Statistical Science

The Statistical Analysis of fMRI Data

Martin A. Lindquist

Full-text: Open access


In recent years there has been explosive growth in the number of neuroimaging studies performed using functional Magnetic Resonance Imaging (fMRI). The field that has grown around the acquisition and analysis of fMRI data is intrinsically interdisciplinary in nature and involves contributions from researchers in neuroscience, psychology, physics and statistics, among others. A standard fMRI study gives rise to massive amounts of noisy data with a complicated spatio-temporal correlation structure. Statistics plays a crucial role in understanding the nature of the data and obtaining relevant results that can be used and interpreted by neuroscientists. In this paper we discuss the analysis of fMRI data, from the initial acquisition of the raw data to its use in locating brain activity, making inference about brain connectivity and predictions about psychological or disease states. Along the way, we illustrate interesting and important issues where statistics already plays a crucial role. We also seek to illustrate areas where statistics has perhaps been underutilized and will have an increased role in the future.

Article information

Statist. Sci. Volume 23, Number 4 (2008), 439-464.

First available in Project Euclid: 11 May 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Lindquist, Martin A. The Statistical Analysis of fMRI Data. Statist. Sci. 23 (2008), no. 4, 439--464. doi:10.1214/09-STS282.

Export citation


  • Aguirre, G. K., Zarahn, E. and D’Esposito, M. (1998). The variability of human, BOLD hemodynamic responses. NeuroImage 8 360–369.
  • Andersen, A., Gash, D. and Avison, M. J. (1999). Principal component analysis of the dynamic response measured by fmri: A generalized linear systems framework. Magnetic Resonance in Medicine 17 785–815.
  • Beckmann, C. F., Jenkinson, M. and Smith, S. M. (2003). General multi-level linear modelling for group analysis in fmri. NeuroImage 20 1052–1063.
  • Beckmann, C. F. and Smith, S. M. (2005). Tensorial extensions of independent component analysis for multisubject fmri analysis. NeuroImage 25 294–311.
  • Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289–300.
  • Birn, R. M., Saad, Z. S. and Bandettini, P. A. (2001). Spatial heterogeneity of the nonlinear dynamics in the fmri bold response. NeuroImage 14 817–826.
  • Bohning, D. E., Pecheny, A. P., Epstein, C. M., Speer, A. M., Vincent, D. J., Dannels, W. and George, M. (1997). Mapping transcranial magnetic stimulation (tms) fields in vivo with mri. Neuroreport 8 2535–2538.
  • Bowman, F., Caffo, B., Bassett, S. and Kilts, C. (2008). Bayesian hierarchical framework for spatial modeling of fmri data. NeuroImage 39 146–156.
  • Bowman, F. D. (2005). Spatio-temporal modeling of localized brain activity. Biostatistics 6 558–575.
  • Boynton, G. M., Engel, S. A., Glover, G. H. and Heeger, D. J. (1996). Linear systems analysis of functional magnetic resonance imaging in human v1. J. Neurosci. 16 4207–4221.
  • Brockwell, P. J. and Davis, R. A. (1998). Time Series: Theory and Methods. Springer.
  • Buxton, R. B., Wong, E. C. and Frank, L. R. (1998). Dynamics of blood flow and oxygenation changes during brain activation: The balloon model. Magnetic Resonance in Medicine 39 855–864.
  • Calhoun, V. D., Adali, T., McGinty, V. B., Pekar, J. J., Watson, T. D. and Pearlson, G. D. (2001a). fmri activation in a visual-perception task: Network of areas detected using the general linear model and independent components analysis. Neuroimage 14 1080–1088.
  • Calhoun, V. D., Adali, T., Pearlson, G. and Pekar, J. (2001b). Spatial and temporal independent component analysis of functional mri data containing a pair of task-related waveforms. Human Brain Mapping 13 43–53.
