Bayesian Analysis

Bayesian Inference and Testing of Group Differences in Brain Networks

Daniele Durante and David B. Dunson

Full-text: Open access


Network data are increasingly collected along with other variables of interest. Our motivation is drawn from neurophysiology studies measuring brain connectivity networks for a sample of individuals along with their membership to a low or high creative reasoning group. It is of paramount importance to develop statistical methods for testing of global and local changes in the structural interconnections among brain regions across groups. We develop a general Bayesian procedure for inference and testing of group differences in the network structure, which relies on a nonparametric representation for the conditional probability mass function associated with a network-valued random variable. By leveraging a mixture of low-rank factorizations, we allow simple global and local hypothesis testing adjusting for multiplicity. An efficient Gibbs sampler is defined for posterior computation. We provide theoretical results on the flexibility of the model and assess testing performance in simulations. The approach is applied to provide novel insights on the relationships between human brain networks and creativity.

Article information

Bayesian Anal., Volume 13, Number 1 (2018), 29-58.

First available in Project Euclid: 15 November 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

brain network mixture model multiple testing nonparametric Bayes

Creative Commons Attribution 4.0 International License.


Durante, Daniele; Dunson, David B. Bayesian Inference and Testing of Group Differences in Brain Networks. Bayesian Anal. 13 (2018), no. 1, 29--58. doi:10.1214/16-BA1030.

