent network topology of phenotype-related edges. Our statistical inferential procedure and theories are constructed based on graph combinatorics. We apply the proposed approach to a brain connectome study to identify latent brain functional subnetworks that are associated with schizophrenia and verify the findings using an independent replicate data set. The results demonstrate that the proposed method achieves superior performance with remarkably increased replicability.
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