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 Hayasaki8, Jonathan Taylor9
1Department
of Mathematics and Statistics, McGill University,
Imaging Centre, Montreal Neurological Institute,
3Statistica Sinica, Taipei,
4Neurospin, CEA, Orsay,
5Centre de Recherche en Sciences Neurologiques, Université de Montréal,
6Cuban Neuroscience Centre
7GlaxoSmithKline and FMRIB, Oxford
8Wake Forest University
9Université de Montréal and Stanford
2Brain
Before you start: PCA of time  space
Component
Temporal components (sd, % variance explained)
1
0.68, 46.9%
2
0.29, 8.6%
3
0.17, 2.9%
4
0.15, 2.4%
0
20
40
60
80
100
120
140
Frame
Spatial components
1
Component
1
0.5
2
0
3
-0.5
1: exclude
first frames
2: drift
3: long-range
correlation
or anatomical
effect: remove
by converting
to % of brain
4
0
2
4
6
8
Slice (0 based)
10
12
-1
4: signal?
Bad design:
2 mins rest
2 mins Mozart
2 mins Eminem
2 mins James Brown
Rest
Mozart
Eminem
J. Brown
Temporal components
Component
Period:
5.2
16.1
(sd, % variance explained)
15.6
11.6
seconds
1
0.41, 17%
2
0.31, 9.5%
3
0.24, 5.6%
0
50
100
Frame
Spatial components
150
200
1
Component
1
0.5
2
0
-0.5
3
0
2
4
6
8
10
12
Slice (0 based)
14
16
18
-1
Effect of stimulus on brain response
Alternating hot and warm stimuli separated by rest (9 seconds each).
2
1
0
-1
0
50
100
150
200
250
300
350
Hemodynamic response function: difference of two gamma densities
Stimulus is
delayed and
dispersed by ~6s
0.4
Modeled by convolving
the stimulus with the
“hemodynamic
response function”
0.2
0
-0.2
0
50
Responses = stimuli * HRF, sampled every 3 seconds
2
1
0
-1
0
50
100
150
200
Time, seconds
250
300
350
fMRI data, pain experiment, one slice
First scan of fMRI data
Highly significant effect, T=6.59
1000
hot
rest
warm
890
880
870
500
0
100
200
300
No significant effect, T=-0.74
820
hot
rest
warm
0
800
T statistic for hot - warm effect
5
0
-5
T = (hot – warm effect) / S.d.
~ t110 if no effect
0
100
0
100
200
Drift
300
810
800
790
200
Time, seconds
300
How fMRI differs from other
repeated measures data





Many reps (~200 time points)
Few subjects (~15)
Df within subjects is high, so not worth
pooling sd across subjects
Df between subjects low, so use spatial
smoothing to boost df
Data sets are huge ~4GB, not easy to use
statistics packages such as R
FMRISTAT (Matlab) /
BRAINSTAT (Python)
statistical analysis strategy

Analyse each voxel separately


Break up analysis into stages




Borrow strength from neighbours when needed
1st level: analyse each time series separately
2nd level: combine 1st level results over runs
3rd level: combine 2nd level results over subjects
Cut corners: do a reasonable analysis in a
reasonable time (or else no one will use it!)
1st level:
Linear model with AR(p) errors

Data



Model



Yt = fMRI data at time t
xt = (responses,1, t, t2, t3, … )’ to allow for drift
Yt = xt’β + εt
εt = a1εt-1 + … + apεt-p + σFηt,
ηt ~ N(0,1) i.i.d.
Fit in 2 stages:


1st pass: fit by least squares, find residuals,
estimate AR parameters a1 … ap
2nd pass: whiten data, re-fit by least squares
Higher levels:
Mixed effects model

Data




Model




Ei = effect (contrast in β) from previous level
Si = sd of effect from previous level
zi = (1, treatment, group, gender, …)’
Ei = zi’γ + SiεiF + σRεiR (Si high df, so assumed fixed)
εiF ~ N(0,1) i.i.d. fixed effects error
εiR ~ N(0,1) i.i.d. random effects error
Fit by ReML

