A general statistical analysis for
fMRI data
Keith Worsley12, Chuanhong Liao1, John Aston123,
Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2,
Frank Morales6, 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,
6Cuban Neuroscience Centre
First scan of fMRI data
fMRI data: 120 scans, 3 scans each of
hot, rest, warm, rest, hot, rest, …
1000
800
(a) Highly significant correlation
fMRI data
960
hot
rest
warm
940
920
+
+
0
50
100
150
200
250
300
350
600
+
400
200
400
fMRI data
940
(b) No significant correlation
920
900
0
hot
rest
warm
0
50
100
150
200
250
300
350
400
Z statistic for hot - warm
6
fMRI data
850
4
840
2
(c) Drift
830
0
50
100
150
200
250
Time, t (seconds)
300
350
400
Z = (effect hot – warm) / S.d.
~ N(0,1) if no effect
+
+
0
+
-2
-4
-6
FMRISTAT: Simple, general, valid,
robust, fast analysis of fMRI data
• Linear model:
?
?
Yt = (stimulust * HRF) b + driftt c + errort
• AR(p) errors:
unknown parameters
?
?
?
errort = a1 errort-1 + … + ap errort-p + s WNt
FMRIDESIGN example: Pain perception
(a) Stimulus, s(t): alternating hot and warm stimuli on forearm, separated by rest (9 seconds each).
2
1
0
-1
0
50
100
150
200
250
300
350
(b) Hemodynamic response function, h(t): difference of two gamma densities (Glover, 1999)
400
0
50
350
400
0
50
350
400
0.4
0.2
0
-0.2
100
150
200
250
300
(c) Response, x(t): sampled at the slice acquisition times every 3 seconds
2
1
0
-1
100
150
200
Time, t (seconds)
250
300
FMRILM step 1: estimate temporal correlation
?
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.
• Bias correction is very quick and effective:
Raw autocorrelation
Smoothed 15mm
~ -0.05
Bias corrected â1
~0
0.4
0.3
0.2
0.1
0
-0.1
FMRILM step 2: refit the linear model
Pre-whiten: Yt* = Yt – â1 Yt-1, then fit using least squares:
Effect: hot – warm
Sd of effect
6
2.5
4
2
2
1.5
0
1
-2
-4
0.5
-6
0
T statistic = Effect / Sd
6
4
2
0
-2
-4
-6
T > 4.90
(P < 0.05,
corrected)
Higher order AR model? Try AR(4):
â1
0.4
~0
0.4
0.3
0.3
0.2
0.2
0.1
â3
â2
~0
0.1
0
0
-0.1
-0.1
0.4
â4
0.4
0.3
0.3
0.2
0.2
0.1
~0
0.1
0
0
-0.1
-0.1
AR(1) seems to be adequate
… has no effect on the T statistics:
AR(1)
AR(2)
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
But ignoring correlation …
AR(4)
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
biases T up ~12%
more false positives
Results from 4 runs on the same subject
Run 1
Run 2
Run 3
Run 4
5
Effect
Ei
0
-5
2.5
Sd
Si
2
1.5
1
0.5
0
5
T stat
Ei / Si
0
-5
MULTISTAT combines effects from
different runs/sessions/subjects:
• Ei = effect for run/session/subject i
from
• Si = standard error of effect
FMRILM
• Mixed effects model:
?
?
F
Ei = covariatesi c + Si WNi +  WNiR
}
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
REML estimation of the mixed effects
model using the EM algorithm
•
•
•
•
Slow to converge (10 iterations by default).
^2 > 0 ), but
Stable (maintains estimate 
^2 biased if 2 (random effect) is small, so:

Re-parametrise the variance model:
?2
2
Var(Ei) = Si + 
= (Si2 – minj Sj2) + (2 + minj Sj2)
? 2
2
=
Si*
+
*
^2 = *
^ 2 – min S 2 (less biased estimate)
• 
j j
Problem: 4 runs, 3 df for random effects sd
Run 1
Run 2
Run 3
Run 4
...
MULTISTAT
5
Effect
Ei
0
… very noisy sd:
-5
2
Sd
Si
1
… and T>15.96 for P<0.05 (corrected):
0
5
T stat
Ei / Si
0
-5
… so no response is detected …
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,
sd = smooth
random effects sd
fixed effects sd
) fixed effects sd
which removes the anatomical structure
before smoothing.