  • Dale, A. M. (1999). Optimal experimental design for event-related fmri. Human Brain Mapping 8 109–114.
  • Duong, T., Kim, D., Ugurbil, K. and Kim, S. (2000). Spatio-temporal dynamics of the bold fmri signals: Toward mapping columnar structures using the early negative response. Magnetic Resonance in Medicine 44 231–242.
  • Friston, K. (1994). Functional and effective connectivity in neuroimaging: A synthesis. Human Brain Mapping 2 56–78.
  • Friston, K. J., Harrison, L. and Penny, W. (2003). Dynamic causal modelling. NeuroImage 19 1273–1302.
  • Friston, K. J., Mechelli, A., Turner, R. and Price, C. J. (2000). Nonlinear responses in fmri: The balloon model, volterra kernels, and other hemodynamics. NeuroImage 12 466–477.
  • Friston, K. J., Penny, W., Phillips, C., Kiebel, S., Hinton, G. and Ashburner, J. (2002). Classical and Bayesian inference in neuroimaging: Theory. NeuroImage 16 465–483.
  • Genovese, C., Lazar, N. and Nichols, T. (2002). Thresholding of statistical maps in functional neuroimaging using the false discovery rate. NeuroImage 15 870–878.
  • Glahn, D., C., Thompson, P., M. and Blangero, J. (2007). Neuroimaging endophenotypes: Strategies for finding genes influencing brain structure and function. Human Brain Mapping 28 488–501.
  • Glover, G. H. (1999a). Deconvolution of impulse response in event-related BOLD fMRI. NeuroImage 9 416–429.
  • Glover, G. H. (1999b). Simple analytic spiral k-space algorithm. Magnetic Resonance in Medicine 42 412–415.
  • Glover, G. H., Li, T. and Ress, D. (2000). Image-based method for retrospective correction of physiological motion effects in fmri: Retroicor. Magnetic Resonance in Medicine 44 162–167.
  • Goldman, R. I., Stern, J. M., Engel, J., J. and Cohen, M. S. (2000). Acquiring simultaneous eeg and functional mri. Clin Neurophysiol 11 1974–1980.
  • Gossl, C., Auer, D. and Fahrmeir, L. (2001). Bayesian spatiotemporal inference in functional magnetic resonance imaging. Biometrics 57 554–562.
  • Goutte, C., Nielsen, F. A. and Hansen, L. K. (2000). Modeling the haemodynamic response in fmri using smooth fir filters. IEEE Trans. Med. Imaging 19 1188–1201.
  • Grinband, J., Wager, T. D., Lindquist, M., Ferrera, V. P. and Hirsch, J. (2008). Detection of time-varying signals in event-related fmri designs. NeuroImage 43 509–520.
  • Gudbjartsson, H. and Patz, S. (1995). The Rician distribution of noisy MRI data. Magnetic Resonance in Medicine 34 910–914.
  • Guo, Y. and Pagnoni, G. (2008). A unified framework for group independent component analysis for multi-subject fmri data. NeuroImage 42 1078–1093.
  • Haacke, M. E., Brown, R. W., Thompson, M. R. and Venkatesan, R. (1999). Magnetic Resonance Imaging: Physical Principles and Sequence Design. Wiley.
  • Hayasaka, S. and Nichols, T. (2004). Combining voxel intensity and cluster extent with permutation test framework. NeuroImage 23 54–63.
  • Horwitz, B. (2003). The elusive concept of brain connectivity. NeuroImage 19 466–470.
  • Huettel, S. A., Song, A. W. and Mccarthy, G. (2004). Functional Magnetic Resonance Imaging. Sinauer Associates.
  • Hyvarinen, A., Karhunen, J. and Oja, E. (2001). Independent Component Analysis. Wiley, New York.
  • Jackson, J., Meyer, C., Nishimura, D. and Macovski, A. (1991). Selection of a convolution function for Fourier inversion using gridding [computerised tomography application]. Medical Imaging, IEEE Transactions on 10 473–478.
  • Katanoda, K., Matsuda, Y. and Sugishita, M. (2002). A spatio-temporal regression model for the analysis of functional mri data. NeuroImage 17 1415–1428.
  • Kim, D., Duong, T. and Kim, S. (2000). High-resolution mapping of iso-orientation columns by fmri. Nature Neuroscience 3 164–169.
  • Le Bihan, D., Mangin, J.-F., Poupon, C., Clark, C. A., Pappata, S., Molko, N. and Chabriat, H. (2001). Diffusion tensor imaging: Concepts and applications. Journal of Magnetic Resonance Imaging 13 534–546.