Use EM for stability, 10 iterations
Where we use spatial
information

1st level: smooth AR parameters to lower
variability and increase “df”

Higher levels: smooth Random / Fixed effects
sd ratio to lower variability and increase “df”

Final level: use random field theory to correct
for multiple comparisons
1st level: Autocorrelation
AR(1) model: εt = a1 εt-1 + σFηt
 Fit the linear model using least squares
 εt = Y t – Y t
 â1 = Correlation (εt , εt-1)
 Estimating εt changes their correlation structure slightly,
so â1 is slightly biased:

Raw autocorrelation Smoothed 12.4mm
~ -0.05
Bias corrected â1
~0
0.3
0.2
0.1
0
-0.1
How much smoothing?
• Variability in
â lowers df
• Df depends
on contrast
• Smoothing â
brings df back up:
(
FWHMâ2
+1
2
FWHMdata
dfâ = dfresidual 2
1
dfeff
Hot stimulus
=
1
+
2 acor(contrast of data)2
dfresidual
dfâ
FWHMdata = 8.79
Residual df = 110
100
Target = 100 df
Contrast of data, acor = 0.61
50
dfeff
0
0
10
20
30
FWHM = 10.3mm
FWHMâ
)
3/2
Hot-warm stimulus
Residual df = 110
100
Target = 100 df
Contrast of data, acor = 0.79
50
dfeff
0
0
10
20
30
FWHM = 12.4mm FWHMâ
Higher order AR model? Try AR(3):
â
1
â
2
â
3
0.3
0.2
AR(1) seems
to be adequate
0.1
0
… has little effect on the T statistics:
No correlation
AR(1), df=100
AR(2), df=99
-0.1
AR(3), df=98
5
0
-5
biases T up ~12% → more false positives
2nd level: 4 runs, 3 df for random effects sd
Run 1
Run 2
Run 3
Run 4
2nd level
Effect,
Ei
1
0
… very noisy sd:
-1
0.2
Sd,
Si
0.1
… and T>15.96 for P<0.05 (corrected):
0
5
T stat,
E i / Si
0
… so no response is detected …
-5
Solution:
Spatial smoothing of the sd ratio
• 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,

sd = smooth
random effects sd
 fixed effects sd
fixed effects sd
)
which removes the anatomical structure
before smoothing.
^

