## The Annals of Applied Statistics

### Hypothesis testing for network data in functional neuroimaging

#### Abstract

In recent years, it has become common practice in neuroscience to use networks to summarize relational information in a set of measurements, typically assumed to be reflective of either functional or structural relationships between regions of interest in the brain. One of the most basic tasks of interest in the analysis of such data is the testing of hypotheses, in answer to questions such as “Is there a difference between the networks of these two groups of subjects?” In the classical setting, where the unit of interest is a scalar or a vector, such questions are answered through the use of familiar two-sample testing strategies. Networks, however, are not Euclidean objects, and hence classical methods do not directly apply. We address this challenge by drawing on concepts and techniques from geometry and high-dimensional statistical inference. Our work is based on a precise geometric characterization of the space of graph Laplacian matrices and a nonparametric notion of averaging due to Fréchet. We motivate and illustrate our resulting methodologies for testing in the context of networks derived from functional neuroimaging data on human subjects from the 1000 Functional Connectomes Project. In particular, we show that this global test is more statistically powerful than a mass-univariate approach. In addition, we have also provided a method for visualizing the individual contribution of each edge to the overall test statistic.

#### Article information

Source
Ann. Appl. Stat., Volume 11, Number 2 (2017), 725-750.

Dates
Revised: November 2016
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.aoas/1500537721

Digital Object Identifier
doi:10.1214/16-AOAS1015

Mathematical Reviews number (MathSciNet)
MR3693544

Zentralblatt MATH identifier
06775890

#### Citation

Ginestet, Cedric E.; Li, Jun; Balachandran, Prakash; Rosenberg, Steven; Kolaczyk, Eric D. Hypothesis testing for network data in functional neuroimaging. Ann. Appl. Stat. 11 (2017), no. 2, 725--750. doi:10.1214/16-AOAS1015. https://projecteuclid.org/euclid.aoas/1500537721