  • Liao, C., Worsley, K. J., Poline, J.-B., Duncan, G. H. and Evans, A. C. (2002). Estimating the delay of the response in fMRI data. NeuroImage 16 593–606.
  • Lindquist, M., Zhang, C., Glover, G., Shepp, L. and Yang, Q. (2006). A generalization of the two dimensional prolate spheroidal wave function method for non-rectilinear mri data acquisition methods. IEEE Transactions in Image Processing 15 2792–2804.
  • Lindquist, M., Zhang, C., Glover, G. and Shepp, L. (2008a). Acquisition and statistical analysis of rapid 3d fmri data. Statist. Sinica 18 1395–1419.
  • Lindquist, M., Zhang, C., Glover, G. and Shepp, L. (2008b). Rapid three-dimensional functional magnetic resonance imaging of the negative bold response. Journal of Magnetic Resonance 191 100–111.
  • Lindquist, M. A., Loh, J., Atlas, L. and Wager, T. D. (2008c). Modeling the hemodynamic response function in fmri: Efficiency, bias and mis-modeling. NeuroImage. To appear.
  • Lindquist, M. A. and Wager, T. D. (2007a). Modeling state-related fMRI activity using change-point theory. NeuroImage 35 1125–1141.
  • Lindquist, M. A. and Wager, T. D. (2007b). Validity and power in hemodynamic response modeling: A comparison study and a new approach. Human Brain Mapping 28 764–784.
  • Liu, T. and Frank, L. (2004). Efficiency, power, and entropy in event-related fmri with multiple trial types: Part i: Theory. NeuroImage 21 387–400.
  • Logothetis, N. K. (2000). Can current fmri techniques reveal the micro-architecture of cortex? Nature Neuroscience 3 413.
  • Loh, J., Lindquist, M. A. and Wager, T. D. (2008). Residual analysis for detecting mis-modeling in fmri. Statist. Sinica.
  • 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.
  • Luo, W.-L. and Nichols, T. E. (2003). Diagnosis and exploration of massively univariate neuroimaging models. NeuroImage 19 1014–1032.
  • Malonek, D. and Grinvald, A. (1996). The imaging spectroscopy reveals the interaction between electrical activity and cortical microcirculation: Implication for optical, pet and mr functional brain imaging. Science 272 551–554.
  • Mansfield, P., Coxon, R. and Hykin, J. (1995). Echo-volumar imaging (evi) at 3.0 t: First normal volunteer and functional imaging results. Journal of Computer Assisted Tomography 19 847–852.
  • Mansfield, P., Howseman, A. and Ordidge, R. (1989). Volumar imaging using nmr spin echos: Echo-volumar imaging (evi) at 0.1 t. J. Phys. E 22 324–330.
  • McIntosh, A. and Gonzalez-Lima, F. (1994). Structural equation modeling and its application to network analysis in functional brain imaging. Human Brain Mapping 2 2–22.
  • McKeown, M. J. and Makeig, S. (1998). Analysis of fmri data by blind separation into independant spatial components. Human Brain Mapping 6 160–188.
  • Menon, R., Luknowsky, D. C. and Gati, J. S. (1998). Mental chronometry using latency resolved functional mri. Proc. Natl. Acad Sci. USA 95 10902–10907.
  • Menon, R., Ogawa, S., Hu, X., Strupp, J., Andersen, P. and Ugurbil, K. (1995). Bold based functional mri at 4 tesla includes a capillary bed contribution: Echo-planar imaging mirrors previous optical imaging using intrinsic signals. Magnetic Resonance in Medicine 33 453–459.
  • Miezin, F., Maccotta, L., Ollinger, J., Petersen, S. and Buckner, R. (2000). Characterizing the hemodynamic response: Effects of presentation rate, sampling procedure, and the possibility of ordering brain activity based on relative timing. NeuroImage 11 735–759.
  • Nichols, T. E. and Holmes, A. P. (2002). Nonparametric permutation tests for functional neuroimaging: A primer with examples. Human Brain Mapping 15 1–25.
  • Ogawa, S., Tank, D., Menon, R., Ellerman, J., Kim, S., Merkle, H. and Ugurbil, K. (1992). Intrinsic signal changes accompanying sensory simulation: Functional brain mapping and magnetic resonance imaging. Proc. Nat. Acad. Sci. 89 5951–5955.
  • Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge Univ. Press.
  • Penny, W., Trujillo-Barreto, N. and Friston, K. (2005). Bayesian fMRI time series analysis with spatial priors. NeuroImage 24 350–362.
  • Pruessmann, K., Weiger, M., Scheidegger, M. and Boesiger, P. (1999). Sense: Sensitivity encoding for fast mri. Magnetic Resonance in Medicine 42 952–956.
  • Purdon, P. L., Solo, V., Weissko, R. M. and Brown, E. (2001). Locally regularized spatiotemporal modeling and model comparison for functional MRI. NeuroImage 14 912–923.
  • Riera, J. J., Watanabe, J., Kazuki, I., Naoki, M., Aubert, E., Ozaki, T. and Kawashima, R. (2004). A state-space model of the hemodynamic approach: Nonlinear filtering of bold signals. NeuroImage 21 547–567.
  • Roberts, S. W. (1959). Control chart tests based on geometric moving averages. Technometrics 1 239–250.
  • Robinson, L. F., Wager, T. and Lindquist, M. (2009). Change point estimation in multi-subject fmri studies. To appear.
  • Roebroeck, A., Formisano, E. and Goebel, R. (2005). Mapping directed influence over the brain using granger causality and fmri. NeuroImage 25 230–242.
  • Rowe, D. B. and Logan, B. R. (2004). A complex way to compute fmri activation. NeuroImage 23 1078–1092.
  • Rubin, D. (1974). Estimating causal effects of treatments in randomized and non-randomized studies. Journal of Educational Psychology 66 688–701.
  • Schacter, D. L., Buckner, R. L., Koutstaal, W., Dale, A. M. and Rosen, B. R. (1997). Rectangular confidence regions for the means of multivariate normal distributions. Late onset of anterior prefrontal activity during true and false recognition: An event-related fMRI study. NeuroImage 6 259–269.
  • Sodickson, D. and Manning, W. (1997). Simultaneous acquisition of spatial harmonics (smash): Fast imaging with radiofrequency coil arrays. Magnetic Resonance in Medicine 38 591–603.
  • Thompson, J., Peterson, M. and Freeman, R. (2004). High-resolution neurometabolic coupling revealed by focal activation of visual neurons. Nature Neuroscience 7 919–920.
  • Vazquez, A. L., Cohen, E. R., Gulani, V., Hernandez-Garcia, L., Zheng, Y., Lee, G. R., Kim, S. G., Grotberg, J. B. and Noll, D. C. (2006). Vascular dynamics and bold fmri: Cbf level effects and analysis considerations. NeuroImage 32 1642–1655.
  • Wager, T. D., Davidson, M. L., Hughes, B. L., Lindquist, M. A. and Ochsner, K. N. (2008). Prefrontal-subcortical pathways mediating successful emotion regulation. Neuron 59 1037–1050.
  • Wager, T. D. and Nichols, T. E. (2003). Optimization of experimental design in fmri: A general framework using a genetic algorithm. NeuroImage 18 293–309.
  • Wager, T. D., Vazquez, A., Hernandez, L. and Noll, D. C. (2005). Accounting for nonlinear BOLD effects in fMRI: Parameter estimates and a model for prediction in rapid event-related studies. NeuroImage 25 206–218.
  • Woolrich, M. W., Behrens, T. E., Beckmann, C. F. and Smith, S. M. (2005). Mixture models with adaptive spatial regularization for segmentation with an application to fmri data. IEEE Transactions on Medical Imaging 24 1–11.
  • Woolrich, M. W., Behrens, T. E. and Smith, S. M. (2004). Constrained linear basis sets for HRF modelling using variational Bayes. NeuroImage 21 1748–1761.
  • Worsley, K. J. and Friston, K. J. (1995). Analysis of fMRI time-series revisited-again. NeuroImage 2 173–181.
  • Worsley, K. J., Taylor, J. E., Tomaiuolo, F. and Lerch, J. (2004). Unified univariate and multivariate random field theory. NeuroImage 23 189–195.
  • Yacoub, E., Le, T. and Hu, X. (1998). Detecting the early response at 1.5 tesla. NeuroImage 7 S266.
  • Zarahn, E. (2002). Using larger dimensional signal subspaces to increase sensitivity in fmri time series analyses. Hum. Brain Mapp. 17 13–16.