1. The statistical analysis of fMRI data
Keith Worsley12, Chuanhong Liao1, John Aston123,
Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2,
Frank Morales6, Alan Evans2, Tom Nichols7, Satoru Hayasaki7
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,
6Cuban Neuroscience Centre
7University of Michigan

3. 500
1000
First scan of fMRI data -5 5
T statistic for hot - warm effect
100
200
300
870
880
890
hot
rest
warm
Highly significant effect, T=6.59
800
820
No significant effect, T=-0.74
790
810
Drift
Time, seconds
fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, …
T = (hot – warm effect) / S.d.
~ t110 if no effect

4. 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?

5. More importantly ... Fast execution / slow execution? Matlab / C?
Script (batch) / GUI?
Lazy / hard working … ?
Why not just use SPM?
Develop new ideas ...
FMRISTAT: Simple, general, valid, robust, fast analysis of fMRI data

7. PCA_IMAGE: PCA of time  space:20 40 60 80
100
120
140 4 3 2 1
Component
Frame
Temporal components (sd, % variance explained)
0.68, 46.9%
0.29, 8.6%
0.17, 2.9%
0.15, 2.4% -1
-0.5
0.5
Slice (0 based)
Spatial components 6 8 10 12
1: exclude
first frames
2: drift
3: long-range
correlation
or anatomical
effect: remove
by converting
to % of brain
4: signal?

8. FMRILM: fits a linear model for fMRI time series with AR(p) errors
? ?
Yt = (stimulust * HRF) b + driftt c + errort
AR(p) errors:
? ? ?
errort = a1 errort-1 + … + ap errort-p + s WNt
unknown parameters

10. FMRIDESIGN example: pain perception50
100
150
200
250
300
350 -1 1 2
Alternating hot and warm stimuli separated by rest (9 seconds each).
hot
warm
-0.2
0.2
0.4
Hemodynamic response function: difference of two gamma densities
Responses = stimuli * HRF, sampled every 3 seconds
Time, seconds

12. FMRILM 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: which_stats = ‘_cor’
Raw autocorrelation Smoothed 12.4mm Bias corrected â1
~ ~ 0

13. Effective df depends on smoothing
Variability in acor lowers df
Df depends
on contrast
Smoothing acor brings df back up:
dfacor = dfresidual( )
acor(contrast of data)2
dfeff dfresidual dfacor
FWHMacor /2
FWHMdata2 =
Hot stimulus
Hot-warm stimulus
FWHMdata = 8.79
Residual df = 110
Residual df = 110
100
100
Target = 100 df
Target = 100 df 50
Contrast of data, acor = 0.61 50
Contrast of data, acor = 0.79
dfeff
dfeff 10 20 30 10 20 30
FWHM = 10.3mm
FWHM = 12.4mm
FWHMacor
FWHMacor

14. FMRILM second step: refit the linear model
Pre-whiten: Yt* = Yt – â1 Yt-1, then fit using least squares:
Hot - warm effect, % ‘_mag_ef’
Sd of effect, % ‘_mag_sd’ 1
0.25
0.2
0.5
0.15
0.1
-0.5
0.05 -1
which_stats = ‘_mag_ef _mag_sd _mag_t’
T = effect / sd, 110 df ‘_mag_t’ 6 4
T > 4.93
(P < 0.05,
corrected) 2 -2 -4 -6

15. Higher order AR model? Try AR(3): ‘_AR’1 2 3
0.3
0.2
AR(1) seems
to be adequate
0.1
… has little effect on the T statistics:
-0.1
No correlation
biases T up ~12% → more false positives
AR(1), df=100
AR(2), df=98
AR(3), df=98 5 -5

17. Results from 4 runs on the same subject1
Effect, E i
‘_mag_ef’ -1
0.2
Sd, S i
0.1
‘_mag_sd’ 5
T stat, E
/ S i i
‘_mag_t’ -5

18. MULTISTAT: mixed effects linear model for combining 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

19. 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-parameterize the variance model:
Var(Ei) = Si2 + 2
= (Si2 – minj Sj2) + (2 + minj Sj2)
= Si* *2
2 = *2 – minj Sj2 (less biased estimate) ^ ^ ? ? ^ ^

20. Problem: 4 runs, 3 df for random effects sd  ...
Problem: 4 runs, 3 df for random effects sd  ...
Run 1
Run 2
Run 3
Run 4
Effect, E i
Sd, S
T stat,
/ S -1 1
MULTISTAT
0.1
0.2 -5 5
‘_mag_ef’
… very noisy sd:
‘_mag_sd’
… and T>15.96 for P<0.05 (corrected):
‘_mag_t’
… so no response is detected …

