Statistical methods for comparing brain connectomes at different scales Djalel-E. Meskaldji a,b , Stephan Morgenthaler c and Dimitri Van De Ville a,b a Institute of Bioengineering, ´ Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), Lausanne, Switzerland; b Department of Radiology and Medical Informatics,University of Geneva, Geneva, Switzerland; c FSB/MATHA, Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), Lausanne, Switzerland. ABSTRACT Physiological Brain connectivity and spontaneous interaction between regions of interest of the brain can be rep- resented by a matrix (full or sparse) or equivalently by a complex network called connectome. This representation of brain connectivity is adopted when comparing different patterns of structural and functional connectivity to null models or between groups of individuals. Two levels of comparison could be considered when analyzing brain connectivity: the global level and the local level. In the global level, the whole brain information is summarized by one summary statistic, whereas in the local analysis, each region of interest of the brain is summarized by a specific statistic. We show that these levels are mutually informatively integrative in some extent. We present different methods of analysis at both levels, the most relevant global and local network measures. We discuss as well the assumptions to be satisfied for each method; the error rates controlled by each method, and the challenges to overcome, especially, in the local case. We also highlight the possible factors that could influence the statistical results and the questions that have to be addressed in such analyses. Keywords: Brain connectivity, complex networks, multiple testing, type I error, family wise error rate, false discovery rate. 1. INTRODUCTION Detecting sparse signals from data is one of the most important goals of data analysis. The detection difficulty is influenced by the complexity of the data. In brain connectivity context, we seek to detect differences between groups of brain networks. These differences are expected to be sparse. A common representation of brain connectivity is via connectivity/adjacency matrices or equivalently by networks in the graph theory sense. In the brain connectivity literature, this representation is referred to the brain connectome . 1 In such representation, nodes/vertices of the network, or lines/columns of the connectivity matrix represent brain regions of interest (ROIs), while (weighted) edges/connections of the network or cells of the connectivity matrix characterizes a measure of connectivity between pairs of ROIs, i.e., how much two nodes/vertices are related/connected. This representation helped to understand brain organization and function, and has become an important aspect of neuroscience. 2, 3 Depending on the imaging modality, different types of connectivity can be obtained. Investigating differences in connectivity between distinct populations based on connectivity matrices is attractive, but also comes with a certain number of problems, 2, 4 among them, the high number of multiple comparisons. Different levels of analysis are considered to formulate the hypotheses, from the local level, where hypotheses are formulated on single units such as nodes and connections, up to the global level, where the brain network information is summarized in one measure. Statistical methods used for comparing connectomes depend on the level of the analysis as well as the modality and the multi-modality of the data. Further author information: (Send correspondence to D.E.M.) D.E.M.: E-mail: djalel.meskaldji@epfl.ch, Telephone: +41 (0)21 693 72 19 S.M.: E-mail: stephan.morgenthaler@epfl.ch, Telephone: +41 (0)21 693 42 32 D.V.D.V.: E-mail: dimitri.vandeville@epfl.ch, Telephone: +41 (0)21 693 96 69