Export citation


  • Agresti, A. (2002). Categorical Data Analysis. Second edition. New York: Wiley.
  • Airoldi, E. M., Blei, D. M., Fienberg, S. E., and Xing, E. P. (2008). “Mixed membership stochastic blockmodels.” Journal of Machine Learning Research, 9: 1981–2014.
  • Arden, R., Chavez, R. S., Grazioplene, R., and Jung, R. E. (2010). “Neuroimaging creativity: A psychometric view.” Behavioural Brain Research, 214(2): 143–156.
  • Begg, M. D. and Lagakos, S. (1990). “On the consequences of model misspecification in logistic regression.” Environmental Health Perspectives, 87: 69–75.
  • Benjamini, Y. and Hochberg, Y. (1995). “Controlling the false discovery rate: A practical and powerful approach to multiple testing.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 57(1): 289–300.
  • Berger, J. O. and Delampady, M. (1987). “Testing precise hypotheses.” Statistical Science, 2(3): 317–335.
  • Berger, J. O. and Sellke, T. (1987). “Testing a point null hypothesis: The irreconcilability of P values and evidence.” Journal of the American Statistical Association, 82(397): 112–122.
  • Bhattacharya, A. and Dunson, D. B. (2011). “Sparse Bayesian infinite factor models.” Biometrika, 98(2): 291–306.
  • Bressler, S. L. and Menon, V. (2010). “Large-scale brain networks in cognition: Emerging methods and principles.” Trends in Cognitive Sciences, 14(6): 277–290.
  • Bullmore, E. and Sporns, O. (2009). “Complex brain networks: Graph theoretical analysis of structural and functional systems.” Nature Reviews Neuroscience, 10(3): 186–198.
  • Bullmore, E. and Sporns, O. (2012). “The economy of brain network organization.” Nature Reviews Neuroscience, 13(5): 336–349.
  • Canale, A. and Dunson, D. B. (2011). “Bayesian kernel mixtures for counts.” Journal of the American Statistical Association, 106(496): 1528–1539.
  • Carlsson, I., Wendt, P. E., and Risberg, J. (2000). “On the neurobiology of creativity. Differences in frontal activity between high and low creative subjects.” Neuropsychologia, 38(6): 873–885.
  • Craddock, R. C., Jbabdi, S., Yan, C.-G., Vogelstein, J. T., Castellanos, F. X., Martino, A. D., Kelly, C., Heberlein, K., Colcombe, S., and Milham, M. P. (2013). “Imaging human connectomes at the macroscale.” Nature Methods, 10(6): 524–539.
  • Daianu, M., Jahanshad, N., Nir, T. M., Toga, A. W., Jack, C. R., Weiner, M. W., and Thompson, P. M. (2013). “Breakdown of brain connectivity between normal aging and Alzheimer’s disease: A structural k-core network analysis.” Brain Connectivity, 3(4): 407–422.
  • Deegan, J. (1976). “The consequences of model misspecification in regression analysis.” Multivariate Behavioral Research, 11(2): 237–248.
  • Desikan, R. S., Ségonne, F., Fischl, B., Quinn, B. T., Dickerson, B. C., Blacker, D., Buckner, R. L., Dale, A. M., Maguire, R. P., Hyman, B. T., Albert, M. S., and Killiany, R. J. (2006). “An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest.” NeuroImage, 31(3): 968–980.
  • DiRienzo, A. G. and Lagakos, S. W. (2001). “Effects of model misspecification on tests of no randomized treatment effect arising from Cox’s proportional hazards model.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(4): 745–757.
  • Dunson, D. B. and Herring, A. H. (2005). “Bayesian latent variable models for mixed discrete outcomes.” Biostatistics, 6(1): 11–25.
  • Dunson, D. B. and Xing, C. (2009). “Nonparametric Bayes modeling of multivariate categorical data.” Journal of the American Statistical Association, 104(487): 1042–1051.
  • Durante, D. and Dunson, D. B. (2016). “Supplementary material for “Bayesian Inference and Testing of Group Differences in Brain Networks”.” Bayesian Analysis.
  • Durante, D., Dunson, D. B., and Vogelstein, J. T. (2016). “Nonparametric Bayes modeling of populations of networks.” Journal of the American Statistical Association, in press.
  • Fornito, A., Zalesky, A., and Breakspear, M. (2013). “Graph analysis of the human connectome: Promise, progress, and pitfalls.” NeuroImage, 80: 426–444.
  • Frank, O. and Strauss, D. (1986). “Markov graphs.” Journal of the American Statistical Association, 81(395): 832–842.
  • Fuster, J. M. (2000). “The Module: Crisis of a paradigm.” Neuron, 26(1): 51–53.
  • Fuster, J. M. (2006). “The cognit: A network model of cortical representation.” International Journal of Psychophysiology, 60(2): 125–132.
  • Gelman, A. and Rubin, D. B. (1992). “Inference from iterative simulation using multiple sequences.” Statistical Science, 7(4): 457–472.
  • Gelman, A., Van Dyk, D. A., Huang, Z., and Boscardin, J. W. (2008). “Using redundant parameterizations to fit hierarchical models.” Journal of Computational and Graphical Statistics, 17(1): 95–122.
  • Genovese, C. R., Lazar, N. A., and Nichols, T. (2002). “Thresholding of statistical maps in functional neuroimaging using the false discovery rate.” NeuroImage, 15(4): 870–878.
  • Ghosh, J. and Dunson, D. B. (2009). “Default prior distributions and efficient posterior computation in Bayesian factor analysis.” Journal of Computational and Graphical Statistics, 18(2): 306–320.
  • Ginestet, C. E., Balanchandran, P., Rosenberg, S., and Kolaczyk, E. D. (2014). “Hypothesis testing for network data in functional neuroimaging.” arXiv.1407.5525.
  • Gray Roncal, W., Koterba, Z. H., Mhembere, D., Kleissas, D. M., Vogelstein, J. T., Burns, R., Bowles, A. R., Donavos, D. K., Ryman, S., Jung, R. E., Wu, L., Calhoun, V., and Vogelstein, R. J. (2013). “MIGRAINE: MRI graph reliability analysis and inference for connectomics.” In IEEE Global Conference on Signal and Information Processing, 313–316. IEEE.
  • Heilman, K. M., Nadeau, S. E., and Beversdorf, D. O. (2003). “Creative innovation: Possible brain mechanisms.” Neurocase, 9(5): 369–379.