Random effects sd
(3 df)
Fixed effects sd
(448 df)
Regularized sd
(112 df)
4
3
2
1
0
Random effects sd
Fixed effects sd
Fixed effects sd
Over
runs
Over subjects
3
Smooth
15mm
~1
~1.6
2
~3
1
0
Effective df depends on the smoothing
(
FWHMratio2
dfratio = dfrandom 2
+1
2
FWHMdata
1
1
1
=
+
dfeff dfratio dffixed
e.g. dfrandom = 3,
)
3/2
dffixed = 112, FWHMdata = 6mm:
FWHMratio (mm) 0 5 10 15 20 infinite
dfeff
3 11 45 112 192 448
Random effects
Fixed effects
variability
bias
compromise!
Final result: 15mm smoothing, 112 effective df …
Run 1
Run 2
Run 3
Run 4
MULTISTAT
5
Effect
Ei
0
… less noisy sd:
-5
2
Sd
Si
1
… and T>4.90 for P<0.05 (corrected):
0
5
T stat
Ei / Si
0
-5
… and now we can detect a response!
Conjunction: All Ti > threshold = Min Ti > threshold
‘Minimum of Ti’
For P=0.05,
threshold = 1.82
Efficiency = 82%
‘Average of Ti’
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
1
For P=0.05,
threshold = 4.90
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
If the conjunction is significant,
does it mean that all effects > 0?
• Problem: for the conjunction of 20 effects, the threshold
can be negative!?!?!
• Reason: significance is based on the wrong null
hypothesis, namely: all effects = 0
• Correct null hypothesis is: at least one effect = 0.
Unfortunately the P-value depends on the unknown > 0
effects …
• If the effects are random, all effects > 0 is meaningless.
The only parameter is the (single) population effect, so
that the conjunction just tests if population effect > 0.
• P-values now depend on the random effects sd, not the
fixed effects sd. But the minimum (i.e. the conjunction) is
less efficient (sensitive) than the average (the usual test).
FWHM – the local smoothness of the noise
• Used by STAT_THRESHOLD to find the P-value of local maxima and
the spatial extent of clusters of voxels above a threshold.
• u = normalised residuals from linear model = residuals / sd
• u· = vector of spatial derivatives of u
· 1/2 (mm-3)
• λ = |Var(u)|
• FWHM = (4 log 2)1/2 λ-1/3 (mm)
(If residuals are modeled as white noise smoothed with a Gaussian kernel,
this would be its FWHM).
• λ and FWHM are corrected for low df and large voxel size so they are
approximately unbiased.
• For a search region S, the number of ‘resolution elements’ is
Resels(S) = Vol(S) AvgS(FWHM-3) = Vol(S) AvgS(λ) (4 log 2)-3/2
• For local maxima in S, P_value = Resels(S) x (function of threshold).
• For a cluster C, P-value depends on Resels(C) instead of Vol(C), so that
clusters in smooth regions are less significant.
• Need a correction for the randomness of λ and FWHM - depends on df .
• Correction is more important for small clusters C than for large search
regions S.
…. FWHM depends on the spatial correlation between neighbours
FWHM (mm) over scans (448 df)
20
Resels=1.90
P=0.007
Resels=0.57
P=0.387
smoothed FWHM over runs
FWHM (mm) over runs (3 df)
20
15
15
10
10
5
5
0
0
20
smoothed (runs FWHM / scans FWHM)
1.5
15
10
1
5
0
0.5
T > 4.90
T>4.86
(P < 0.05,
corrected)
Smooth the data before analysis?
• Temporal smoothing or low-pass filtering is used by
SPM’99 to validate a global AR(1) model. For our local
AR(p) model, it is not necessary (but ~ harmless).
• Spatial smoothing is used by SPM’99 to validate
random field theory. Can be harmful for focal signals.
Should fix the theory! STAT_THRESHOLD uses the
better of the Bonferroni or the random field theory.
• A better reason for spatial smoothing is greater
detectability of extensive activation: choose the FWHM
to match the activation (e.g. 10mm FWHM for 10mm
activations) – or try a range of FWHM’s i.e. scale space
– but thresholds are higher …
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
Signal + Gaussian
white noise
Signal
Noise
P < 0.05 (uncorrected), Z > 1.64
5% of volume is false +
4
4
2
2
0
-2
-4
FDR < 0.05, Z > 2.82
5% of discoveries is false +
True +
False +
0
-2
-4
P < 0.05 (corrected), Z > 4.22
5% probability of any false +
4
4
2
2
0
0
-2
-2
-4
-4
Comparison of thresholds
• FDR depends on the ordered P-values (not smoothness):
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 + 1
0.1 0.01 0.001 0.0001
Threshold Z 1.64 2.56 3.28 3.88 4.41
• Bonferroni thresholds the P-values at a/n:
Number of voxels 1
10 100 1000 10000
Threshold Z 1.64 2.58 3.29 3.89 4.42
• Random field theory: resels = volume / FHHM3:
Number of resels
0
1
10 100 1000
Threshold Z 1.64 2.82 3.46 4.09 4.65
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?
