The Annals of Applied Statistics

Investigating differences in brain functional networks using hierarchical covariate-adjusted independent component analysis

Ran Shi and Ying Guo

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Human brains perform tasks via complex functional networks consisting of separated brain regions. A popular approach to characterize brain functional networks in fMRI studies is independent component analysis (ICA), which is a powerful method to reconstruct latent source signals from their linear mixtures. In many fMRI studies, an important goal is to investigate how brain functional networks change according to specific clinical and demographic variabilities. Existing ICA methods, however, cannot directly incorporate covariate effects in ICA decomposition. Heuristic post-ICA analysis to address this need can be inaccurate and inefficient. In this paper, we propose a hierarchical covariate-adjusted ICA (hc-ICA) model that provides a formal statistical framework for estimating covariate effects and testing differences between brain functional networks. Our method provides a more reliable and powerful statistical tool for evaluating group differences in brain functional networks while appropriately controlling for potential confounding factors. We present an analytically tractable EM algorithm to obtain maximum likelihood estimates of our model. We also develop a subspace-based approximate EM that runs significantly faster while retaining high accuracy. To test the differences in functional networks, we introduce a voxel-wise approximate inference procedure which eliminates the need of computationally expensive covariance matrix estimation and inversion. We demonstrate the advantages of our methods over the existing method via simulation studies. We apply our method to an fMRI study to investigate differences in brain functional networks associated with post-traumatic stress disorder (PTSD).

Article information

Ann. Appl. Stat., Volume 10, Number 4 (2016), 1930-1957.

Received: November 2014
Revised: April 2016
First available in Project Euclid: 5 January 2017

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fMRI blind source separation brain functional networks EM algorithm subspace concentration


Shi, Ran; Guo, Ying. Investigating differences in brain functional networks using hierarchical covariate-adjusted independent component analysis. Ann. Appl. Stat. 10 (2016), no. 4, 1930--1957. doi:10.1214/16-AOAS946.

