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A general statistical analysis for fMRI data

A general statistical analysis for fMRI data. Keith Worsley 12 , Chuanhong Liao 1 , John Aston 13 , Jean-Baptiste Poline 4 , Gary Duncan 5 , Vali Petre 2 , Alan Evans 2 1 Department of Mathematics and Statistics, McGill University, 2 Brain Imaging Centre, Montreal Neurological Institute,

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A general statistical analysis for fMRI data

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  1. A general statistical analysis for fMRI data Keith Worsley12, Chuanhong Liao1, John Aston13, Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2, Alan Evans2 1Department of Mathematics and Statistics, McGill University, 2Brain Imaging Centre, Montreal Neurological Institute, 3Imperial College, London, 4Service Hospitalier Frédéric Joliot, CEA, Orsay, 5Centre de Recherche en Sciences Neurologiques, Université de Montréal

  2. Choices … • Time domain / frequency domain? • AR / ARMA / state space models? • Linear / non-linear time series model? • Fixed HRF / estimated HRF? • Voxel / local / global parameters? • Fixed effects / random effects? • Frequentist / Bayesian?

  3. More importantly ... • Fast execution / slow execution? • Matlab / C? • Script (batch) / GUI? • Lazy / hard working … ? • Why not just use SPM? • Develop new ideas ...

  4. Aim: Simple, general, valid, robust, fast analysis of fMRI data Linear model, AR(p) errors: ? ? Yt = (stimulust * HRF) b + driftt c + errort unknown parameters ? ? ? errort = a1 errort-1 + … + ap errort-p + s WNt

  5. MATLAB: reads MINC or analyze format (www/math.mcgill.ca/keith/fmristat) • FMRIDESIGN: Sets up stimulus, convolves it with the HRF and its derivatives (for estimating delay). • FMRILM: Fits model, estimates effects (contrasts in the magnitudes, b), standard errors, T and F statistics. • MULTISTAT: Combines effects from separate scans/sessions/subjects in a hierarchical fixed / random effects analysis. • TSTAT_THRESHOLD: Uses random field theory / Bonferroni to find thresholds for corrected P-values for peaks and clusters of T and F maps.

  6. Example: Pain perception

  7. First step: estimate the autocorrelation ? AR(1) model: errort = a1 errort-1 + s WNt • Fit the linear model using least squares • êrrort = Yt – fitted Yt • â1 = Correlation ( êrrort , êrrort-1) • Estimating the errors êrrort changes their correlation structure slightly, so â1 is slightly biased: Raw autocorrelation Smoothed 15mm Bias corrected ~ -0.05 ~ 0

  8. Second step: refit the linear model Pre-whiten: Yt* = Yt – â1 Yt-1, then fit using least squares: Effect: hot – warm Sd of effect T statistic = Effect / Sd T > 4.86 (P < 0.05, corrected)

  9. Higher order AR model? Try AR(4): â1 â2 â3 â4 â2, â3, â4 ~ 0, so AR(1) seems to be adequate

  10. … has no effect on the T statistics: AR(1) AR(2) AR(4) But using zero correlation … biases T up ~12%  more false positives

  11. Results from 4 scans on the same subject Scan 1 Scan 2 Scan 3 Scan 4 Effect Ei Sd Si T stat Ei / Si

  12. MULTISTAT: combines effects from different scans/sessions/subjects: • Ei = effect for scan/session/subject i • Si = standard error of effect • Mixed effects model: Ei = covariatesi c + fi + ri }from FMRILM Usually 1, but could add group, treatment, age, sex, ... ‘Fixed effects’ error, due to variability within the same scan, known sd Si Random effect, due to variability from scan to scan, unknown sd 

  13. Fitted using the EM algorithm • Slow to converge (10 iterations by default). • Stable (maintains positive variances). • 2 biased if random effect is small, so: • Sj2  Sj2 -minjSj2 2  2 +minjSj2 • Fit the model • 2  2 -minjSj2 ^ ^ ^

  14. Problem: 4 scans, 3 df for random effects sd  ... Scan 1 Scan 2 Scan 3 Scan 4 MULTISTAT Effect Ei … very noisy sd: Sd Si … and no response is detected: T stat Ei / Si

  15. Solution: Spatial regularization of the sd • Basic idea: increase df by spatial smoothing (local pooling) of the sd. • Can’t smooth the random effects sd directly, - too much anatomical structure. • Instead, • random effects sd • fixed effects sd • which removes the anatomical structure before smoothing.  ) sd= smooth  fixed effects sd

  16. Random effects sd (3 df) Fixed effects sd (448 df) Regularized sd (112 df) Random effects sd Fixed effects sd  Fixed effects sd Over scans Over subjects Smooth 15mm

  17. Effective df dfratio = dfrandom ( 2 ( FWHMratio / FWHMdata )2 + 1 )3/2 dfeff = 1 / ( 1 / dfratio + 1 / dffixed ) e.g. dfrandom = 3, dffixed = 112, FWHMdata = 6mm: FWHMratio (mm) 0 5 10 15 20 infinite dfeff 3 11 45 112 192 448 variability bias compromise!

  18. Final result: 15mm smoothing, 112 effective df … Scan 1 Scan 2 Scan 3 Scan 4 MULTISTAT Effect Ei … less noisy sd: Sd Si … and now we can detect a response: T stat Ei / Si

  19. T > 4.86 (P < 0.05, corrected) T>4.86

  20. + + Conclusion • Largest portion of variance comes from the last stage i.e. combining over subjects: sdscan2 sdsess2 sdsubj2 nscan nsess nsubj nsess nsubj nsubj • If you want to optimize total scanner time, take more subjects. • What you do at early stages doesn’t matter very much!

  21. P.S. Estimating the delay of the response • Delays or latency in the neuronal response are modeled as a temporal scale shift in the reference HRF: • Fast voxel-wise delay estimator is found by adding the derivative of the reference HRF with respect to the log scale shift as an extra term to the linear model. • Bias correction using the second derivative. • Shrunk to the reference delay by a factor of 1/(1+1/T2), T is the T statistic for the magnitude.

  22. Delay of the hot stimulus T stat for magnitude = 0 T stat for delay = 5.4 secs Delay (secs) Sd of delay (secs)

  23. Varying the delay of the reference HRF T stat for mag T stat for delay Delay Sd of delay Ref. delay = 4.0 ~5.4s Ref. delay = 5.4 Ref. delay = 7.3 ~5.4s >4.86 ~0 ~5.4s 0.6s

  24. T > 4.86 (P < 0.05, corrected) Delay (secs) 6.5 5 5.5 4 4.5

  25. T > 4.86 (P < 0.05, corrected) Delay (secs) 6.5 5 5.5 4 4.5

  26. References • http:/www.math.mcgill.ca/keith/fmristat • Worsley et al. (2000). A general statistical analysis for fMRI data. NeuroImage, 11:S648, and submitted. • Liao et al. (2001). Estimating the delay of the fMRI response. NeuroImage, 13:S185 (Poster #185, Tuesday morning), and submitted.

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