The Annals of Applied Statistics

Meta-analysis of functional neuroimaging data using Bayesian nonparametric binary regression

Yu Ryan Yue, Martin A. Lindquist, and Ji Meng Loh

Full-text: Open access


In this work we perform a meta-analysis of neuroimaging data, consisting of locations of peak activations identified in 162 separate studies on emotion. Neuroimaging meta-analyses are typically performed using kernel-based methods. However, these methods require the width of the kernel to be set a priori and to be constant across the brain. To address these issues, we propose a fully Bayesian nonparametric binary regression method to perform neuroimaging meta-analyses. In our method, each location (or voxel) has a probability of being a peak activation, and the corresponding probability function is based on a spatially adaptive Gaussian Markov random field (GMRF). We also include parameters in the model to robustify the procedure against miscoding of the voxel response. Posterior inference is implemented using efficient MCMC algorithms extended from those introduced in Holmes and Held [Bayesian Anal. 1 (2006) 145–168]. Our method allows the probability function to be locally adaptive with respect to the covariates, that is, to be smooth in one region of the covariate space and wiggly or even discontinuous in another. Posterior miscoding probabilities for each of the identified voxels can also be obtained, identifying voxels that may have been falsely classified as being activated. Simulation studies and application to the emotion neuroimaging data indicate that our method is superior to standard kernel-based methods.

Article information

Ann. Appl. Stat., Volume 6, Number 2 (2012), 697-718.

First available in Project Euclid: 11 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Binary response data augmentation fMRI Gaussian Markov random fields Markov chain Monte Carlo meta-analysis Spatially adaptive smoothing


Yue, Yu Ryan; Lindquist, Martin A.; Loh, Ji Meng. Meta-analysis of functional neuroimaging data using Bayesian nonparametric binary regression. Ann. Appl. Stat. 6 (2012), no. 2, 697--718. doi:10.1214/11-AOAS523.

