Annals of Applied Statistics

A Bayesian hierarchical spatial point process model for multi-type neuroimaging meta-analysis

Jian Kang, Thomas E. Nichols, Tor D. Wager, and Timothy D. Johnson

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Neuroimaging meta-analysis is an important tool for finding consistent effects over studies that each usually have 20 or fewer subjects. Interest in meta-analysis in brain mapping is also driven by a recent focus on so-called “reverse inference”: where as traditional “forward inference” identifies the regions of the brain involved in a task, a reverse inference identifies the cognitive processes that a task engages. Such reverse inferences, however, require a set of meta-analysis, one for each possible cognitive domain. However, existing methods for neuroimaging meta-analysis have significant limitations. Commonly used methods for neuroimaging meta-analysis are not model based, do not provide interpretable parameter estimates, and only produce null hypothesis inferences; further, they are generally designed for a single group of studies and cannot produce reverse inferences. In this work we address these limitations by adopting a nonparametric Bayesian approach for meta-analysis data from multiple classes or types of studies. In particular, foci from each type of study are modeled as a cluster process driven by a random intensity function that is modeled as a kernel convolution of a gamma random field. The type-specific gamma random fields are linked and modeled as a realization of a common gamma random field, shared by all types, that induces correlation between study types and mimics the behavior of a univariate mixed effects model. We illustrate our model on simulation studies and a meta-analysis of five emotions from 219 studies and check model fit by a posterior predictive assessment. In addition, we implement reverse inference by using the model to predict study type from a newly presented study. We evaluate this predictive performance via leave-one-out cross-validation that is efficiently implemented using importance sampling techniques.

Article information

Ann. Appl. Stat., Volume 8, Number 3 (2014), 1800-1824.

First available in Project Euclid: 23 October 2014

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Bayesian spatial point processes classification hierarchical model random intensity measure neuorimage meta-analysis


Kang, Jian; Nichols, Thomas E.; Wager, Tor D.; Johnson, Timothy D. A Bayesian hierarchical spatial point process model for multi-type neuroimaging meta-analysis. Ann. Appl. Stat. 8 (2014), no. 3, 1800--1824. doi:10.1214/14-AOAS757.