Average Si
Random effects sd, 3 df
Fixed effects sd, 440 df
Mixed effects sd, ~100 df
0.2
0.15
0.1
0.05
divide
Random sd / fixed sd
0
multiply
Smoothed sd ratio
1.5
1
0.5
random
effect, sd
ratio ~1.3
How much smoothing?
(
dfratio = dfrandom
FWHMratio2
2
+1
2
FWHMdata
)
1
1
1
=
+
dfeff dfratio dffixed
3/2
dfrandom = 3,
dffixed = 4  110
= 440,
FWHMdata = 8mm:
fixed effects
analysis,
dfeff = 440
400
300
dfeff
Target = 100 df
random effects
analysis,
dfeff = 3
200
FWHM
= 19mm
100
0
0
20
40
FWHMratio
Infinity
Final result: 19mm smoothing, 100 df
Run 1
Run 2
Run 3
Run 4
2nd level
Effect,
Ei
1
0
… less noisy sd:
-1
0.2
Sd,
Si
0.1
… and T>4.93 for P<0.05 (corrected):
0
5
T stat,
E i / Si
0
… and now we can detect a response!
-5
Final level: Multiple comparisons correction
Bonferroni
4.7
4.6
Gaussianized threshold
4.5
True
4.4
4.3
4.2
T, 10 df
Random Field Theory
T, 20 df
Discrete Local Maxima (DLM)
4.1
Gaussian
4
3.9
3.8
3.7
0
Low FWHM:
use Bonferroni
1
2
3
4
5
6
7
8
FWHM of smoothing kernel (voxels)
In between: use Discrete
Local Maxima (DLM)
9
10
High FWHM: use
Random Field Theory
0.12
Gaussian
T, 20 df
T, 10 df
0.1
Random Field Theory
Bonferroni
P-value
0.08
0.06
0.04
True
DLM can ½ P-value
when FWHM ~3 voxels
0.02
0
Discrete Local Maxima
0
Low FWHM:
use Bonferroni
1
2
3
4
5
6
7
FWHM of smoothing kernel (voxels)
In between: use Discrete
Local Maxima (DLM)
8
9
10
High FWHM: use
Random Field Theory
Example: single run, hot-warm
Detected by BON and
DLM but not by RFT
Detected by DLM,
but not by BON or RFT
FWHM – the local smoothness of the noise
FWHM =
voxel size
(2 log 2)1/2
1/2
(1 – correlation)
(If the noise is modeled as white noise smoothed
with a Gaussian kernel, this would be its FWHM)
P-values depend on Resels:
0.1
Clusters above t = 3.0, search volume resels = 500
0.1
P value of cluster
P value of local max
Local maximum T = 4.5
0.08
0.06
0.04
0.02
0
0
Volume
Resels =
FWHM3
500
1000
Resels of search volume
0.08
0.06
0.04
0.02
0
0
0.5
1
1.5
Resels of cluster
2
Non-isotropic data (spatially varying FWHM)
FWHM (mm) of scans (110 df)
Cluster Resels=1.90
P=0.007
Cluster Resels=0.57
P=0.387


20
15
15
10
10
5
5
0
0
use ‘average’ FWHM inside search region, but
Has a big effect on cluster P-values



20
fMRI data is smoother in GM than WM
Has little effect on peak P-values


FWHM (mm) of effects (3 df)
smooth regions → big clusters,
rough regions → small clusters, so
Replace cluster volume by cluster resels = volume / FWHM3
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
0.6
0.4
basis1
0.2
HRF
basis2
0
-0.2
-0.4
-5
0
shift
5
10
t (seconds)
15
20
25
HRF(t + shift) ~ basis1(t) w1(shift) + basis2(t) w2(shift)
• Convolve bases with the stimulus, then add to the linear model
• Fit linear model,
estimate w1 and w2
3
w2 / w1
2
1
• Equate w2 / w1 to estimates, then
solve for shift (Hensen et al., 2002)
w1
• To reduce bias when the magnitude
is small, use
0
w2
shift / (1 + 1/T2)
-1
where T = w1 / Sd(w1) is the T statistic
for the magnitude
-2
-3
-5
0
shift (seconds)
5
• Shrinks shift to 0 where there is little
evidence for a response.
Shift of the hot stimulus
T stat for magnitude
T stat for shift
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
Shift (secs)
Sd of shift (secs)
4
2
2
1.5
0
1
-2
0.5
-4
0
Shift of the hot stimulus
T stat for magnitude
T>4
T stat for shift
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
Shift (secs)
~1 sec
T~2
Sd of shift (secs)
4
2
2
1.5
0
+/- 0.5 sec
1
-2
0.5
-4
0
Combining shifts of the hot stimulus
(Contours are T stat for magnitude > 4)
Run 1
Run 2
Run 3
Run 4
MULTISTAT
4
2
Effect,
Ei
0
-2
-4
2
Sd,
Si
1
0
5
T stat,
E i / Si
0
-5
Shift of the hot stimulus
Shift (secs)
T stat for
magnitude
> 4.93
Contrasts in the data used for effects
2
Hot, Sd = 0.16
Warm, Sd = 0.16
9 sec
1
blocks,
9 sec
gaps 0
-1
0
50
100
150
200
Hot-warm, Sd = 0.19
250
300
350
Time (secs)
2
Hot, Sd = 0.28
90 sec
blocks, 1
90 sec
gaps 0
Warm, Sd = 0.43
Only using data near block transitions
Ignoring data in the middle of blocks
-1
0
50
100
150
200
Hot-warm, Sd = 0.55
250
300
350
Time (secs)
Optimum block design
Sd of hot stimulus
0.5
20
0.4
15
Magnitude
Best
design
10
15
20
0.8
15
10
5
0
(secs)
1
20
Delay
5
0
5
X
10
15
0.1
20
20
0.8
15
0.6
Best
design
X
0.4
0.2
15
20
0
0
(secs)
1
10
(Not enough signal)
10
0.2
Best
design
0.6
Best
design
X
5
0.3
10
0
10
0.4
15
0.2
0.1
5
0.5
20
0.3
X
5
Gap
(secs)
Sd of hot-warm
5
0
0.4
0.2
(Not enough signal)
Block (secs)
5
10
15
20
0
Optimum event design
0.5
(Not
enough
signal)
____ magnitudes
……. delays
uniform . . . . . . . . .
random .. . ... .. .
concentrated :
0.4
Sd of
effect
(secs
for
delays)
0.3
0.2
12 secs best for
magnitudes
0.1
0
5
15
7 secs best for 10
delays Average time between events (secs)
20
How many subjects?

