  Gaussian Graphical Models (GGM) are popularly used in neuroimaging studies
based on fMRI, EEG or MEG to estimate functional connectivity, or relationships
between remote brain regions. In multi-subject studies, scientists seek to
identify the functional brain connections that are different between two groups
of subjects, i.e. connections present in a diseased group but absent in
controls or vice versa. This amounts to conducting two-sample large scale
inference over network edges post graphical model selection, a novel problem we
call Population Post Selection Inference. Current approaches to this problem
include estimating a network for each subject, and then assuming the subject
networks are fixed, conducting two-sample inference for each edge. These
approaches, however, fail to account for the variability associated with
estimating each subject's graph, thus resulting in high numbers of false
positives and low statistical power. By using resampling and random
penalization to estimate the post selection variability together with proper
random effects test statistics, we develop a new procedure we call $R^{3}$ that
solves these problems. Through simulation studies we show that $R^{3}$ offers
major improvements over current approaches in terms of error control and
statistical power. We apply our method to identify functional connections
present or absent in autistic subjects using the ABIDE multi-subject fMRI
study.