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  • Adolphs, R. (1999). The human amygdala and emotion. Neuroscientist 5 125–137.
  • Baddeley, A. J., Møller, J. and Waagepetersen, R. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat. Neerl. 54 329–350.
  • Best, N. G., Ickstadt, K. and Wolpert, R. L. (2000). Spatial Poisson regression for health and exposure data measured at disparate resolutions. J. Amer. Statist. Assoc. 95 1076–1088.
  • Best, N. G., Ickstadt, K., Wolpert, R. L., Cockings, S., Elliott, P., Bennett, J., Bottle, A. and Reed, S. (2002). Modeling the impact of traffic-related air pollution on childhood respiratory illness. In Case Studies in Bayesian Statistics, Vol. V (Pittsburgh, PA, 1999) (C. Gatsonis, R. Kass, B. Carlin, A. Carriquiry, A. Gelman, I. Verdinelli and M. West, eds.). Lecture Notes in Statist. 162 183–259. Springer, New York.
  • Bondesson, L. (1982). On simulation from infinitely divisible distributions. Adv. in Appl. Probab. 14 855–869.
  • Brooks, S. P. and Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Statist. 7 434–455.
  • Costafreda, S. G., Brammer, M. J., David, A. S. and Fu, C. H. Y. (2008). Predictors of amygdala activation during the processing of emotional stimuli: A meta-analysis of 385 PET and fMRI studies. Brain Research Reviews 58 57–70.
  • Cox, D. R. (1955). Some statistical methods connected with series of events. J. R. Stat. Soc. Ser. B Stat. Methodol. 17 129–157; discussion, 157–164.
  • Damien, P., Laud, P. W. and Smith, A. F. M. (1995). Approximate random variate generation from infinitely divisible distributions with applications to Bayesian inference. J. R. Stat. Soc. Ser. B Stat. Methodol. 57 547–563.
  • Eickhoff, S. B., Laird, A. R., Grefkes, C., Wang, L. E., Zilles, K. and Fox, P. T. (2009). Coordinate-based activation likelihood estimation meta-analysis of neuroimaging data: A random-effects approach based on empirical estimates of spatial uncertainty. Hum. Brain Mapp. 30 2907–2926.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209–230.
  • Fox, P. T., Lancaster, J. L., Parsons, L. M., Xiong, J. H. and Zamarripa, F. (1997). Functional volumes modeling: Theory and preliminary assessment. Hum. Brain Mapp. 5 306–311.
  • Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457–472.
  • Iacoboni, M., Freedman, J., Kaplan, J., Jamieson, K. H., Freedman, T., Knapp, B. and Fitzgerald, K. (2007). This is your brain on politics. The New York Times.
  • Ickstadt, K. and Wolpert, R. L. (1999). Spatial regression for marked point processes. In Bayesian Statistics, 6 (Alcoceber, 1998) (J. Bernardo, J. Berger, A. Dawid and A. Smith, eds.) 323–341. Oxford Univ. Press, New York.
  • Illian, J. B., Møller, J. and Waagepetersen, R. P. (2009). Hierarchical spatial point process analysis for a plant community with high biodiversity. Environ. Ecol. Stat. 16 389–405.
  • Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Wiley, Chichester.
  • Kang, J., Nichols, T. E., Wager, T. D. and Johnson, T. D. (2014). Supplement to “A Bayesian hierarchical spatial point process model for multi-type neuroimaging meta-analysis.” DOI:10.1214/14-AOAS757SUPP.
  • 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.
  • Lindquist, K. A., Wager, T. D., Kober, H., Bliss-Moreau, E. and Barrett, L. F. (2012). The brain basis of emotion: A meta-analytic review. Behav. Brain Sci. 35 121–143.
  • Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Stat. 25 451–482.
  • Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability 100. Chapman & Hall/CRC, Boca Raton, FL.
  • Møller, J. and Waagepetersen, R. P. (2007). Modern statistics for spatial point processes. Scand. J. Stat. 34 643–684.
  • Nielsen, F. A. and Hansen, L. K. (2002). Modeling of activation data in the BrainMap database: Detection of outliers. Hum. Brain Mapp. 15 146–156.
  • Niemi, A. and Fernández, C. (2010). Bayesian spatial point process modeling of line transect data. J. Agric. Biol. Environ. Stat. 15 327–345.
  • Paton, J. J., Belova, M. A., Morrison, S. E. and Salzman, C. D. (2006). The primate amygdala represents the positive and negative value of visual stimuli during learning. Nature 439 865–870.
  • Poldrack, R. A. (2011). Inferring mental states from neuroimaging data: From reverse inference to large-scale decoding. Neuron 72 692–697.
  • Radua, J. and Mataix-Cols, D. (2009). Voxel-wise meta-analysis of grey matter changes in obsessive-compulsive disorder. Br. J. Psychiatry 195 393–402.
  • Russell, J. A. and Barrett, L. F. (1999). Core affect, prototypical emotional episodes, and other things called emotion: Dissecting the elephant. J. Pers. Soc. Psychol. 76 805–819.
  • Stoyan, D. and Penttinen, A. (2000). Recent applications of point process methods in forestry statistics. Statist. Sci. 15 61–78.
  • Teh, Y. W., Jordan, M. I., Beal, M. J. and Blei, D. M. (2006). Hierarchical Dirichlet processes. J. Amer. Statist. Assoc. 101 1566–1581.
  • Turkeltaub, P. E., Eden, G. F., Jones, K. M. and Zeffiro, T. A. (2002). Meta-analysis of the functional neuroanatomy of single-word reading: Method and validation. Neuroimage 16 765–780.
  • van Lieshout, M. N. M. and Baddeley, A. J. (2002). Extrapolating and interpolating spatial patterns. In Spatial Cluster Modelling (A. B. Lawson and D. G. T. Denison, eds.) 61–86. Chapman & Hall/CRC, Boca Raton, FL.
  • Wager, T. D., Jonides, J. and Reading, S. (2004). Neuroimaging studies of shifting attention: A meta-analysis. Neuroimage 22 1679–1693.
  • Wager, T. D., Phan, K., Liberzon, I. and Taylor, S. F. (2003). Valence, gender, and lateralization of functional brain anatomy in emotion: A meta-analysis of findings from neuroimaging. NeuroImage 19 513–531.
  • Wager, T. D., Barrett, L. F., Bliss-moreau, E., Lindquist, K. A., Duncan, S., Kober, H., Joseph, J., Davidson, M. and Mize, J. (2008). The neuroimaging of emotion. In Handbook of Emotions, Chapter 15 (M. Lewis, J. M. Haviland-Jones and L. F. Barrett, eds.) 848. Guilford, New York.
  • Walker, S. G. (2007). Sampling the Dirichlet mixture model with slices. Comm. Statist. Simulation Comput. 36 45–54.
  • Wolpert, R. L. and Ickstadt, K. (1998a). Poisson/gamma random field models for spatial statistics. Biometrika 85 251–267.
  • Wolpert, R. L. and Ickstadt, K. (1998b). Simulation of Lévy random fields. In Practical Nonparametric and Semiparametric Bayesian Statistics (D. Dey, P. Müller and D. Sinha, eds.). Lecture Notes in Statist. 133 227–242. Springer, New York.
  • Woodard, D. B., Wolpert, R. L. and O’Connell, M. A. (2010). Spatial inference of nitrate concentrations in groundwater. J. Agric. Biol. Environ. Stat. 15 209–227.
  • Yarkoni, T., Poldrack, R. A., Van Essen, D. C. and Wager, T. D. (2010). Cognitive neuroscience 2.0: Building a cumulative science of human brain function. Trends in Cognitive Sciences 14 489–496.
  • Yarkoni, T., Poldrack, R. A., Nichols, T. E., Van Essen, D. C. and Wager, T. D. (2011). Large-scale lexical decoding of human brain activity. Unpublished manuscript.
  • 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.

Supplemental materials

  • Supplementary material: Supplement to “A Bayesian hierarchical spatial point process model for multi-type neuroimaging meta-analysis”. In this online supplemental article, we provide (1) proofs of main theorems for the HPGRF model, (2) details on posterior computations, (3) additional figures to assess the posterior variabilities of intensity functions in simulation studies and data application, (4) sensitivity analysis, and (5) details of a Bayesian spatial point process classifier.