Transcript DCM - UZH
Multiple comparison correction
Klaas Enno Stephan
Branco Weiss Laboratory (BWL)
Institute for Empirical Research in Economics
University of Zurich
Functional Imaging Laboratory (FIL)
Wellcome Trust Centre for Neuroimaging
University College London
With many thanks for slides & images to:
FIL Methods group
Methods & models for fMRI data analysis
29 October 2008
Overview of SPM
Image time-series
Realignment
Kernel
Design matrix
Smoothing
General linear model
Statistical parametric map (SPM)
Statistical
inference
Normalisation
Gaussian
field theory
p <0.05
Template
Parameter estimates
Voxel-wise time series analysis
model
specification
Time
parameter
estimation
hypothesis
statistic
BOLD signal
single voxel
time series
SPM
Inference at a single voxel
NULL hypothesis
H0: activation is zero
t-distribution
= p(t > u | H0)
p-value: probability of getting a value
of t at least as extreme as u.
If is small we reject the null
hypothesis.
u
contrast of
estimated
parameters
t=
variance
estimate
We can choose u to ensure a voxelwise significance level of .
cT ˆ
t
T
ˆ
stˆd (c )
cT ˆ
ˆ 2cT X T X c
1
~ tN p
t-contrast: a simple example
Passive word listening versus rest
cT = [ 1
0 ]
1
Q: activation during
listening ?
10
Null hypothesis:
X
20
30
1 0
c ˆ
t
Std (cT ˆ )
T
40
50
60
70
80
0.5
1
1.5
Design matrix
2
2.5
p ( y | cT ˆ 0)
Statistics:
set-level
p
c
0.000 10
SPMresults:
Height threshold T = 3.2057 {p<0.001}
voxel-level
mm mm mm
( Z)
T
p uncorrected
p-values adjusted for search volume
13.94
12.04
11.82
0.000
520
13.72
12.29
0.000
426
9.89
7.39
0.000
35
6.84
0.000
9
0.002
3
6.36
0.000
8
6.19
0.000
9
0.005
2
5.96
0.015
1
5.84
0.015
1
5.44
5.32
cluster-level
p corrected
kE
p
Inf 0.000
voxel-level
Infp 0.000
p FDR-corr T
uncorrected
FWE-corr
Inf 0.000
0.000
0.000
13.94
Inf0.000
0.000
0.000
0.000
12.04
0.000 0.000
0.000
11.82
Inf
0.000
0.000
13.72
7.830.000
0.000
0.000
0.000
12.29
6.360.000 0.000
0.000
9.89
0.000
0.000
0.000
7.39
5.99
0.000
0.000
0.000
0.000
6.84
0.024
0.001
0.000
5.65 0.000 6.36
0.001
0.001
0.000
6.19
5.530.003 0.000
0.000
0.000
5.96
0.058
0.000
5.84
5.360.004 0.000
0.166
0.000
5.44
5.270.022
0.166
0.036 0.000
0.000
5.32
4.97 0.000
4.87 0.000
-63 -27 15
mm mm mm
-48 p -33 12
(Z )
uncorrected
-66
-21
6
Inf
-63 -27 15
57 0.000
-21 12
Inf
0.000
-48 -33 12
Inf
-66 -21
6
63 0.000
-12 -3
Inf
0.000
57 -21 12
57
-39
6
Inf
0.000
63 -12 -3
36 0.000
-30 -15
7.83
57 -39
6
6.36
0.000
36 -30 -15
51
0
48
5.99
0.000
51
0 48
5.65
-63 -54 -3
-63 0.000
-54 -3
5.53
0.000
-30 -33 -18
-30 0.000
-33 -18
5.36
36 -27
9
5.27
36 0.000
-27 -459 42 9
4.97
48 27 24
-45 0.000
42
9 -27 42
4.87
0.000
36
48 27 24
36 -27 42
Inference on images
Noise
Signal
Signal+Noise
Use of ‘uncorrected’ p-value, =0.1
11.3%
11.3%
12.5%
10.8%
11.5%
10.0%
10.7%
11.2%
Percentage of Null Pixels that are False Positives
10.2%
9.5%
Using an ‘uncorrected’ p-value of 0.1 will lead us to conclude on
average that 10% of voxels are active when they are not.
This is clearly undesirable. To correct for this we can define
a null hypothesis for images of statistics.
Family-wise null hypothesis
FAMILY-WISE NULL HYPOTHESIS:
Activation is zero everywhere.
If we reject a voxel null hypothesis
at any voxel, we reject the family-wise
null hypothesis
A false-positive anywhere in the image
gives a Family Wise Error (FWE).
Family-Wise Error (FWE) rate = ‘corrected’ p-value
Use of ‘uncorrected’ p-value, =0.1
Use of ‘corrected’ p-value, =0.1
FWE
The Bonferroni correction
The family-wise error rate (FWE), , for a family of N
independent voxels is
α = Nv
where v is the voxel-wise error rate.
Therefore, to ensure a particular FWE set
v=α/N
BUT ...
The Bonferroni correction
Independent voxels
Spatially correlated voxels
Bonferroni is too conservative for smooth brain images !
Smoothness
• intrinsic smoothness
– some vascular effects have extended spatial support
• extrinsic smoothness
– resampling during preprocessing
– matched filter theorem
deliberate additional smoothing to increase SNR
• described in resolution elements: "resels"
• resel = size of image part that corresponds to the FWHM (full width half
maximum) of the Gaussian convolution kernel that would have produced the
observed image when applied to independent voxel values
• # resels is similar, but not identical to the number of independent
observations
• can be computed from spatial derivatives of the residuals
• critical for corrected p-values based on Gaussian random field theory
Random Field Theory
• Consider a statistic image as a discretisation of a
continuous underlying random field
• Use results from continuous random field theory
Discretisation
(“lattice
approximation”)
Euler Characteristic (EC)
Topological measure
– threshold an image at u
- EC = # blobs
- at high u:
p (blob) = E [EC]
therefore
FWE, = E [EC]
Euler Characteristic (EC)
EEC R(4 log 2)(2 )
R
ZT
3 / 2
ZT exp( 0.5Z )
2
T
= number of resels
= Z value threshold
We can determine that Z threshold for which E[EC] = 0.05.
At this threshold, every remaining voxel represents a significant
activation, corrected for multiple comparisons across the search
volume.
Voxel, cluster and set level tests
Sensitivity
Regional
specificity
Voxel level test:
intensity of a voxel
Cluster level test:
spatial extent above u
Set level test:
number of clusters
above u
Small volume correction (SVC)
• When we have a good hypothesis about where an
activation should be, we can reduce the search
volume:
–
–
–
–
mask defined by (probabilistic) anatomical atlases
mask defined by functional localisers
mask defined by orthogonal contrasts
spherical search volume around known coordinates
• Computing EC is relatively insensitive to the shape of
the search volume.
Thank you