Bayesian Analysis

Spatial Bayesian Variable Selection Models on Functional Magnetic Resonance Imaging Time-Series Data

Kuo-Jung Lee, Galin L. Jones, Brian S. Caffo, and Susan S. Bassett

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A common objective of fMRI (functional magnetic resonance imaging) studies is to determine subject-specific areas of increased blood oxygenation level dependent (BOLD) signal contrast in response to a stimulus or task, and hence to infer regional neuronal activity. We posit and investigate a Bayesian approach that incorporates spatial and temporal dependence and allows for the task-related change in the BOLD signal to change dynamically over the scanning session. In this way, our model accounts for potential learning effects in addition to other mechanisms of temporal drift in task-related signals. We study the properties of the model through its performance on simulated and real data sets.

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Bayesian Anal., Volume 9, Number 3 (2014), 699-732.

First available in Project Euclid: 5 September 2014

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Bayesian variable selection fMRI Ising distribution Markov chain Monte Carlo


Lee, Kuo-Jung; Jones, Galin L.; Caffo, Brian S.; Bassett, Susan S. Spatial Bayesian Variable Selection Models on Functional Magnetic Resonance Imaging Time-Series Data. Bayesian Anal. 9 (2014), no. 3, 699--732. doi:10.1214/14-BA873.

