1. Model-driven statistical analysis of fMRI data
Keith Worsley
Department of Mathematics and Statistics,
Brain Imaging Centre, Montreal Neurological Institute, McGill University

2. References
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:
FMRISTAT: MATLAB package from

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. Exploring the data: PCA of time  space20 40 60 80
100
120 4 3 2 1
Component
Frame
Temporal components (sd, % variance explained)
105.7, 77.8%
26.1, 4.8%
15.8, 1.7%
14.8, 1.5%
Slice
Spatial components 6 8 10
1: exclude
first frames
2: drift
3: long-range
correlation
or anatomical
effect: remove
by converting
to % of brain
4: signal?

5. Modeling the data: 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?
Compromise:
Simple, general, valid, robust, fast statistical analysis

7. Covariates 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

9. Linear model for fMRI time series with AR(p) correlated errors
? ?
Yt = (stimulust * HRF) b + driftt c + errort
AR(p) errors:
? ? ?
errort = a1 errort-1 + … + ap errort-p + s WNt
‘White Noise’
unknown parameters

10. 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
-0.1
0.1
0.2
0.3

11. Pre-whiten: Yt* = Yt – â1 Yt-1, then refit using least squares:
Second step: pre-whiten, refit the linear model
Pre-whiten: Yt* = Yt – â1 Yt-1, then refit using least squares: -1
-0.5
0.5 1
Hot - warm effect, %
0.05
0.1
0.15
0.2
0.25
Sd of effect, % -6 -4 -2 2 4 6
T = effect / sd, 110 df
T > 4.93
(P < 0.05,
corrected)

12. Higher order AR model? Try AR(3):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)
AR(2)
AR(3) 5 -5

14. Results from 4 runs on the same subject

15. 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
Lin. Mod. ? ?
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-parameterize 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 1
Run 2
Run 3
Run 4
Effect, E i
Sd, S
T stat,
/ S -1 1
MULTISTAT
0.1
0.2 -5 5
… very noisy sd:
… and T>15.96 for P<0.05 (corrected):
… 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. ^ 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
1.5
random
effect, sd
ratio ~1.3 1
0.5

20. 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
Why 100?
If out by 50%,
dbn of T not
much affected
Target = 100 df
random effects
analysis,
dfeff = 3
FWHMratio

21. 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 …
… less noisy sd:
… and T>4.93 for P<0.05 (corrected):
… and now we can detect a response!

23. P-values assessed for:
Peaks or local maxima
Spatial extent of clusters of neighbouring voxels above a pre-chosen threshold (~3)
Correct for searching over a pre-specified region (usually the whole brain), which depends on:
number of voxels in the search region (Bonferroni) or
number of resels = volume / FWHM3 in the search region (random field theory)
in practice, take the minimum of the two!

25. FWHM is spatially varying (non-isotropic)
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

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. Resels=1.90
P=0.007
Resels=0.57
P=0.387

28. Statistical summary: clusters
clus vol resel p-val (one)
( 0)
( 0)
( 0)
(0.001)
(0.001)
(0.002)
(0.006)
(0.007)
(0.011)
(0.013)

29. Statistical summary: peaks
clus peak p-val (one) q-val (i j k) ( x y z )
( 0) ( ) ( )
( 0) ( ) ( )
( 0) ( ) ( )
( 0) ( ) ( )
( 0) ( ) ( )
( 0) ( ) ( )
(0.004) ( ) ( )
(0.004) ( ) ( )
(0.004) ( ) ( )
(0.005) ( ) ( )
(0.007) ( ) ( )
(0.007) ( ) ( )
(0.007) ( ) ( )
(0.008) ( ) ( )
(0.009) ( ) ( )
(0.009) ( ) ( )
( 0.01) ( ) ( )
(0.011) ( ) ( )
(0.012) ( ) ( )
(0.012) ( ) ( )

30. T>4.86

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

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

33. T>4.86

34. 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)

35. 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)

36. 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! +

37. References
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:
FMRISTAT: MATLAB package from

38. 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

39. 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

40. Shift of the hot stimulus
T stat for magnitude T stat for shift
Shift (secs) Sd of shift (secs)

41. Shift of the hot stimulus
T stat for magnitude T stat for shift
T>4
T~2
Shift (secs) Sd of shift (secs)
~1 sec
+/- 0.5 sec

42. 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

43. Shift of the hot stimulus
Shift (secs)
T stat for
magnitude > 4.93

44. References
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:
FMRISTAT: MATLAB package from

45. 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

46. 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 +

47. 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

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

49. 5% of discoveries is false +
FDR < 0.05, T > 2.67
5% of discoveries is false +

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

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

52. 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 + +

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