1. TSTAT_THRESHOLD (~1 secs execution) Calculates P=0.05 (corrected) threshold t for the T statistic using the minimum given by a Bonferroni correction and non-isotropic random field theory (Worsley et al., 1996, 1999). For this example, t=4.86, and voxels where T>t are shown in green: FMRILM MULTISTAT (~3 mins execution) Combines results from separate runs of FMRILM using REML estimation with a regularized random effects analysis. Model: E i = effect for run i; x i = vector of regressors (= (1, 1, …, 1)´ to average the effects);  = unknown vector of regression parameters; S i = standard deviation of effect;  = unknown random effects standard deviation, WN i f, WN i r = Gaussian white noises,  = random/fixed sd: E i = x i ´  + S i WN i f +  WN i r,  2 = (S 2 +  2 ) / S 2, S 2 = Average i S i 2. Step 1: Fit model by EM algorithm: Summary Many methods are available for the statistical analysis of fMRI data that range from a simple linear model for the response and a global autoregressive model for the temporal errors (Bullmore, et al., 1996; SPM), to a more sophisticated non-linear model for the response with a local state space model for the temporal errors (Purdon, et al., 1998). We have written Matlab programs FMRIDESIGN, FMRILM, MULTISTAT and TSTAT_THRESHOLD (available at http://www.bic.mni.mcgill.ca/users/keith ) that seek a compromise between validity, generality, simplicity and execution speed. A General Statistical Analysis for fMRI Data K. J. Worsley 12, C. Liao 1, M. Grabove 1, V. Petre 2, B. Ha 2, A.C. Evans 2 1 Department of Mathematics and Statistics, McGill University 2 McConnell Brain Imaging Centre, Montreal Neurological Institute, McGill University FMRIDESIGN FMRILM    TSTAT_ THRESHOLD FMRIDESIGN (~2 secs execution) Sets up stimuli s t and convolutes it with the hemodynamic response function h t (difference of two gamma densities, Glover, 1999) to create the response design matrix x t for the linear model: FMRILM (~6 mins execution) Fits linear model with AR(1) errors. Model: Y t = fMRI data at time t; (h*s) t = hemodynamic response function h convoluted with vector of stimuli s, at time t;  = vector of linear model parameters; d t = polynomial drift (1, t, t 2, …,t q )’;  = vector of drift parameters;  = standard deviation parameter;  t = AR(p) errors (p=1); a j = autoregressive parameters; WN t = Gaussian white noise : Y t = (h*s) t ´  + d t ´  +  t ;  t = a 1  t-1 + …+ a p  t-p + WN t Step 1: Fit model by least squares, calculate lag 1 autocorrelation a 1, then smooth it: fMRI data Run 1 Run 2... Run m Run 1 Run 2... Run m Run 1 Run 2... Run m SUBJECT 1 SUBJECT 2 SUBJECT n Smooth 15 mm Step 2: Model fitting biases correlation by ~ –0.05, so bias correction is needed: Step 3: Whiten data and design matrix with a 1, fit linear model again by least squares to get estimates , . For a contrast c, find effect c  and its standard deviation Sd(c  ): Step 4: T statistic T = c  / Sd(c  ), thresholded at P<0.05 (see TSTAT_THRESHOLD) FMRILM_ARP (>30 mins execution) Fits linear model with AR(p) errors for p>1. Run 1 Run 2 Run 3 Run 4 Sd Ratio Final Combining the runs: Conclusions The simple AR(1) model appears to be adequate. The FWHM ratio parameter acts as a convenient way of providing an analysis mid- way between a random effects and a fixed effects analysis; setting FWHM ratio = 0 (no smoothing) produces a random effects analysis; setting FWHM ratio to infinity, which smoothes the sd ratio  to one everywhere, produces a fixed effects analysis. In practice, we choose FWHM ratio to produce a final df final which is at least 100, so that errors in its estimation do not greatly affect the distribution of test statistics. Ignoring the correlation If the temporal correlation is ignored completely, that is, the observations are treated as independent and a least squares analysis is used, then the T statistic T 0 is ~11% larger than T 1, the T statistic assuming AR(1) errors. This has the effect of increasing the number of false positives: FMRILM fMRI data FMRILM    MULTISTAT    fMRI data FMRILM fMRI data fMRI data MULTISTAT Combining the subjects: T = Effect / Sd T statistics T p for AR(p) models: for p  1 they are very similar, again indicating that the AR(1) model is adequate Autoregressive coefficients a p for AR(3): for p  2, a p ~0, so that the AR(1) model fitted by FMRILM seems to be adequate Drift removal by adding polynomial variables 1, t, t 2, …,t q to the model (q=3 by default). MULTISTAT   = References Bullmore, E.T. et al. (1996). Magnetic Resonance in Medicine, 35:261-277. Glover, G.H. (1999). NeuroImage, 9:416-429. Purdon, P.L. et al. (1998). NeuroImage, 7:S618. Worsley, K.J. et al. (1996). Human Brain Mapping, 4:58-73. Worsley, K.J. et al. (1999). Human Brain Mapping, 8:98-101. Worsley, K.J. et al. (2000). NeuroImage (submitted).  = There was little evidence of random effects between runs on the same subject (  ~ 1), but there were substantial random effects between subjects (  ~ 3): Smooth 15 mm ^ ^ ^^^ ^ ^^^ ^^ ^^^ ^ ^ ^ Step 2: ^ ^ ^ ^ Resample to Talairach space after linear or non-linear transformations