  • Hoff, P. (2008). “Modeling homophily and stochastic equivalence in symmetric relational data.” In Advances in Neural Information Processing Systems, 657–664.
  • Hoff, P. D., Raftery, A. E., and Handcock, M. S. (2002). “Latent space approaches to social network analysis.” Journal of the American Statistical Association, 97(460): 1090–1098.
  • Hunter, D. R., Goodreau, S. M., and Handcock, M. S. (2008a). “Goodness of fit of social network models.” Journal of the American Statistical Association, 103(481): 248–258.
  • Hunter, D. R., Handcock, M. S., Butts, C. T., Goodreau, S. M., and Morris, M. (2008b). “ergm: A package to fit, simulate and diagnose exponential-family models for networks.” Journal of Statistical Software, 24(3).
  • Jung, R. E., Segall, J. M., Bockholt, H. J., Flores, R. A., Smith, S. M., Chavez, R. S., and Haier, R. J. (2010). “Neuroanatomy of creativity.” Human Brain Mapping, 31(3): 398–409.
  • Kantarci, B. and Labatut, V. (2013). “Classification of complex networks based on topological properties.” In 2013 International Conference on Cloud and Green Computing, 297–304. IEEE.
  • Kass, R. E. and Raftery, A. E. (1995). “Bayes factors.” Journal of the American Statistical Association, 90(430): 773–795.
  • Krzanowski, W. (1988). Principles of Multivariate Analysis: A User’s Perspective. Oxford University Press.
  • Newton, M. A., Noueiry, A., Sarkar, D., and Ahlquist, P. (2004). “Detecting differential gene expression with a semiparametric hierarchical mixture method.” Biostatistics, 5(2): 155–176.
  • Nowicki, K. and Snijders, T. A. B. (2001). “Estimation and prediction for stochastic blockstructures.” Journal of the American Statistical Association, 96(455): 1077–1087.
  • Olde Dubbelink, K. T. E., Hillebrand, A., Stoffers, D., Deijen, J. B., Twisk, J. W. R., Stam, C. J., and Berendse, H. W. (2014). “Disrupted brain network topology in Parkinson’s disease: A longitudinal magnetoencephalography study.” Brain, 137(1): 197–207.
  • Polson, N. G., Scott, J. G., and Windle, J. (2013). “Bayesian inference for logistic models using Pólya–Gamma latent variables.” Journal of the American Statistical Association, 108(504): 1339–1349.
  • Ramsey, J. D., Hanson, S. J., Hanson, C., Halchenko, Y. O., Poldrack, R. A., and Glymour, C. (2010). “Six problems for causal inference from fMRI.” NeuroImage, 49(2): 1545–1558.
  • Rousseau, J. and Mengersen, K. (2011). “Asymptotic behaviour of the posterior distribution in overfitted mixture models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(5): 689–710.
  • Rubinov, M. and Sporns, O. (2010). “Complex network measures of brain connectivity: Uses and interpretations.” NeuroImage, 52(3): 1059–1069.
  • Sawyer, K. R. (2012). Explaining Creativity: The Science of Human Innovation. Oxford University Press.
  • Scott, J. G., Kelly, R. C., Smith, M. A., Zhou, P., and Kass, R. E. (2015). “False discovery rate regression: An application to neural synchrony detection in primary visual cortex.” Journal of the American Statistical Association, 110(510): 459–471.
  • Sellke, T., Bayarri, M. J., and Berger, J. O. (2001). “Calibration of p values for testing precise null hypotheses.” The American Statistician, 55(1): 62–71.
  • Shobe, E. R., Ross, N. M., and Fleck, J. I. (2009). “Influence of handedness and bilateral eye movements on creativity.” Brain and Cognition, 71(3): 204–214.
  • Simpson, S. L., Bowman, F. D., and Laurienti, P. J. (2013). “Analyzing complex functional brain networks: Fusing statistics and network science to understand the brain.” Statistics Surveys, 7: 1–36.
  • Simpson, S. L., Hayasaka, S., and Laurienti, P. J. (2011). “Exponential random graph modeling for complex brain networks.” PLoS ONE, 6(5): e20039.
  • Simpson, S. L., Moussa, M. N., and Laurienti, P. J. (2012). “An exponential random graph modeling approach to creating group-based representative whole-brain connectivity networks.” NeuroImage, 60(2): 1117–1126.
  • Smith, S. M., Miller, K. L., Salimi-Khorshidi, G., Webster, M., Beckmann, C. F., Nichols, T. E., Ramsey, J. D., and Woolrich, M. W. (2011). “Network modelling methods for FMRI.” NeuroImage, 54(2): 875–891.
  • Sporns, O. (2013). “Structure and function of complex brain networks.” Dialogues in Clinical Neuroscience, 15(3): 247–262.
  • Stam, C. J. (2014). “Modern network science of neurological disorders.” Nature Reviews Neuroscience, 15(10): 683–695.
  • Stirling, J. and Elliott, R. (2008). Introducing Neuropsychology. Routledge.
  • Takeuchi, H., Taki, Y., Sassa, Y., Hashizume, H., Sekiguchi, A., Fukushima, A., and Kawashima, R. (2010). “White matter structures associated with creativity: Evidence from diffusion tensor imaging.” NeuroImage, 51(1): 11–18.
  • Tansey, W., Koyejo, O., Poldrack, R. A., and Scott, J. G. (2014). “False discovery rate smoothing.” arXiv.1411.6144.
  • Wang, H. and Marron, J. (2007). “Object oriented data analysis: Sets of trees.” The Annals of Statistics, 35(5): 1849–1873.
  • Wang, J., He, L., Zheng, H., and Lu, Z.-L. (2014). “Optimizing the magnetization-prepared rapid gradient-echo (MP-RAGE) sequence.” PLoS ONE, 9(5): e96899.
  • Wasserman, S. and Pattison, P. (1996). “Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and $p^{*}$.” Psychometrika, 61(3): 401–425.
  • Zalesky, A., Fornito, A., and Bullmore, E. T. (2010). “Network-based statistic: Identifying differences in brain networks.” NeuroImage, 53(4): 1197–1207.

Supplemental materials

  • Supplement A: Supplementary Materials for “Bayesian Inference and Testing of Group Differences in Brain Networks”. The online supplementary material contains proofs of the Propositions 1, 2 and 3, providing theoretical support for our methodology.