1. A general statistical analysis for fMRI data
Keith Worsley12, Chuanhong Liao1, John Aston123,
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:
? ?
Yt = (stimulust * HRF) b + driftt c + errort
AR(p) errors:
? ? ?
errort = a1 errort-1 + … + ap errort-p + s WNt
unknown parameters

5. MATLAB: reads MINC or analyze format (www/math. mcgill
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 runs/sessions/subjects in a hierarchical fixed / random effects analysis.
STAT_THRESHOLD: Uses random field theory / Bonferroni to find thresholds for corrected P-values for peaks and clusters of T and F maps.

7. Example: Pain perception50
100
150
200
250
300
350
400 -1 1 2
(a) Stimulus, s(t): alternating hot and warm stimuli on forearm, separated by rest (9 seconds each).
hot
warm
-0.2
0.2
0.4
(b) Hemodynamic response function, h(t): difference of two gamma densities (Glover, 1999)
(c) Response, x(t): sampled at the slice acquisition times every 3 seconds
Time, t (seconds)

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

10. 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)

11. Higher order AR model? Try AR(4):
â â2
~ 0
â â4
~ 0
~ 0
AR(1) seems to be adequate

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

14. Results from 4 runs on the same subject
Run Run Run Run 4
Effect Ei Sd Si
T stat
Ei / Si

15. MULTISTAT: combines effects from different runs/sessions/subjects:
Ei = effect for run/session/subject i
Si = standard error of effect
Mixed effects model:
Ei = covariatesi c + Si WNiF +  WNiR
}from
FMRILM ? ?
Usually 1, but
could add group,
treatment, age,
sex, ...
‘Fixed effects’ error,
due to variability
within the same run
Random effect,
due to variability
from run to run

16. REML estimation using the EM algorithm
Slow to converge (10 iterations by default).
Stable (maintains estimate 2 > 0 ), but
2 biased if 2 (random effect) is small, so:
Re-parametrise the variance model:
Var(Ei) = Si2 + 2
= (Si2 – minj Sj2) + (2 + minj Sj2)
= Si* *2
2 = *2 – minj Sj2 (less biased estimate) ^ ^ ? ? ^ ^

17. Problem: 4 runs, 3 df for random effects sd  ...
Problem: 4 runs, 3 df for random effects sd  ...
Run Run Run Run MULTISTAT
Effect Ei
… very noisy sd: Sd Si
… and T>15.96 for P<0.05 (corrected):
T stat
Ei / Si
… so no response is detected …

18. 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

19. ~1 ~3 ~1.6 Random effects sd (3 df) Fixed effects sd (448 df)
Regularized sd
(112 df)
 Fixed effects sd
Random effects sd
Fixed effects sd
Over scans ~3
Over subjects
Smooth
15mm ~1
~1.6

20. Effective df dfeff 3 11 45 112 192 448 dfratio = dfrandom(2 + 1)
dfeff dfratio dffixed
e.g. dfrandom = 3, dffixed = 112, FWHMdata = 6mm:
FWHMratio (mm) infinite
dfeff
FWHMratio /2
FWHMdata2 =
Random effects Fixed effects
variability bias
compromise!

21. Final result: 15mm smoothing, 112 effective df …
Run Run Run Run MULTISTAT
Effect Ei
… less noisy sd: Sd Si
… and T>4.86 for P<0.05 (corrected):
T stat
Ei / Si
… and now we can detect a response!

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

24. Conclusion
Largest portion of variance comes from the last stage i.e. combining over subjects:
sdrun sdsess sdsubj2
nrun 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! +

25. 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.
0.6
Delay = 4.0 seconds, log scale shift = -0.3
0.4
Delay = 5.4 seconds, log scale shift = 0 (reference hrf, h0)
Delay = 7.3 seconds, log scale shift = +0.3
0.2
-0.2 5 10 15 20 25
t (seconds)

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

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

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

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

30. Comparison: SPM’99: fmristat: Different slice acquisition times:
Drift removal:
Temporal correlation:
Estimation of effects:
Rationale:
Random effects:
Map of the delay:
SPM’99:
Adds a temporal derivative
Low frequency cosines (flat at the ends)
AR(1), global parameter, bias reduction not necessary
Band pass filter, then least-squares, then correction for temporal correlation
More robust,
but lower df
No regularization,
low df No
fmristat:
Shifts the model
Polynomials
(free at the ends)
AR(p), voxel parameters, bias reduction
Pre-whiten, then least squares (no further corrections needed)
More accurate, higher df
Regularization, high df
Yes

35. 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 and submitted.

36. False Discovery Rate (FDR)
Benjamini and Hochberg (1995), Journal of the Royal Statistical Society
Benjamini and Yekutieli (2001), Annals of Statistics
Genovese et al. (2001), NeuroImage
FDR controls the expected proportion of false positives amongst the discoveries, whereas
Bonferroni / random field theory controls the probability of any false positives
No correction controls the proportion of false positives in the volume

37. Signal True + Noise False + Signal + Gaussian white noise
P < 0.05 (uncorrected), Z > 1.64
5% of volume is false +
Signal
True +
Noise
False +
FDR < 0.05, Z > 2.82
5% of discoveries is false +
P < 0.05 (corrected), Z > 4.22
5% probability of any false +

38. Comparison of thresholds
FDR depends on the ordered P-values:
P1 < P2 < … < Pn. To control the FDR at a = 0.05, find
K = max {i : Pi < (i/n) a}, threshold the P-values at PK
Proportion of true
Threshold Z
Bonferroni thresholds the P-values at a/n:
Number of voxels
Threshold Z
Random field theory: resels = volume / FHHM3:
Number of resels
Threshold Z

39. Which do you prefer? P < 0.05 (uncorrected), Z > 1.64
5% of volume is false +
FDR < 0.05, Z > 2.91
5% of discoveries is false +
P < 0.05 (corrected), Z > 4.86
5% probability of any false +
Which do you prefer?