Largest portion of variance comes from the
last stage i.e. combining over subjects:
sdrun2
sdsess2
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!
Features special to
FMRISTAT / BRAINSTAT


Bias correction for AR coefficients
Df boosting due to smoothing:



P-value adjustment for:




AR coefficients
random/fixed effects variance
peaks due to small FWHM (DLM)
clusters due to spatially varying FWHM
Delays analysed the same way as magnitudes
Sd of effects before collecting data
Our entry in the
Functional Imaging
Analysis contest
Jonathan Taylor
Stanford
Keith Worsley
McGill
Why a Functional Imaging
Analysis Contest (FIAC)?






Competing packages produce slightly
different results, which is “correct”?
Simulated data?
Real data, compare analyses
“Contest” session at 2005 Human Brain Map
conference
9 entrants
Results in a special issue of Human Brain
Mapping in May, 2006
The main participants






SPM (Statistical Parametric Mapping, 1993),
University College, London, “SAS”, (MATLAB)
AFNI (1995), NIH, more display and
manipulation, not much stats (C)
FSL (2000), Oxford, the “upstart” (C)
….
FMRISTAT (2001), McGill, stats only (MATLAB)
BRAINSTAT (2005), Stanford/McGill, Python
version of FMRISTAT
FIAC paradigm


16 subjects
4 runs per subject



4 conditions per run





2 runs: event design
2 runs: block design
Same sentence, same speaker
Same sentence, different speaker
Different sentence, same speaker
Different sentence, different speaker
3T, 191 frames, TR=2.5s
Response

Events
0.4
0.2
0
-0.2
0
50
100
150
200
250
300
350
400
450
500
350
400
450
500
Beginning of block/run

Blocks
0.4
0.2
0
-0.2
0
50
100
150
200
250
Seconds
300
Design matrix for block expt

B1, B2 are basis functions for magnitude and delay:
1st snt in block
S snt, S spk, B1
S snt, S spk, B2
S snt, D spk, B1
S snt, D spk, B2
D snt, S spk, B1
D snt, S spk, B2
D snt, D spk, B1
D snt, D spk, B2
Constant
Linear
Quadratic
Cubic
Spline
Whole brain avg
1st level analysis


Motion and slice time correction (using FSL)
5 conditions
3 contrasts
Beginning
of block/run
Same sent, Same sent,
same speak diff speak
Diff sent,
same speak
Diff sent,
diff speak
Sentence
Speaker
0
0
-0.5
-0.5
-0.5
0.5
0.5
-0.5
0.5
0.5
1
-1
-1
1
Interaction 0