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  • Anand, A., Li, Y., Wang, Y., Wu, J., Gao, S., Bukhari, L., Mathews, V. P., Kalnin, A. and Lowe, M. J. (2005). Activity and connectivity of brain mood regulating circuit in depression: A functional magnetic resonance study. Biol. Psychiatry 57 1079–1088.
  • Attias, H. (1999). Independent factor analysis. Neural Comput. 11 803–851.
  • Attias, H. (2000). A variational Bayesian framework for graphical models. Adv. Neural Inf. Process. Syst. 12 209–215.
  • Beck, A. T., Steer, R. A. and Carbin, M. G. (1988). Psychometric properties of the beck depression inventory: Twenty-five years of evaluation. Clin. Psychol. Rev. 8 77–100.
  • Beck, A. T., Steer, R. A., Brown, G. K. et al. (1996). Manual for the beck depression inventory-II.
  • Beckmann, C. F. and Smith, S. M. (2004). Probabilistic independent component analysis for functional magnetic resonance imaging. IEEE Trans. Med. Imag. 23 137–152.
  • Beckmann, C. F. and Smith, S. M. (2005). Tensorial extensions of independent component analysis for multisubject FMRI analysis. NeuroImage 25 294–311.
  • 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.
  • Beckmann, C. F., Mackay, C. E., Filippini, N. and Smith, S. M. (2009). Group comparison of resting-state FMRI data using multi-subject ICA and dual regression. NeuroImage 47 S148.
  • Biswal, B. B. and Ulmer, J. L. (1999). Blind source separation of multiple signal sources of fMRI data sets using independent component analysis. J. Comput. Assist. Tomogr. 23 265–271.
  • 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., Brammer, M., Williams, S. C., Rabe-Hesketh, S., Janot, N., David, A., Mellers, J., Howard, R. and Sham, P. (1996). Statistical methods of estimation and inference for functional MR image analysis. Magn. Reson. Med. 35 261–277.
  • Calhoun, V. D., Adali, T., Pearlson, G. D. and Pekar, J. J. (2001). A method for making group inferences from functional MRI data using independent component analysis. Hum. Brain Mapp. 14 140–151.
  • Campbell, D. G., Felker, B. L., Liu, C.-F., Yano, E. M., Kirchner, J. E., Chan, D., Rubenstein, L. V. and Chaney, E. F. (2007). Prevalence of depression–PTSD comorbidity: Implications for clinical practice guidelines and primary care-based interventions. Journal of General Internal Medicine 22 711–718.
  • Chen, C.-H., Ridler, K., Suckling, J., Williams, S., Fu, C. H., Merlo-Pich, E. and Bullmore, E. (2007). Brain imaging correlates of depressive symptom severity and predictors of symptom improvement after antidepressant treatment. Biol. Psychiatry 62 407–414.
  • Cole, L. J., Farrell, M. J., Gibson, S. J. and Egan, G. F. (2010). Age-related differences in pain sensitivity and regional brain activity evoked by noxious pressure. Neurobiol. Aging 31 494–503.
  • Daniels, J. K., Frewen, P., McKinnon, M. C. and Lanius, R. A. (2011). Default mode alterations in posttraumatic stress disorder related to early-life trauma: A developmental perspective. J. Psychiatry Neurosci. 36 56–59.
  • Daubechies, I., Roussos, E., Takerkart, S., Benharrosh, M., Golden, C., D’ardenne, K., Richter, W., Cohen, J. D. and Haxby, J. (2009). Independent component analysis for brain fMRI does not select for independence. Proc. Natl. Acad. Sci. USA 106 10415–10422.
  • Filippini, N., MacIntosh, B. J., Hough, M. G., Goodwin, G. M., Frisoni, G. B., Smith, S. M., Matthews, P. M., Beckmann, C. F. and Mackay, C. E. (2009). Distinct patterns of brain activity in young carriers of the APOE-$\varepsilon$4 allele. Proc. Natl. Acad. Sci. USA 106 7209–7214.
  • First, M. B. (1995). Structured clinical interview for the DSM (SCID). In The Encyclopedia of Clinical Psychology. American Psychiatric Press, Washington, DC.
  • Genovese, C. R., Lazar, N. A. and Nichols, T. (2002). Thresholding of statistical maps in functional neuroimaging using the false discovery rate. NeuroImage 15 870–878.
  • Greicius, M. D., Flores, B. H., Menon, V., Glover, G. H., Solvason, H. B., Kenna, H., Reiss, A. L. and Schatzberg, A. F. (2007). Resting-state functional connectivity in major depression: Abnormally increased contributions from subgenual cingulate cortex and thalamus. Biol. Psychiatry 62 429–437.
  • Griffanti, L., Salimi-Khorshidi, G., Beckmann, C. F., Auerbach, E. J., Douaud, G., Sexton, C. E., Zsoldos, E., Ebmeier, K. P., Filippini, N., Mackay, C. E. et al. (2014). ICA-based artefact removal and accelerated fMRI acquisition for improved resting state network imaging. NeuroImage 95 232–247.
  • Guo, Y. (2011). A general probabilistic model for group independent component analysis and its estimation methods. Biometrics 67 1532–1542.
  • Guo, Y. and Pagnoni, G. (2008). A unified framework for group independent component analysis for multi-subject fMRI data. NeuroImage 42 1078–1093.
  • Guo, Y. and Tang, L. (2013). A hierarchical model for probabilistic independent component analysis of multi-subject fMRI studies. Biometrics 69 970–981.
