The Annals of Applied Statistics

Spatial Bayesian variable selection and grouping for high-dimensional scalar-on-image regression

Fan Li, Tingting Zhang, Quanli Wang, Marlen Z. Gonzalez, Erin L. Maresh, and James A. Coan

Full-text: Open access

Abstract

Multi-subject functional magnetic resonance imaging (fMRI) data has been increasingly used to study the population-wide relationship between human brain activity and individual biological or behavioral traits. A common method is to regress the scalar individual response on imaging predictors, known as a scalar-on-image (SI) regression. Analysis and computation of such massive and noisy data with complex spatio-temporal correlation structure is challenging. In this article, motivated by a psychological study on human affective feelings using fMRI, we propose a joint Ising and Dirichlet Process (Ising-DP) prior within the framework of Bayesian stochastic search variable selection for selecting brain voxels in high-dimensional SI regressions. The Ising component of the prior makes use of the spatial information between voxels, and the DP component groups the coefficients of the large number of voxels to a small set of values and thus greatly reduces the posterior computational burden. To address the phase transition phenomenon of the Ising prior, we propose a new analytic approach to derive bounds for the hyperparameters, illustrated on 2- and 3-dimensional lattices. The proposed method is compared with several alternative methods via simulations, and is applied to the fMRI data collected from the KLIFF hand-holding experiment.

Article information

Source
Ann. Appl. Stat., Volume 9, Number 2 (2015), 687-713.

Dates
Received: October 2014
Revised: February 2015
First available in Project Euclid: 20 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1437397107

Digital Object Identifier
doi:10.1214/15-AOAS818

Mathematical Reviews number (MathSciNet)
MR3371331

Zentralblatt MATH identifier
06499926

Keywords
Bayesian Dirichlet Process fMRI Ising model phase transition scalar-on-image regression stochastic search variable selection

Citation

Li, Fan; Zhang, Tingting; Wang, Quanli; Gonzalez, Marlen Z.; Maresh, Erin L.; Coan, James A. Spatial Bayesian variable selection and grouping for high-dimensional scalar-on-image regression. Ann. Appl. Stat. 9 (2015), no. 2, 687--713. doi:10.1214/15-AOAS818. https://projecteuclid.org/euclid.aoas/1437397107