Smoothing of temporal autocorrelation to
control the effective df
Magnitude sd (relative to error)
Efficiency
2
1.5
Sd of contrasts (lower is
better) for a single run,
assuming additivity of
responses
Event
Block
1
0.5
0
Diff sente Diff speak
• For the magnitudes,
event and block have
similar efficiency
• For the delays, event is
much better.
Interac
Delay sd (seconds)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Event
Block
Diff sente Diff speak
Interac
2nd level analysis


Analyse events and blocks separately
Register contrasts to Talairach (using FSL)


Bad registration on 2 subjects - dropped
Combine 2 runs using fixed FX
3rd level analysis

Combine remaining 14 subjects using random FX


3 contrasts × event/block × magnitude/delay = 12
Threshold using best of Bonferroni, random field
theory, and discrete local maxima (new!)
Part of slice
z = -2 mm
Magnitude (%BOLD), diff - same sentence, event experiment
Subj 0
1
3
4
6
7
8
9
10
11
12
13
14
Mixed
effects
15
Left
2
1
0
Ef
Right
-1
Random
/fixed
effects sd
smoothed
7.0105mm
1.5
-2
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: min fMRI > 6214
1
Left
Sd
Right
271
272
271
265
264
132
270
275
269
274
248
256
264
278
1
0
0.5
5
FWHM (mm)
15
40
Left
-50
T
0
x (mm)
df
0.5
10
0
5
50
Ant.
Post.
Right
P=0.05 threshold for local maxima is +/- 5.68
-5
-40-20 0
y (mm)
0
Magnitude (%BOLD), diff - same sentence, block experiment
Subj 0
1
3
4
6
7
8
9
10
11
12
13
14
Mixed
effects
15
Left
2
1
0
Ef
Right
-1
Random
/fixed
effects sd
smoothed
7.103mm
1.5
-2
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: min fMRI > 5904
1
Left
Sd
Right
202
202
204
205
204
203
201
202
200
206
205
202
204
200
1
0
0.5
5
FWHM (mm)
15
40
Left
-50
T
0
x (mm)
df
0.5
10
0
5
50
Ant.
Post.
Right
P=0.05 threshold for local maxima is +/- 5.67
-5
-40-20 0
y (mm)
0
Delay shift (secs), diff - same sentence, event experiment
Subj 0
1
3
4
6
7
8
9
10
11
12
13
14
Mixed
effects
15
Left
0.2
0.1
0
-0.1
-0.2
Ef
Right
Random
/fixed
effects sd
smoothed
10.6778mm
1.5
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5
Left
0.4
Sd
1
0.2
Right
df
271
272
271
265
264
132
270
275
269
274
248
256
264
278
0
0.5
40
FWHM (mm)
15
T
0
-2
Ant.
Post.
Right
P=0.05 threshold for local maxima is +/- 4.31
-50
x (mm)
Left
2
10
0
5
50
-40-20 0
y (mm)
0
Delay shift (secs), diff - same sentence, block experiment
Subj 0
1
3
4
6
7
8
9
10
11
12
13
14
Mixed
effects
15
Left
1
0.5
0
Ef
Right
-0.5
Random
/fixed
effects sd
smoothed
8.8952mm
1.5
-1
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5
2
Left
1.5
Sd
1
1
0.5
Right
df
202
202
204
205
204
203
201
202
200
206
205
202
204
200
0
0.5
40
FWHM (mm)
15
T
0
-2
Ant.
Post.
Right
P=0.05 threshold for local maxima is +/- 4.3
-50
x (mm)
Left
2
10
0
5
50
-40-20 0
y (mm)
0
Event
Block
Subj 0
1
3
4
6
7
8
9
10
11
12
13
14
Magnitude (%BOLD), diff - same sentence, block experiment
Mixed
effects
15
1
3
4
6
7
8
9
10
11
12
13
14
Left
Left
0
0
Random
/fixed
effects sd
smoothed
7.0105mm
1.5
-2
1
1
Left
271
265
264
132
270
275
269
274
248
256
264
278
Sd
0.5
1
0
0.5
40
Right
Right
272
Random
/fixed
effects sd
smoothed
7.103mm
1.5
-2
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: min fMRI > 5904
Left
Sd
271
-1
Right
Right
-1
2
1
Ef
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: min fMRI > 6214
df
Mixed
effects
15
1
Ef
Magnitude
Subj 0
2
df
202
202
204
205
204
203
201
202
200
206
205
202
204
200
x (mm)
10
FWHM (mm)
15
-50
T
0
0.5
Left
Left
0
0
5
-50
T
1
40
FWHM (mm)
15
5
0.5
0
5
x (mm)
Magnitude (%BOLD), diff - same sentence, event experiment
50
1
3
4
6
7
8
9
10
11
12
13
14
-40-20 0
y (mm)
P=0.05 threshold for local maxima is +/- 5.67
Delay shift (secs), diff - same sentence, block experiment
Mixed
effects
15
Subj 0
1
3
4
6
7
8
9
10
11
12
13
14
Mixed
effects
15
Left
Left
0.2
0.1
0
-0.1
-0.2
Ef
-0.5
2
1.5
Sd
1
0.2
271
272
271
265
264
132
270
275
269
274
248
256
264
278
1
0.5
0
Right
Right
df
Random
/fixed
effects sd
smoothed
8.8952mm
1.5
-1
Left
Left
1
0.5
40
df
202
202
204
205
204
203
201
202
200
206
205
202
204
200
0
0.5
40
-2
10
T
0
2
-50
0
5
50
0
P=0.05 threshold for local maxima is +/- 4.3
Ant.
-40-20 0
y (mm)
-2
Post.
Right
Ant.
Post.
Right
P=0.05 threshold for local maxima is +/- 4.31
x (mm)
0
-50
Left
Left
T
FWHM (mm)
15
x (mm)
FWHM (mm)
15
2
0
1
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5
0.4
Sd
-40-20 0
y (mm)
0
Right
Right
Random
/fixed
effects sd
smoothed
10.6778mm
1.5
-5
0.5
Ef
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5
Delay
0
Ant.
Subj 0
5
50
Post.
Right
Delay shift (secs), diff - same sentence, event experiment
-5
Ant.
Post.
Right
P=0.05 threshold for local maxima is +/- 5.68
10
0
10
0
5
50
-40-20 0
y (mm)
0
Delays in different – same
sentence