#### References

• Achard, S., Salvador, R., Whitcher, B., Suckling, J. and Bullmore, E. (2006). A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. J. Neurosci. 26 63–72.
• Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, Hoboken, NJ.
• Arsigny, V., Fillard, P., Pennec, X. and Ayache, N. (2007). Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29 328–347.
• Aydin, B., Pataki, G., Wang, H., Bullitt, E. and Marron, J. S. (2009). A principal component analysis for trees. Ann. Appl. Stat. 3 1597–1615.
• Barden, D., Le, H. and Owen, M. (2013). Central limit theorems for Frechet means in the space of phylogenetic trees. Electron. J. Probab. 18 1–25.
• Beckmann, C. F., DeLuca, M., Devlin, J. T. and Smith, S. M. (2005). Investigations into resting-state connectivity using independent component analysis. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 360 1001–1013.
• Bhatia, R. (1997). Matrix Analysis. Graduate Texts in Mathematics 169. Springer, New York.
• Bhatia, R. (2007). Positive Definite Matrices. Princeton Univ. Press, Princeton, NJ.
• Bhattacharya, A. and Bhattacharya, R. (2012). Nonparametric Inference on Manifolds with Applications to Shape Spaces. Cambridge Univ. Press, New York.
• Bhattacharya, R. and Lin, L. (2017). Omnibus CLTs for Fréchet means and nonparametric inference on non-Euclidean spaces. Proc. Amer. Math. Soc. 145 413–428.
• Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 1–29.
• Bhattacharya, R. and Patrangenaru, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds. II. Ann. Statist. 33 1225–1259.
• Bhattacharya, R., Buibas, M., Dryden, I., Ellingson, L., Groisser, D., Hendriks, H., Huckemann, S., Le, H., Liu, X. and Marron, J. (2011). Extrinsic data analysis on sample spaces with a manifold stratification. In Advances in Mathematics, Invited Contributions at the Seventh Congress of Romanian Mathematicians, Brasov 148–156.
• Bickel, P. J. and Levina, E. (2008a). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604.
• Bickel, P. J. and Levina, E. (2008b). Regularized estimation of large covariance matrices. Ann. Statist. 36 199–227.
• Billera, L. J., Holmes, S. P. and Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Adv. in Appl. Math. 27 733–767.
• Biswal, B. B., Mennes, M., Zuo, X.-N., Gohel, S. and Kelly, C. et al. (2010). Toward discovery science of human brain function. Proc. Natl. Acad. Sci. USA 107 4734–4739.
• Bonnabel, S. and Sepulchre, R. (2009). Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank. SIAM J. Matrix Anal. Appl. 31 1055–1070.
• Bookstein, F. (1978). The Measurement of Biological Shape and Shape Change. Springer, London.
• Buckner, R. L., Andrews-Hanna, J. R. and Schacter, D. L. (2008). The brain’s default network: Anatomy, function and relevance to disease. Ann. N.Y. Acad. Sci. 1124 1–38.
• Bullmore, E. and Sporns, O. (2009). Complex brain networks: Graph theoretical analysis of structural and functional systems. Nat. Rev., Neurosci. 10 186–198.
• Bullmore, E. and Sporns, O. (2012). The economy of brain network organization. Nat. Rev., Neurosci. 13 336–349.
• Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation. J. Amer. Statist. Assoc. 106 672–684.
• Cai, T., Liu, W. and Luo, X. (2011). A constrained ${L}_{1}$ minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106 594–607.
• Chavel, I. (1984). Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics 115. Academic Press, Inc., Orlando, FL. Including a chapter by Burton Randol. With an appendix by Jozef Dodziuk.
• Cheng, S. H. and Higham, N. J. (1998). A modified Cholesky algorithm based on a symmetric indefinite factorization. SIAM J. Matrix Anal. Appl. 19 1097–1110.
• Chung, F. R. K. (1997). Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92. Amer. Math. Soc., Providence, RI.
• Dryden, I. L., Koloydenko, A. and Zhou, D. (2009). Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Ann. Appl. Stat. 3 1102–1123.
• El Karoui, N. (2008). Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Statist. 36 2717–2756.
• Ellegren, H. and Parsch, J. (2007). The evolution of sex-biased genes and sex-biased gene expression. Nat. Rev., Genet. 8 689–698.
• Fisher, R. (1953). Dispersion on a sphere. Proc. R. Soc. Lond. Ser. A 217 295–305.
• Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1987). Statistical Analysis of Spherical Data. Cambridge Univ. Press, Cambridge.
• Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincaré 10 215–310.
• Fu, Y. and Ma, Y. (2013). Graph Embedding for Pattern Analysis. Springer, New York.
• Ginestet, C. E., Fournel, A. P. and Simmons, A. (2014). Statistical network analysis for functional MRI: Summary networks and group comparisons. Front. Comput. Neurosci. 8 Art. ID 51.
• Ginestet, C. E. and Simmons, A. (2011). Statistical parametric network analysis of functional connectivity dynamics during a working memory task. NeuroImage 5 688–704.
• Ginestet, C. E., Li, J., Balachandran, P., Rosenberg, P. and Kolaczyk, E. D. (2017). Supplement to “Hypothesis testing for network data in functional neuroimaging.” DOI:10.1214/16-AOAS1015SUPP.
• Greicius, M. D., Krasnow, B., Reiss, A. L. and Menon, V. (2003). Functional connectivity in the resting brain: A network analysis of the default mode hypothesis. Proc. Natl. Acad. Sci. USA 100 253–258.
• Gromov, M. (2007). Metric Structures for Riemannian and Non-Riemannian Spaces, English ed. Birkhäuser, Inc., Boston, MA.