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  • Aguirre, G. K., Zarahn, E., and D’Esposito, M. (1997). “Empirical analyses of BOLD fMRI statistics. II. Spatially smoothed data collected under null-hypothesis and experimental conditions.” NeuroImage, 5: 199–212.
  • Bassett, S. S., Yousem, D. M., Cristinzio1, C., Kusevic1, I., Yassa1, M. A., Caffo, B. S., and Zeger, S. L. (2006). “Familial risk for Alzheimer’s disease alters fMRI activation patterns.” Brain, 129: 1229–1239.
  • Bench, C. J., Frith, C. D., Grasby, P. M., Friston, K. J., Paulesu, E., Frackowiak, R. S. J., and Dolan, R. J. (1993). “Investigations of the functional anatomy of attention using the Stroop test.” Neuropsychologia, 31: 907–922.
  • Bowman, F., Caffo, B., Bassett, S., and Kilts, C. (2008). “A Bayesian hierarchical framework for spatial modeling of fMRI data.” NeuroImage, 39: 146–156.
  • Caffo, B., Bowman, F., Eberly, L., and Bassett, S. (2011). “A Markov chain Monte Carlo based analysis of a multilevel model for functional MRI data.” In Brooks, S., Gelman, A., Jones, G., and Meng, X.-L. (eds.), Handbook of Markov Chain Monte Carlo. Boca Raton, FL: CRC Press.
  • Carter, C. S., Mintum, M., and Cohen, J. D. (1995). “Interference and facilitation effects during selective attention: H$_{2}^{15}$O study of Stroop task performance.” NeuroImage, 280: 747–749.
  • Chen, S., Wang, C., Eberly, L., Caffo, B., and Schwartz, B. (2009). “Adaptive control of the false discovery rate in voxel-based morphometry.” Human Brain Mapping, 30: 2304–2311.
  • Davidson, D. J., Zacks, R. T., and Williams, C. C. (2003). “Stroop interference, practice, and aging.” Neuropsychology, 10: 85–98.
  • Fernández, C., Ley, E., and Steel, M. F. J. (2001). “Benchmark priors for Bayesian model averaging.” Journal of Econometrics, 100: 381–427.
  • Fisher, L. M., Freed, D. M., and Corkin, S. (1990). “Stroop color-word test performance in patients with Alzheimer’s disease.” Journal of Clinical and Experimental Neuropsychology, 12: 745–758.
  • Flegal, J. M. and Gong, L. (2013). “Relative fixed-width stopping rules for Markov chain Monte Carlo simulations.” arXiv:1303.0238.
  • Flegal, J. M., Haran, M., and Jones, G. L. (2008). “Markov chain Monte Carlo: Can we trust the third significant figure?” Statistical Science, 23: 250–260.
  • Flegal, J. M. and Jones, G. L. (2011). “Implementing Markov chain Monte Carlo: estimating with confidence.” In Brooks, S., Gelman, A., Jones, G., and Meng, X.-L. (eds.), Handbook of Markov Chain Monte Carlo. Boca Raton, FL: CRC Press.
  • Friston, K. J., Ashburner, J. T., Kiebel, S. J., Nichols, T. E., and Penny, W. D. (2007). Statistical parametric mapping: The analysis of functional brain images. Elsevier/Academic Press.
  • Friston, K. J., Fletcher, P., Josephs, O., Holmes, A., Rugg, M., and Turner, R. (1998). “Event-related fMRI: characterizing differential responses.” Neuroimage, 7: 30–40.
  • Friston, K. J., Holmes, A., Worsley, K. J., Polin, J. B., Frith, C., and Frackowik, R. (1995). “Statistical parametric maps in functional imaging: A general linear approach.” Human Brain Mapping, 2: 189–210.
  • Friston, K. J., Worsley, K., Frackowiak, R., Mazziotta, J., and Evans, A. (1994). “Assessing the significance of focal activations using their spatial extent.” Human Brain Mapping, 1: 210–220.
  • Gelman, A. (1998). “Simulating normalizing constants: From importance sampling to bridge sampling to path sampling.” Statistical Science, 13: 163–185.
  • Genovese, C. (2000). “A Bayesian time-course model for functional magnetic resonance imaging data.” Journal of the American Statistical Association, 95: 691–703.
  • George, E. I. (2000). “The variable selection problem.” Journal of the American Statistical Association, 95: 1304–1308.
  • George, E. I. and McCulloch, R. E. (1993). “Variable selection via Gibbs sampling.” Journal of the American Statistical Association, 88: 881–889.
  • — (1997). “Approaches for Bayesian variable selection.” Statistica Sinica, 7: 339–374.
  • Glover, G. H. (1999). “Deconvolution of impulse response in event-related BOLD.” NeuroImage, 9: 416–429.
  • Goldsmith, J., Huang, L., and Crainiceanu, C. M. (2014). “Smooth scalar-on-image regression via spatial Bayesian variable selection.” Journal of Computational and Graphical Statistics, 23: 46–64.
  • Gössl, C., Fahrmeir, L., and Auer, D. P. (2001). “Bayesian spatiotemporal inference in functional magnetic resonance images.” Biometric, 57: 554–562.
  • Johnson, A. A., Jones, G. L., and Neath, R. C. (2013). “Component-wise Markov chain Monte Carlo: Uniform and geometric ergodicity under mixing and composition.” Statistical Science, 28: 360–375.
  • Jones, G. L., Haran, M., Caffo, B. S., and Neath, R. (2006). “Fixed-width output analysis for Markov chain Monte Carlo.” Journal of the American Statistical Association, 101: 1537–1547.
  • Kang, J., Johnson, T. D., Nichols, T. E., and Wager, T. D. (2011). “Meta analysis of functional neuroimaging data via Bayesian spatial point processes.” Journal of the American Statistical Association, 106: 124–134.