Export citation

References

  • Allen, J. P., Porter, M., McFarland, F. C., McElhaney, K. B. and Marsh, P. (2007). The relation of attachment security to adolescents’ paternal and peer relationships, depression, and externalizing behavior. Child Development 78 1222–1239.
  • Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2 1152–1174.
  • Beckes, L. and Coan, J. A. (2011). Social baseline theory: The role of social proximity in emotion and economy of action. Social and Personality Psychology Compass 5 976–988.
  • Bowman, F. D. (2007). Spatiotemporal models for region of interest analyses of functional neuroimaging data. J. Amer. Statist. Assoc. 102 442–453.
  • Bowman, F. D., Caffo, B., Bassett, S. S. and Kilts, C. (2008). A Bayesian hierarchical framework for spatial modeling of fMRI data. NeuroImage 39 146–156.
  • Bradley, M. M. and Lang, P. J. (1994). Measuring emotion: The self-assessment mankin and the semantic differential. J. Behav. Ther. Exp. Psychiatry 25 49–59.
  • Coan, J. A. (2010). Adult attachment and the brain. J. Soc. Pers. Relatsh. 27 210–217.
  • Coan, J. A. (2011). The social regulation of emotion. In Oxford Handbook of Social Neuroscience 614–623. Oxford Univ. Press, New York.
  • Coan, J. A., Beckes, L. and Allen, J. P. (2013). Childhood maternal support and social capital moderate the regulatory impact of social relationships in adulthood. Int. J. Psychophysiol. 88 224–231.
  • Coan, J. A. and Maresh, E. L. (2014). Social baseline theory and the social regulation of emotion. In The Handbook of Emotion Regulation, 2nd ed. (J. Gross, ed.) 221–236. The Guilford Press, New York.
  • Coan, J. A., Schaefer, H. S. and Davidson, R. J. (2006). Lending a hand: Social regulation of the neural response to threat. Psychol. Sci. 17 1032–1039.
  • Craig, A. D. (2009). How do you fell now? The anterior insula and human awareness. Nat. Rev. Neurosci. 10 59–70.
  • Critchley, H. D., Corfield, D. R., Chandler, M. P., Mathias, C. J. and Dolan, R. J. (2000). Cerebral correlates of autonomic cardiovascular arousal: A functional neuroimaging investigation in humans. J. Physiol. (Lond.) 523 259–270.
  • Derado, G., Bowman, F. D. and Kilts, C. D. (2010). Modeling the spatial and temporal dependence in fMRI data. Biometrics 66 949–957.
  • Dunson, D. B., Herring, A. H. and Engel, S. M. (2008). Bayesian selection and clustering of polymorphisms in functionally related genes. J. Amer. Statist. Assoc. 103 534–546.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209–230.
  • Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615–629.
  • Friston, K. J., Holmes, A. P., Worsley, K., Poline, P. J., Frith, C. and Frackowiak, R. (1995). Statistical parametric maps in functional imaging: A general linear approach. Hum. Brain Mapp. 2 189–210.
  • Ge, T., Müller-Lenke, N., Bendfeldt, K., Nichols, T. E. and Johnson, T. D. (2014). Analysis of multiple sclerosis lesions via spatially varying coefficients. Ann. Appl. Stat. 8 1095–1118.
  • Gelman, A. E. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457–472.
  • George, E. and McCulloch, R. E. (1993). Variable selection via Gibbs sampling. J. Amer. Statist. Assoc. 88 881–889.
  • George, E. and McCulloch, R. E. (1997). Approaches for Bayesian variable selection. Statist. Sinica 7 339–373.
  • Goldsmith, J., Huang, L. and Crainiceanu, C. M. (2014). Smooth scalar-on-image regression via spatial Bayesian variable selection. J. Comput. Graph. Statist. 23 46–64.
  • Gössl, C., Auer, D. P. and Fahrmeir, L. (2001). Bayesian spatiotemporal inference in functional magnetic resonance imaging. Biometrics 57 554–562.
  • Huang, L., Goldsmith, J., Reiss, P. T., Reich, D. S. and Crainiceanu, C. M. (2013). Bayesian scalar-on-image regression with application to association between intracranial DTI and cognitive outcomes. NeuroImage 83 210–223.
  • Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161–173.
  • Ishwaran, H. and Zarepour, M. (2000). Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models. Biometrika 87 371–390.
  • Jenkinson, M., Bannister, P., Brady, M. and Smith, S. (2002). Improved optimization for the robust and accurate linear registration and motion correction of brain images. NeuroImage 17 825–841.
  • Johnson, T. D., Liu, Z., Bartsch, A. J. and Nichols, T. E. (2013). A Bayesian non-parametric Potts model with application to pre-surgical FMRI data. Stat. Methods Med. Res. 22 364–381.
  • Kalus, S., Sämann, P. G. and Fahrmeir, L. (2014). Classification of brain activation via spatial Bayesian variable selection in fMRI regression. Adv. Data Anal. Classif. 8 63–83.
  • Kang, J., Johnson, T. D., Nichols, T. E. and Wager, T. D. (2011). Meta analysis of functional neuroimaging data via Bayesian spatial point processes. J. Amer. Statist. Assoc. 106 124–134.
  • Kim, S., Tadesse, M. G. and Vannucci, M. (2006). Variable selection in clustering via Dirichlet process mixture models. Biometrika 93 877–893.