21. 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

22. ^ Average Si  divide multiply random effect, sd ratio ~1.3
Random effects sd, 3 df
Fixed effects sd, 440 df
Mixed effects sd, ~100 df
0.2
0.15
0.1
0.05
divide
multiply
Random sd / fixed sd
Smoothed sd ratio ‘_sdratio’
1.5
random
effect, sd
ratio ~1.3 1
0.5

23. Effective df depends on smoothing
dfratio = dfrandom( )
dfeff dfratio dffixed
FWHMratio /2
FWHMdata2
e.g. dfrandom = 3,
dffixed = 4  110
= 440,
FWHMdata = 8mm: = 20 40
Infinity
100
200
300
400
fixed effects
analysis,
dfeff = 440
dfeff
FWHM
= 19mm
Target = 100 df
random effects
analysis,
dfeff = 3
FWHMratio

24. Final result: 19mm smoothing, 100 effective df …
Run 1
Run 2
Run 3
Run 4
Effect, E i
Sd, S
T stat,
/ S -1 1
MULTISTAT
0.1
0.2 -5 5
Final result: 19mm smoothing, 100 effective df …
‘_mag_ef’
‘_ef’
… less noisy sd:
‘_sd’
‘_mag_sd’
… and T>4.93 for P<0.05 (corrected):
‘_t’
‘_mag_t’
… and now we can detect a response!

26. FWHM – the local smoothness of the noise
voxel size
(1 – correlation)1/2
FWHM = (2 log 2)1/2
(If the noise is modeled as white noise smoothed with a Gaussian kernel, this would be its FWHM)
P-values depend on Resels:
Volume
FWHM3
Resels =
Local maximum T = 4.5
Clusters above t = 3.0, search volume resels = 500
0.1
0.1
0.08
0.08
0.06
0.06
P value of local max
P value of cluster
0.04
0.04
0.02
0.02
500
1000
0.5 1
1.5 2
Resels of search volume
Resels of cluster

27. Non-isotropic data (spatially varying FWHM)
fMRI data is smoother in GM than WM
VBM data is highly non-isotropic
Has little effect on P-values for local maxima (use ‘average’ FWHM inside search region), but
Has a big effect on P-values for spatial extents: smooth regions → big clusters, rough regions → small clusters, so
Replace cluster volume by cluster resels = volume / FWHM3

28. Resels=1.90 P=0.007 Resels=0.57 P=0.387 ‘_fwhm’ ‘_fwhm’
FWHM (mm) of scans (110 df)
FWHM (mm) of effects (3 df) 20 20
Resels=1.90
P=0.007 15 15 10 10
Resels=0.57
P=0.387 5 5
FWHM of effects (smoothed)
effects / scans FWHM (smoothed) 20
1.5 15 10 1 5
0.5

29. In between use Discrete Local Maxima (DLM)
STAT_SUMMARY
Low FWHM
use Bonferroni
In between use Discrete Local Maxima (DLM)
High FWHM use Random Field Theory
4.7
Bonferroni
4.6
4.5
True
T, 10 df
4.4
Random Field Theory
4.3
T, 20 df
Gaussianized threshold
4.2
Discrete Local Maxima (DLM)
4.1
Gaussian 4
3.9
Bonferroni, N=Resels
3.8
3.7 1 2 3 4 5 6 7 8 9 10
FWHM of smoothing kernel (voxels)

30. In between use Discrete Local Maxima (DLM)
STAT_SUMMARY
Low FWHM
use Bonferroni
In between use Discrete Local Maxima (DLM)
High FWHM use Random Field Theory
0.12
Gaussian
T, 20 df
T, 10 df
0.1
Bonferroni
Random Field Theory
0.08
DLM
can ½
P-value
when
FWHM
~3 voxels
P-value
0.06
0.04
True
Discrete Local Maxima
0.02
Bonferroni, N=Resels 1 2 3 4 5 6 7 8 9 10
FWHM of smoothing kernel (voxels)

32. STAT_SUMMARY example: single run, hot-warm
Detected by BON and
DLM but not by RFT
Detected by DLM,
but not by BON or RFT

33. T>4.86

34. T>4.86
T > 4.93
(P < 0.05, corrected)

35. T>4.86
T > 4.93
(P < 0.05, corrected)

36. T>4.86

37. Conjunction: Minimum Ti > threshold
Minimum of Ti ‘_conj’
Average of Ti ‘_mag_t’
For P=0.05,
threshold = 1.82
For P=0.05,
threshold = 4.93
Efficiency = 82%