  • Hendler, T., Rotshtein, P., Yeshurun, Y., Weizmann, T., Kahn, I., Ben-Bashat, D., Malach, R. and Bleich, A. (2003). Sensing the invisible: Differential sensitivity of visual cortex and amygdala to traumatic context. NeuroImage 19 587–600.
  • Himberg, J., Hyvärinen, A. and Esposito, F. (2004). Validating the independent components of neuroimaging time series via clustering and visualization. NeuroImage 22 1214–1222.
  • Hyvärinen, A., Karhunen, J. and Oja, E. (2001). Independent Component Analysis 46. Wiley, New York.
  • Hyvärinen, A. and Oja, E. (2000). Independent component analysis: Algorithms and applications. Neural Netw. 13 411–430.
  • Kessler, R. C., Sonnega, A., Bromet, E., Hughes, M. and Nelson, C. B. (1995). Posttraumatic stress disorder in the national comorbidity survey. Arch. Gen. Psychiatry 52 1048–1060.
  • Kostantinos, N. (2000). Gaussian mixtures and their applications to signal processing. In Advanced Signal Processing Handbook. CRC Press, New York.
  • Lee, S., Shen, H., Truong, Y., Lewis, M. and Huang, X. (2011). Independent component analysis involving autocorrelated sources with an application to functional magnetic resonance imaging. J. Amer. Statist. Assoc. 106 1009–1024.
  • Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44 226–233.
  • Mckeown, M. J., Makeig, S., Brown, G. G., Jung, T.-P., Kindermann, S. S., Kindermann, R. S., Bell, A. J. and Sejnowski, T. J. (1998). Analysis of fMRI data by blind separation into independent spatial components. Hum. Brain Mapp. 6 160–188.
  • McLachlan, G. and Peel, D. (2004). Finite Mixture Models. Wiley, New York.
  • Meilijson, I. (1989). A fast improvement to the EM algorithm on its own terms. J. Roy. Statist. Soc. Ser. B 51 127–138.
  • Meng, X.-L. and Rubin, D. B. (1991). Using EM to obtain asymptotic variance-covariance matrices: The SEM algorithm. J. Amer. Statist. Assoc. 86 899–909.
  • Minka, T. P. (2000). Automatic choice of dimensionality for PCA. In NIPS 13 598–604. MIT Press, Cambridge, MA.
  • Quiton, R. L. and Greenspan, J. D. (2007). Sex differences in endogenous pain modulation by distracting and painful conditioning stimulation. Pain 132 Suppl 1 S134–S149.
  • Raichle, M. E., MacLeod, A. M., Snyder, A. Z., Powers, W. J., Gusnard, D. A. and Shulman, G. L. (2001). A default mode of brain function. Proc. Natl. Acad. Sci. USA 98 676–682.
  • Reineberg, A. E., Andrews-Hanna, J. R., Depue, B. E., Friedman, N. P. and Banich, M. T. (2015). Resting-state networks predict individual differences in common and specific aspects of executive function. NeuroImage 104 69–78.
  • Seber, G. A. and Lee, A. J. (2012). Linear Regression Analysis 936. Wiley, New York.
  • Sheline, Y. I., Barch, D. M., Price, J. L., Rundle, M. M., Vaishnavi, S. N., Snyder, A. Z., Mintun, M. A., Wang, S., Coalson, R. S. and Raichle, M. E. (2009). The default mode network and self-referential processes in depression. Proc. Natl. Acad. Sci. USA 106 1942–1947.
  • Shi, R. and Guo, Y. (2016). Supplement to “Investigating differences in brain functional networks using hierarchical covariate-adjusted independent component analysis.” DOI:10.1214/16-AOAS946SUPP.
  • Smith, S. M., Fox, P. T., Miller, K. L., Glahn, D. C., Fox, P. M., Mackay, C. E., Filippini, N., Watkins, K. E., Toro, R., Laird, A. R. et al. (2009). Correspondence of the brain’s functional architecture during activation and rest. Proc. Natl. Acad. Sci. USA 106 13040–13045.
  • Smith, D. V., Utevsky, A. V., Bland, A. R., Clement, N., Clithero, J. A., Harsch, A. E., Carter, R. M. and Huettel, S. A. (2014). Characterizing individual differences in functional connectivity using dual-regression and seed-based approaches. NeuroImage 95 1–12.
  • Tohka, J., Foerde, K., Aron, A. R., Tom, S. M., Toga, A. W. and Poldrack, R. A. (2008). Automatic independent component labeling for artifact removal in fMRI. NeuroImage 39 1227–1245.
  • Whitfield-Gabrieli, S., Thermenos, H. W., Milanovic, S., Tsuang, M. T., Faraone, S. V., McCarley, R. W., Shenton, M. E., Green, A. I., Nieto-Castanon, A., LaViolette, P. et al. (2009). Hyperactivity and hyperconnectivity of the default network in schizophrenia and in first-degree relatives of persons with schizophrenia. Proc. Natl. Acad. Sci. USA 106 1279–1284.
  • Xu, L., Cheung, C., Yang, H. and Amari, S. (1997). Maximum equalization by entropy maximization and mixture of cumulative distribution functions. In Proc. of ICNN’97 1821–1826. Springer, New York.

Supplemental materials

  • Supplement to the paper “Investigating differences in brain functional networks using hierarchical covariate-adjusted independent component analysis”. This document presents the following contents: details about our EM algorithm and the approximate EM algorithm; the proof of Theorem 1; the criteria of selecting activating voxels within each brain network; the specification of initial guesses to our algorithm; additional simulation results with $q=10$; additional simulation results and theoretical results on the robustness of the approximate EM; additional results on the robustness of our method in real data analysis.