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
0.2
basis1
basis2
HRF
0
-0.2
-0.4
-5
0
5
10
t (seconds)
15
20
25
shift
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
1
• Equate w2 / w1 to estimates, then
solve for shift (Hensen et al., 2002)
w2 / w1
2
w1
• To reduce bias when the magnitude
is small, use
0
shift / (1 + 1/T2)
w2
-1
-2
where T = w1 / Sd(w1) is the T statistic
for the magnitude
-3
-5
• Shrinks shift to 0 where there is little
evidence for a response.
0
shift (seconds)
5
Delay of the hot stimulus (= shift + 5.4 sec)
T stat for magnitude
5
T stat for shift
5
0
0
-5
-5
Delay (secs)
Sd of delay (secs)
10
5
8
4
6
3
4
2
2
1
0
0
Varying the delay and dispersion of the reference HRF
T stat for magnitude
10.4
Reference dispersion (seconds)
T stat for shift
5
5
10.4
0
5.2
2.6
0
5.2
2.6
2.4
5.4
8.4
Delay (secs)
10.4
-5
2.4
8.4
Sd of delay (secs)
10
8
5.4
10.4
5
4
6
5.2
-5
3
5.2
4
2
2.6
2.4
5.4
2
1
2.6
0
8.4
2.4
Reference delay (seconds)
5.4
8.4
0
T > 4.86
(P < 0.05,
corrected)
Delay
(secs)
6.5
5
5.5
4
4.5
T > 4.86
(P < 0.05,
corrected)
Delay
(secs)
6.5
5
5.5
4
4.5
EFFICIENCY for optimum block design
Sd of hot stimulus
Sd of hot-warm
0.5
20
Magnitude
0.4
15
Optimum
design
Delay
InterStimulus Interval (secs)
10
10
0
10
15
20
0.8
15
5
10
15
5
0
20
0
15
0.1
20
20
0.8
15
5
0
0
(secs)
1
0.6
Optimum
design
0.4
0.2
X
10
5
10
X
(Not enough signal)
0.2
Optimum
design
0.6
Optimum
design
10
5
0
(secs)
1
20
0.3
0.2
0.1
5
0.4
15
0.3
X
5
0.5
20
0.4
X
(Not enough signal)
5
Stimulus Duration (secs)
10
15
0.2
20
0
EFFICIENCY for optimum event design
0.5
0.45
(Not
enough
signal)
____ magnitudes
……. delays
uniform . . . . . . . . .
random .. . ... .. .
concentrated :
Sd of effect (secs for delays)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
5
10
15
Average time between events (secs)
20
How many subjects?
• Variance =
sdrun2
sdsess2
sdsubj2
nrun nsess nsubj + nsess nsubj + nsubj
• The largest portion of variance comes from the
last stage, i.e. combining over subjects.
• If you want to optimize total scanner time, take
more subjects, rather than more scans per subject.
• What you do at early stages doesn’t matter very
much - any reasonable design will do …
Comparison SPM’99:
FMRISTAT:
• Different slice • Adds a temporal
•Shifts the model
acquisition times: derivative
• Drift removal: • Low frequency cosines • Polynomials
(flat at the ends)
(free at the ends)
• Temporal
• AR(1), global parameter, • AR(p), voxel
correlation:
bias reduction not
parameters, bias
necessary
reduction
• Estimation of
• Band pass filter, then
• Pre-whiten, then least
effects:
least-squares, then
squares (no further
correction for temporal
corrections needed)
correlation
• Rationale:
• More robust, low df
• More efficient, high df
• Random effects: • No regularization, low df • Regularization, high df
• FWHM:
• Global, ~ OK for local • Local, is OK for local
maxima, but not clusters maxima and clusters
• Map of delay: • No
• Yes
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
fMRI response. NeuroImage, 16:593-606.
• http://www.math.mcgill.ca/keith/fmristat
- 200K of MATLAB code
- fully worked example
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