38. Efficiency : optimum block design
Sd of hot stimulus
Sd of hot-warm 20
0.5 20
0.5
0.4
0.4 15 15
Magnitude
Optimum
design
0.3
0.3 10 10
0.2
Optimum
design
0.2 X 5 5
0.1
0.1 X 5 10 15 20 5 10 15 20
InterStimulus Interval (secs)
(secs)
(secs) 20 1 20 1
0.8
0.8 15 15
Delay
0.6
0.6
Optimum
design X
Optimum
design X 10 10
0.4
0.4 5 5
0.2
0.2
(Not enough signal)
(Not enough signal) 5 10 15 20 5 10 15 20
Stimulus Duration (secs)

39. Efficiency : optimum event design
0.5
(Not
enough
signal)
____ magnitudes
……. delays
uniform
0.45
random
concentrated :
0.4
0.35
0.3
Sd of effect (secs for delays)
0.25
0.2
0.15
0.1
0.05 5 10 15 20
Average time between events (secs)

40. How many subjects?
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! +

41. Estimating the delay of the response
Delay or latency to the peak of the HRF is approximated by a linear combination of two optimally chosen basis functions:
delay -5 5 10 15 20 25
-0.4
-0.2
0.2
0.4
0.6
t (seconds)
basis1
basis2
HRF
shift
HRF(t + shift) ~ basis1(t) w1(shift) + basis2(t) w2(shift)
Convolve bases with the stimulus, then add to the linear model

42. Fit linear model, estimate w1 and w2 -5 5 -3 -2 -1 1 2 3
shift (seconds)
Fit linear model, estimate w1 and w2
Equate w2 / w1 to estimates, then solve for shift (Hensen et al., 2002)
To reduce bias when the magnitude is small, use
shift / (1 + 1/T2)
where T = w1 / Sd(w1) is the T statistic for the magnitude
Shrinks shift to 0 where there is little evidence for a response.
w2 / w1 w1 w2

43. Shift of the hot stimulus
T stat for magnitude ‘_mag_t’ T stat for shift ‘_del_t’
Shift (secs) ‘_del_ef’ Sd of shift (secs) ‘_del_sd’

44. Shift of the hot stimulus
T stat for magnitude ‘_mag_t’ T stat for shift ‘_del_t’
T>4
T~2
Shift (secs) ‘_del_ef’ Sd of shift (secs) ‘_del_sd’
~1 sec
+/- 0.5 sec

45. Combining shifts of the hot stimulus
(Contours are T stat for magnitude > 4)
Run 1
Run 2
Run 3
Run 4
Effect, E i
Sd, S
T stat,
/ S -4 -2 2 4
MULTISTAT 1 -5 5
‘_ef’
‘_del_ef’
‘_sd’
‘_del_sd’
‘_t’
‘_del_t’

46. Shift of the hot stimulus
Shift (secs)
‘_del_ef’
T stat for
magnitude
‘_mag_t’ > 4.93

47. 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 conjuncs No
fmristat:
Shifts the model
Splines
(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, conjuncs
Yes

48. References http://www.math.mcgill.ca/keith/fmristat
Worsley et al. (2002). A general statistical analysis for fMRI data. NeuroImage, 15:1-15.
Liao et al. (2002). Estimating the delay of the response in fMRI data. NeuroImage, 16:

49. Functional connectivity
Measured by the correlation between residuals at every pair of voxels (6D data!)
Local maxima are larger than all 12 neighbours
P-value can be calculated using random field theory
Good at detecting focal connectivity, but
PCA of residuals x voxels is better at detecting large regions of co-correlated voxels
Activation only
Correlation only
Voxel 2 + +
Voxel 2 + + + + + + +
Voxel 1 +
Voxel 1 + +

50. Component > threshold
|Correlations| > 0.7,
P<10-10 (corrected)
First Principal
Component > threshold

51. 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

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

53. 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 T
Bonferroni thresholds the P-values at a/n:
Number of voxels
Threshold T
Random field theory: resels = volume / FHHM3:
Number of resels
Threshold T

54. P < 0.05 (uncorrected), T > 1.64
5% of volume is false +

55. 5% of discoveries is false +
FDR < 0.05, T > 2.67
5% of discoveries is false +

56. P < 0.05 (corrected), T > 4.93 5% probability of any false +

57. Random fields and brain mapping
Keith Worsley
Department of Mathematics and Statistics,
McConnell Brain Imaging Centre,
Montreal Neurological Institute,
McGill University