• Higham, N. J. (2002). Computing the nearest correlation matrix: A problem from finance. IMA J. Numer. Anal. 22 329–343.
• Hotz, T., Huckemann, S., Le, H., Marron, J. S., Mattingly, J. C., Miller, E., Nolen, J., Owen, M., Patrangenaru, V. and Skwerer, S. (2013). Sticky central limit theorems on open books. Ann. Appl. Probab. 23 2238–2258.
• Kang, H., Ombao, H., Linkletter, C., Long, N. and Badre, D. (2012). Spatio-spectral mixed-effects model for functional magnetic resonance imaging data. J. Amer. Statist. Assoc. 107 568–577.
• Kendall, D. G. (1977). The diffusion of shape. Adv. in Appl. Probab. 9 428–430.
• Kendall, D. G. (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16 81–121.
• Kendall, W. S. and Le, H. (2011). Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables. Braz. J. Probab. Stat. 25 323–352.
• Krishnamachari, R. T. and Varanasi, M. K. (2013). On the geometry and quantization of manifolds of positive semi-definite matrices. IEEE Trans. Signal Process. 61 4587–4599.
• Le, H. (2001). Locating Fréchet means with application to shape spaces. Adv. in Appl. Probab. 33 324–338.
• Le, H. and Kume, A. (2000). The Fréchet mean shape and the shape of the means. Adv. in Appl. Probab. 32 101–113.
• Lee, J. (2006). Introduction to Smooth Manifolds. Springer, London.
• Leon, P. S., Knock, S. A., Woodman, M. M., Domide, L., Mersmann, J., McIntosh, A. R. and Jirsa, V. (2013). The Virtual Brain: A simulator of primate brain network dynamics. Front. Neuroinform. 7 Art. ID 10.
• Linial, N. (2002). Finite metric spaces: Combinatorics, geometry and algorithms. In Proceedings of the Eighteenth Annual Symposium on Computational Geometry 63.
• Linial, N., London, E. and Rabinovich, Y. (1995). The geometry of graphs and some of its algorithmic applications. Combinatorica 15 215–245.
• Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Wiley, Cichester.
• McEwen, B. S. (1999). Permanence of brain sex differences and structural plasticity of the adult brain. Proc. Natl. Acad. Sci. USA 96 7128–7130.
• Micheloyannis, S., Vourkas, M., Tsirka, V., Karakonstantaki, E., Kanatsouli, K. and Stam, C. J. (2009). The influence of ageing on complex brain networks: A graph theoretical analysis. Hum. Brain Mapp. 30 200–208.
• Moakher, M. (2005). A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26 735–747.
• Moakher, M. and Zéraï, M. (2011). The Riemannian geometry of the space of positive-definite matrices and its application to the regularization of positive-definite matrix-valued data. J. Math. Imaging Vision 40 171–187.
• Newman, M. E. J. (2010). Networks: An Introduction. Oxford Univ. Press, Oxford.
• Pachou, E., Vourkas, M., Simos, P., Smit, D., Stam, C., Tsirka, V. and Micheloyannis, S. (2008). Working memory in schizophrenia: An EEG study using power spectrum and coherence analysis to estimate cortical activation and network behavior. Brain Topogr. 21 128–137.
• Schäfer, J. and Strimmer, K. (2005). A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. Stat. Appl. Genet. Mol. Biol. 4 Art. ID 32.
• Skwerer, S., Bullitt, E., Huckemann, S., Miller, E., Oguz, I., Owen, M., Patrangenaru, V., Provan, S. and Marron, J. (2014). Tree-oriented analysis of brain artery structure. J. Math. Imaging Vision 50 126–143.
• Thirion, B., Flandin, G., Pinel, P., Roche, A., Ciuciu, P. and Poline, J.-B. (2006). Dealing with the shortcomings of spatial normalization: Multi-subject parcellation of fMRI datasets. Hum. Brain Mapp. 27 678–693.
• Tomasi, D. and Volkow, N. D. (2010). Functional connectivity density mapping. Proc. Natl. Acad. Sci. USA 107 9885–9890.
• Tomasi, D. and Volkow, N. D. (2011). Gender differences in brain functional connectivity density. Hum. Brain Mapp. 33 849–860.
• Tzourio-Mazoyer, N., Landeau, B., Papathanassiou, D., Crivello, F., Etard, O., Delcroix, N., Mazoyer, B. and Joliot, M. (2002). Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain. NeuroImage 15 273–289.
• Wang, H. and Marron, J. (2007). Object oriented data analysis: Sets of trees. Ann. Statist. 35 1849–1873.
• Wang, J., Wang, L., Zang, Y., Yang, H., Tang, H., Gong, Q., Chen, Z., Zhu, C. and He, Y. (2009). Parcellation-dependent small-world brain functional networks: A resting-state fMRI study. Hum. Brain Mapp. 30 1511–1523.
• Watson, G. S. (1983). Statistics on Spheres. University of Arkansas Lecture Notes in the Mathematical Sciences 6. Wiley, New York.
• Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature 393 440–442.
• Xia, C. (2013). Eigenvalues in Riemannian Geometry. IMPA, Rio de Janeiro.
• Yan, S., Xu, D., Zhang, B., Zhang, H.-J., Yang, Q. and Lin, S. (2007). Graph embedding and extensions: A general framework for dimensionality reduction. IEEE Trans. Pattern Anal. Mach. Intell. 29 40–51.
• Yan, C.-G., Craddock, R. C., Zuo, X.-N., Zang, Y.-F. and Milham, M. P. (2013). Standardizing the intrinsic brain: Towards robust measurement of inter-individual variation in 1000 functional connectomes. NeuroImage 80 246–262.
• Ziezold, H. (1977). On expected figures and a strong law of large numbers for random elements in quasi-metric spaces. In Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the 1974 European Meeting of Statisticians.
• Zuo, X.-N., Ehmke, R., Mennes, M., Imperati, D., Castellanos, F. X., Sporns, O. and Milham, M. P. (2012). Network centrality in the human functional connectome. Cereb. Cortex 22 1862–1875.

#### Supplemental materials

• Proofs of theorems. Therein we here provide detailed proofs of the main results in this paper.