Export citation


  • Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. J. Amer. Statist. Assoc. 88 669–679.
  • Andrews, D. F. and Mallows, C. L. (1974). Scale mixtures of normal distributions. J. Roy. Statist. Soc. Ser. B 36 99–102.
  • Brezger, A., Fahrmeir, L. and Hennerfeind, A. (2007). Adaptive Gaussian Markov random fields with applications in human brain mapping. J. Roy. Statist. Soc. Ser. C 56 327–345.
  • Carter, C. K. and Kohn, R. (1996). Markov chain Monte Carlo in conditionally Gaussian state space models. Biometrika 83 589–601.
  • Carvalho, C. M., Polson, N. G. and Scott, J. G. (2010). The horseshoe estimator for sparse signals. Biometrika 97 465–480.
  • Choudhuri, N., Ghosal, S. and Roy, A. (2007). Nonparametric binary regression using a Gaussian process prior. Stat. Methodol. 4 227–243.
  • Crainiceanu, C. M., Ruppert, D., Carroll, R. J., Adarsh, J. and Goodner, B. (2007). Spatially adaptive penalized splines with heteroscedastic errors. J. Comput. Graph. Statist. 16 265–288.
  • Dale, A. M., Fischl, B. and Sereno, M. I. (1999). Cortical surface-based analysis. I. Segmentation and surface reconstruction. Neuroimage 9 179–194.
  • Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer, New York.
  • Dey, D. K., Ghosh, S. K. and Mallick, B. K., eds. (2000). Generalized Linear Models: A Bayesian Perspective. Biostatistics 5. Dekker, New York.
  • Fischl, B., Sereno, M. I. and Dale, A. M. (1999). Cortical surface-based analysis. II: Inflation, flattening, and a surface-based coordinate system. Neuroimage 9 195–207.
  • Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Anal. 1 515–533 (electronic).
  • Gelman, A., van Dyk, D. A., Huang, Z. and Boscardin, W. J. (2008). Using redundant parameterizations to fit hierarchical models. J. Comput. Graph. Statist. 17 95–122.
  • Gu, C. (1990). Adaptive spline smoothing in non-Gaussian regression models. J. Amer. Statist. Assoc. 85 801–807.
  • Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Monographs on Statistics and Applied Probability 43. Chapman & Hall, London.
  • Holmes, C. C. and Held, L. (2006). Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Anal. 1 145–168 (electronic).
  • Holmes, C. C. and Mallick, B. K. (2003). Generalized nonlinear modeling with multivariate free-knot regression splines. J. Amer. Statist. Assoc. 98 352–368.
  • 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.
  • Kober, H., Barrett, L. F., Joseph, J., Bliss-Moreau, E., Lindquist, K. and Wager, T. D. (2008). Functional grouping and cortical-subcortical interactions in emotion: A meta-analysis of neuroimaging studies. Neuroimage 42 998–1031.
  • Krivobokova, T., Crainiceanu, C. M. and Kauermann, G. (2008). Fast adaptive penalized splines. J. Comput. Graph. Statist. 17 1–20.
  • Lindquist, M. A. (2008). The statistical analysis of fMRI data. Statist. Sci. 23 439–464.
  • Loader, C. (1999). Local Regression and Likelihood. Springer, New York.
  • McCullagh, P. and Nelder, J. (1989). Generalized Linear Models, 2nd ed. Chapman & Hall/CRC, Boca Raton, FL.
  • O’Sullivan, F., Yandell, B. S. and Raynor, W. J. Jr. (1986). Automatic smoothing of regression functions in generalized linear models. J. Amer. Statist. Assoc. 81 96–103.
  • Penny, W. D., Trujillo-Barreto, N. J. and Friston, K. J. (2005). Bayesian fMRI time series analysis with spatial priors. Neuroimage 24 350–362.
  • Psarakis, S. and Panaretos, J. (1990). The folded $t$ distribution. Comm. Statist. Theory Methods 19 2717–2734.
  • Rue, H. and Held, L. (2005). Gaussian Markov Random Fields: Theory and Applications. Monographs on Statistics and Applied Probability 104. Chapman & Hall/CRC, Boca Raton, FL.
  • Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 319–392.
  • Ruppert, D. and Carroll, R. J. (2000). Spatially-adaptive penalties for spline fitting. Aust. N. Z. J. Stat. 42 205–223.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 583–639.
  • Talairach, J. and Tournoux, P. (1988). Co-planar Stereotaxic Atlas of the Human Brain: 3-Dimensional Proportional System—an Approach to Cerebral Imaging. Thieme Medical Publishers, New York.
  • Tipping, M. E. (2001). Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn. Res. 1 211–244.
  • Trippa, L. and Muliere, P. (2009). Bayesian nonparametric binary regression via random tessellations. Statist. Probab. Lett. 79 2273–2280.
  • Turkeltaub, P., Eden, G., Jones, K. and Zeffiro, T. A. T. (2002). Meta-analysis of the functional neuroanatomy of single-word reading: Method and validation. NeuroImage 16 765–780.
  • Wager, T. D., Jonides, J. and Reading, S. (2004). Neuroimaging studies of shifting attention: A meta-analysis. Neuroimage 22 1679–1693.
  • Wager, T. D., Lindquist, M. A. and Kaplan, L. (2007). Meta-analysis of functional neuroimaging data: Current and future directions. Social Cognitive and Affective Neuroscience 2 150–158.
  • Wager, T. D., Barrett, L. F., Bliss-Moreau, E., Lindquist, K., Duncan, S., Kober, H., Joseph, J., Davidson, M. and Mize, J. (2008). The neuroimaging of emotion. In Handbook of Emotion (M. Lewis, ed.) 249–271. Guilford Press, New York.
  • Wager, T. D., Lindquist, M. A., Nichols, T. E., Kober, H. and Van Snellenberg, J. X. (2009). Evaluating the consistency and specificity of neuroimaging data using meta-analysis. Neuroimage 45 S210–S221.
  • Wahba, G., Wang, Y., Gu, C., Klein, R. and Klein, B. (1995). Smoothing spline ANOVA for exponential families, with application to the Wisconsin Epidemiological Study of Diabetic Retinopathy. Ann. Statist. 23 1865–1895.
  • Wiper, M. P., Girón, F. J. and Pewsey, A. (2008). Objective Bayesian inference for the half-normal and half-$t$ distributions. Comm. Statist. Theory Methods 37 3165–3185.
  • Wood, S. A. and Kohn, R. (1998). A Bayesian approach to robust binary nonparametric regression. J. Amer. Statist. Assoc. 93 203–213.
  • Wood, S. A., Kohn, R., Cottet, R., Jiang, W. and Tanner, M. (2008). Locally adaptive nonparametric binary regression. J. Comput. Graph. Statist. 17 352–372.
  • Yue, Y., Loh, J. M. and Lindquist, M. A. (2010). Adaptive spatial smoothing of fMRI images. Stat. Interface 3 3–13.
  • Yue, Y. R. and Loh, J. M. (2011). Bayesian semiparametric intensity estimation for inhomogeneous spatial point processes. Biometrics 67 937–946.
  • Yue, Y. and Speckman, P. L. (2010). Nonstationary spatial Gaussian Markov random fields. J. Comput. Graph. Statist. 19 96–116.