  • Lang, P. J., Greenwald, M. K., Bradley, M. M. and Hamm, A. O. (1993). Looking at pictures: Affective, facial, visceral, and behavioral reactions. Psychophysiology 30 261–273.
  • Lange, K. (2008). Optimization. Springer Texts in Statistics 95. Springer, New York.
  • Lewis, P. A., Critchley, H. D., Rotshtein, P. and Dolan, R. J. (2007). Neural correlates of processing valence and arousal in affective words. Cereb. Cortex 17 742–748.
  • Li, F. and Zhang, N. R. (2010). Bayesian variable selection in structured high-dimensional covariate spaces with applications in genomics. J. Amer. Statist. Assoc. 105 1202–1214.
  • Li, F., Zhang, T., Wang, Q., Gonzalez, M., Maresh, E. L. and Coan, J. A. (2015). Supplement to “Spatial Bayesian variable selection and grouping for high-dimensional scalar-on-image regression.” DOI:10.1214/15-AOAS818SUPP.
  • Maresh, E. L., Beckes, L. and Coan, J. A. (2013). The social regulation of threat-related attentional disengagement in highly anxious individuals. Front. Human Neurosci. 7 515.
  • Mitchell, T. J. and Beauchamp, J. J. (1988). Bayesian variable selection in linear regression. J. Amer. Statist. Assoc. 83 1023–1036.
  • Park, T. and Casella, G. (2008). The Bayesian lasso. J. Amer. Statist. Assoc. 103 681–686.
  • Penny, W. D., Trujillo-Barreto, N. J. and Friston, K. J. (2005). Bayesian fMRI time series analysis with spatial priors. NeuroImage 24 350–362.
  • Raftery, A. E. (1996). Approximate Bayes factors and accounting for model uncertainty in generalised linear models. Biometrika 83 251–266.
  • Reiss, P. T., Mennes, M., Petkova, E., Huang, L., Hoptman, M. J., Biswal, B. B., Colcombe, S. J., Zuo, X.-N. and Milham, M. P. (2011). Extracting information from functional connectivity maps via function-on-scalar regression. NeuroImage 56 140–148.
  • Reiss, P. T., Huo, L., Zhao, Y., Kelly, C. and Ogden, R. T. (2015). Wavelet-domain regression and predictive inference in psychiatric neuroimaging. Ann. Appl. Stat. 9 1076–1101.
  • Russell, J. (1980). A circumplex model of affect. J. Pers. Soc. Psychol. 39 1161–1178.
  • Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statist. Sinica 4 639–650.
  • Smith, M. and Fahrmeir, L. (2007). Spatial Bayesian variable selection with application to functional magnetic resonance imaging. J. Amer. Statist. Assoc. 102 417–431.
  • Smith, M. and Kohn, R. (1996). Nonparametric regression using Bayesian variable selection. J. Econometrics 75 317–343.
  • Smith, M., Pütz, B., Auer, D. and Fahrmeir, L. (2003). Assessing brain activity through spatial Bayesian variable selection. NeuroImage 20 802–815.
  • Smith, S. M., Jenkinson, M., Woolrich, M. W., Beckmann, C. F., Behrens, T. E. J., Johansen-Berg, H., Bannister, P. R., De Luca, M., Drobnjak, I., Flitney, D. E., Niazy, R., Saunders, J., Vickers, J., Zhang, Y., De Stefano, N., Brady, J. M. and Matthews, P. M. (2004). In advances in functional and structural MR image analysis and implementation as FSL. NeuroImage 23(S1) 208–219.
  • Stanley, H. E. (1987). Introduction to Phase Transitions and Critical Phenomena. Oxford Univ. Press, New York.
  • Stingo, F. C., Chen, Y. A., Tadesse, M. G. and Vannucci, M. (2011). Incorporating biological information into linear models: A Bayesian approach to the selection of pathways and genes. Ann. Appl. Stat. 5 1978–2002.
  • Suchard, M. A., Wang, Q., Chan, C., Frelinger, J., Cron, A. and West, M. (2010). Understanding GPU programming for statistical computation: Studies in massively parallel massive mixtures. J. Comput. Graph. Statist. 19 419–438.
  • Tadesse, M. G., Sha, N. and Vannucci, M. (2005). Bayesian variable selection in clustering high-dimensional data. J. Amer. Statist. Assoc. 100 602–617.
  • Vannucci, M. and Stingo, F. C. (2011). Bayesian models for variable selection that incorporate biological information. In Bayesian Statistics 9 (J. Bernardo, M. Bayarri, J. Berger, A. Dawid, D. Heckerman, A. Smith and M. West, eds.) 659–678. Oxford Univ. Press, Oxford.
  • West, M. (2003). Bayesian factor regression models in the “large $p$, small $n$” paradigm. In Bayesian Statistics 7 (Tenerife, 2002) (J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds.) 733–742. Oxford Univ. Press, New York.
  • Wiech, K., Ploner, M. and Tracey, I. (2008). Neurocognitive aspects of pain perception. Trends Cogn. Sci. 12 306–313.
  • Woolrich, M. W., Jenkinson, M., Brady, J. M. and Smith, S. M. (2004). Fully Bayesian spatio-temporal modeling of fMRI data. IEEE Trans. Med. Imag. 23 213–231.
  • Yue, Y. R., Lindquist, M. A. and Loh, J. M. (2012). Meta-analysis of functional neuroimaging data using Bayesian nonparametric binary regression. Ann. Appl. Stat. 6 697–718.
  • Zhang, T., Li, F., Beckes, L. and Coan, J. A. (2013). A semi-parametric model of the hemodynamic response for multi-subject fMRI data. NeuroImage 75 136–145.

Supplemental materials

  • Heatmaps. We provide the heatmaps of the voxels with top 10% highest posterior selection probabilities obtained, resulting from Ising-DP, Ising-Gaussian and i.i.d.-Gaussian priors, respectively, in three regressions [Li et al. (2015)].