Events: 0.14±0.04s; Blocks: 1.19±0.23s
Both significant, P<0.05 (corrected) (!?!)
Answer: take a look at blocks:
Greater
magnitude
Best fitting block
Greater
delay
Different sentence
(sustained interest)
Same sentence
(lose interest)
SPM
FMRISTAT/
BRAINSTAT
Magnitude increase for
 Sentence, Event
 Sentence, Block
 Sentence, Combined
 Speaker, Combined
at (-54,-14,-2)
Magnitude decrease for
 Sentence, Block
 Sentence, Combined
at (-54,-54,40)
Delay increase for
Sentence, Event
at (58,-18,2)
inside the region where all
conditions are activated
Conclusions

Greater %BOLD response for



different – same sentences (1.08±0.16%)
different – same speaker (0.47±0.0.8%)
Greater latency for

different – same sentences (0.148±0.035 secs)
The main effects of sentence repetition (in red) and of speaker
repetition (in blue). 1: Meriaux et al, Madic; 2: Goebel et al, Brain
voyager; 3: Beckman et al, FSL; 4: Dehaene-Lambertz et al, SPM2.
z=-12
z=2
2
3
z=5
1
1,4
3
Fmristat/
Brainstat:
combined
block and
event,
threshold
at T>5.67,
P<0.05.
3
3
1
3
Functional connectivity

Measured by the correlation between residuals at
every pair of voxels (6D data!)
Activation only
Voxel 2
++
+ +++
Correlation only
Voxel 2
Voxel 1
+
+
++
+
Voxel 1
+




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
|Correlations| > 0.7,
P<10-10 (corrected)
First Principal
Component > threshold