  • Lee, K.-J. (2010). “Computational issues in using Bayesian hierarchical methods for the spatial modeling of fMRI data.” Ph.D. thesis, University of Minnesota, School of Statistics.
  • Li, C., Zheng, J., Wang, J., Gui, L., and Li, C. (2009). “An fMRI Stroop task study of prefrontal cortical function in normal aging, mild cognitive impairment, and Alzheimer’s disease.” Current Alzheimer Research, 6: 525–530.
  • Liang, F., Paulo, R., Molina, G., Clyde, M. A., and Berger, J. O. (2008). “Mixture of g-priors for Bayesian variable selection.” Journal of the American Statistical Association, 410–423.
  • Lindquist, M. A. (2008). “The statistical analysis of fMRI data.” Human Brain Mapping, 23: 439–464.
  • Lindquist, M. A., Loh, J. M., Atlas, L. Y., and Wager, T. D. (2009). “Modeling the hemodynamic response function in fMRI: Efficiency, bias and mis-modeling.” NeuroImage, 45: S187–198.
  • Locascio, J., Jennings, P. J., Moore, C. I., and Corkin, S. (1997). “Time series analysis in the time domain and resampling methods for studies of functional magnetic brain imaging.” Human Brain Mapping, 5: 168–193.
  • Lund, T. E., Madsen, K. H., Sidaros, K., Luo, W. L., and Nichols, T. E. (2006). “Non-white noise in fMRI: Does modelling have an impact?” NeuroImage, 29: 54–66.
  • Møller, J. (2003). Spatial Statistics and Computational Methods. Springer.
  • Nichols, T. and Holmes, A. (2002). “Nonparametric permutation tests for functional neuroimaging: A primer with examples.” Human Brain Mapping, 15: 1–25.
  • Polk, T., Drake, R., Jonides, J., Smith, M., and Smith, E. (2008). “Attention enhances the neural processing of relevant features and suppresses the processing of irrelevant features in humans: A functional magnetic resonance imaging study of the Stroop task.” The Journal of Neuroscience, 28: 13786–13792.
  • Propp, J. G. and Wilson, D. B. (1996). “Exact sampling with coupled Markov chains and applications to statistical mechanics.” Random Structures and Algorithms, 9: 223–252.
  • Raftery, A. (1996). “Hypothesis testing and model selection.” In Gilks, W., Richardson, S., and Spiegelhalter, D. (eds.), Markov Chain Monte Carlo in Practice. London: Chapman & Hall.
  • Rajapakse, J. C., Kruggel, F., Maisog, J. M., and Cramon, D. Y. (1998). “Modeling hemodynamic response for analysis of functional MRI time-series.” Human Brain Mapping, 6: 283–300.
  • Sabsevitz, D. S., Swanson, S. J., Hammeke, T. A., Spanaki, M. V., Possing, E. T., Morris, G. L., Mueller, W. M., and Binder, J. R. (2003). “Use of preoperative functional neuroimaging to predict language deficits from epilepsy surgery.” Neurology, 60: 1788.
  • Smith, M. and Fahrmeir, L. (2007). “Spatial Bayesian variable selection with application to functional magnetic resonance imaging.” Journal of the American Statistical Association, 102: 417–431.
  • Smith, M., Pütz, B., Auer, D., and Fahrmeirc, L. (2003). “Assessing brain activity through spatial Bayesian variable selection.” NeuroImage, 20: 802–815.
  • Stroop, J. R. (1935). “Studies of interference in serial verbal reactions.” Journal of Experimental Psychology, 7: 643–661.
  • Taylor, S. F., Kornblum, S., Lauber, E. J., Minoshima, S., and Koeppe, R. A. (1997). “Isolation of specific interference processing in the Stroop task: PET activation studies.” NeuroImage, 6: 81–92.
  • Tjelmeland, H. and Besag, J. (1998). “Markov random fields with higher-order interactions.” Scandinavian Journal of Statistics, 25: 415–433.
  • Wang, F. and Landau, D. (2001). “Efficient, multiple-range random walk algorithm to calculate the density of states.” Physical Review Letters, 81: 2050 – 2053.
  • Woolrich, M. W., Jenkinson, M., Brady, J. M., and Smith, S. M. (2004). “Fully Bayesian spatio-temporal modeling for fMRI data.” IEEE Transactions on Medical Imaging, 23: 213–231.
  • Woolrich, M. W., Ripley, B. D., Brady, M., and Smith, S. M. (2001). “Temporal autocorrelation in univariate linear modeling of fMRI Data.” NeuroImage, 14: 1370–1386.
  • Worsley, K. (2003). “Detecting activation in fMRI data.” Statistical Methods in Medical Research, 12: 401–418.
  • Worsley, K., Marrett, S., Neelin, P., and Evans, A. (1992). “A three-dimensional statistical analysis for CBF activation studies in human brain.” Journal of Cerebral Blood Flow and Metabolism, 12: 900–918.
  • Worsley, K. J., Liao, C. H., Aston, J., Petre, V., Duncan, G. H., Morales, F., and Evans, A. C. (2002). “A general statistical analysis for fMRI data.” NeuroImage, 15: 1–15.
  • Xia, J., Liang, F., and Wang, Y. M. (2009). “FMRI analysis through Bayesian variable selection with a spatial prior.” IEEE International Symposium on Biomedical Imaging (ISBI), 714–717.
  • Zellner, A. (1996). “On assessing prior distributions and Bayesian regression analysis with $g$-prior distributions.” In Bayesian Inference and Decision Techniques: Essays in Honor of Brunode Finetti North-Holland/Elsevier, 233–243.
  • Zhang, C. and Ma, J. (2007). “Simulation via direct computation of partition functions.” Physical Review Letters E, 76: 036708–1–5.