Random Field Theory
Will Penny
SPM short course, London, May 2005
David Carmichael
MfD 2006
image data
parameter
estimates
design
matrix
kernel
realignment &
motion
correction
General Linear Model
smoothing
model fitting
statistic image
Random Field
Theory
normalisation
anatomical
reference
Statistical
Parametric Map
corrected p-values
Overview
1. Terminology
2. Random Field Theory
3. Cluster level inference
4. SPM Results
5. FDR
Overview
1. Terminology
2. Random Field Theory
3. Cluster level inference
4. SPM Results
5. FDR
Inference at a single voxel
NULL hypothesis, H: activation is zero
a = p(t>u|H)
u=2
t-distribution
p-value: probability of getting
a value of t at least as extreme
as u. If a is small we reject the
null hypothesis.
u=(effect size)/std(effect size)
Sensitivity and Specificity
ACTION
Don’t
Reject
TRUTH
H True TN
H False
Reject
FP
FN
TP
Specificity = TN/(# H True) = TN/(TN+FP) = 1 - a
Sensitivity = TP/(# H False) = TP/(TP+FN) = b = power
a = FP/(# H True) = FP/(TN+FP) = p-value/FP rate/sig level
Sensitivity and Specificity
ACTION
Don’t
Reject
TRUTH
At u1
Reject
H True (o)
TN=7
FP=3
H False (x)
FN=0
TP=10
Spec=7/10=70%
Sens=10/10=100%
Eg. t-scores
from regions
that truly do and
do not activate
Specificity = TN/(# H True)
Sensitivity = TP/(# H False)
oooooooxxxooxxxoxxxx
u1
Sensitivity and Specificity
ACTION
Don’t
Reject
TRUTH
Reject
At u2
H True (o)
TN=9
FP=1
H False (x)
FN=3
TP=7
Spec=9/10=90%
Sens=7/10=70%
Eg. t-scores
from regions
that truly do and
do not activate
Specificity = TN/(# H True)
Sensitivity = TP/(# H False)
oooooooxxxooxxxoxxxx
u2
Inference at a single voxel
NULL hypothesis, H: activation is zero
a = p(t>u|H)
We can choose u to ensure
a voxel-wise significance level of a.
u=2
t-distribution
This is called an ‘uncorrected’ p-value, for
reasons we’ll see later.
We can then plot a map of above threshold
voxels.
Inference for Images
Noise
Signal
Signal+Noise
Use of ‘uncorrected’ p-value, a=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 FP anywhere in the image
gives a Family Wise Error (FWE)
Family-Wise Error (FWE) rate = ‘corrected’ p-value
Use of ‘uncorrected’ p-value, a=0.1
Use of ‘corrected’ p-value, a=0.1
FWE
The Bonferroni correction
The Family-Wise Error rate (FWE), a, 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
Assume Independent Voxels
Independent voxels - a good assumption??
• Voxel Point Spread Function (PSF)
- continuous signal is sampled for a discrete period
- imposes a filter that when FT’d gives a PSF
- Gives spread of signal through the image from point source
..worse in PET
• Physiological noise
• Smoothing
• Normalisation
The Bonferroni correction
Independent Voxels
Spatially Correlated Voxels
Bonferroni is too conservative for brain images
Random Field Theory
• Consider a statistic image as a discretisation of a
continuous underlying random field
• Use results from continuous random field theory
Discretisation
Overview
1. Terminology
2. Random Field Theory
3. Cluster level inference
4. SPM Results
5. FDR
Euler Characteristic (EC)
Topological measure
– threshold an image at u
- EC = # blobs
- at high u:
Prob blob = avg (EC)
So
FWE, a = avg (EC)
Example – 2D Gaussian images
α = R (4 ln 2) (2π) -3/2 u exp (-u2/2)
Voxel-wise threshold, u
Number of Resolution
Elements (RESELS), R
N=100x100 voxels,
Smoothness FWHM=10,
gives R=10x10=100
Example – 2D Gaussian images
α = R (4 ln 2) (2π) -3/2 u exp (-u2/2)
For R=100 and α=0.05
RFT gives u=3.8
How do we know number of resels?
1. We can simply use the
FWHM of the smoothing
kernel
But processes such as
normalisation mean
smoothness will vary
2. Estimate the FWHM at
each voxel using residuals
at each voxel (worsley
1998)
Resel Counts for Brain Structures
volume
Surface
area
diameter
Euler # of
space
FWHM=20mm
(1) Threshold depends on Search Volume
(2) Surface area makes a large contribution
Overview
1. Terminology
2. Theory
3. Imaging Data
4. Levels of Inference
5. SPM Results
Applied Smoothing
Smoothness
smoothness » voxel size
practically
FWHM  3  VoxDim
Typical applied smoothing:
Single Subj fMRI: 6mm
PET: 12mm
Multi Subj fMRI: 8-12mm
PET: 16mm
Overview
1. Terminology
2. Theory
3. Imaging Data
4. Levels of Inference
5. SPM Results
Cluster Level Inference
• We can increase sensitivity by trading off anatomical specificity
• Given a voxel level threshold u, we can compute
the likelihood (under the null hypothesis) of getting a cluster containing at
least n voxels
CLUSTER-LEVEL INFERENCE
• Similarly, we can compute the likelihood of getting c
clusters each having at least n voxels
SET-LEVEL INFERENCE
Levels of inference
voxel-level
P(c  1 | n > 0, t  4.37) = 0.048 (corrected)
At least one
cluster with
unspecified
number of
voxels above
threshold
n=1
2
set-level
P(c  3 | n  12, u  3.09) = 0.019
n=82
n=32
cluster-level
P(c  1 | n  82, t  3.09) = 0.029 (corrected)
At least one cluster with at least 82 voxels above threshold
At least 3 clusters above
threshold
Overview
1. Terminology
2. Theory
3. Imaging Data
4. Levels of Inference
5. SPM Results
SPM results I
Activations
Significant at
Cluster level
But not at
Voxel Level
SPM results II
Activations
Significant at
Voxel and
Cluster level
SPM results...
False Discovery Rate
ACTION
Don’t
Reject
TRUTH
At u1
Reject
H True (o)
TN=7
FP=3
H False (x)
FN=0
TP=10
Eg. t-scores
from regions
that truly do and
do not activate
FDR = FP/(# Reject)
a = FP/(# H True)
FDR=3/13=23%
a=3/10=30%
oooooooxxxooxxxoxxxx
u1
False Discovery Rate
ACTION
Don’t
Reject
TRUTH
At u2
Reject
FDR=1/8=13%
a=1/10=10%
H True (o)
TN=9
FP=1
H False (x)
FN=3
TP=7
Eg. t-scores
from regions
that truly do and
do not activate
FDR = FP/(# Reject)
a = FP/(# H True)
oooooooxxxooxxxoxxxx
u2
False Discovery Rate
Noise
Signal
Signal+Noise
Control of Familywise Error Rate at 10%
Occurrence of Familywise Error
FWE
Control of False Discovery Rate at 10%
6.7%
10.4%
14.9%
9.3% 16.2% 13.8% 14.0% 10.5% 12.2%
Percentage of Activated Pixels that are False Positives
8.7%
Summary
• We should not use uncorrected p-values
• We can use Random Field Theory (RFT) to ‘correct’ p-values
• RFT requires FWHM > 3 voxels
• We only need to correct for the volume of interest
• Cluster-level inference
• False Discovery Rate is a viable alternative
Functional Imaging Data
• The Random Fields are the component fields,
Y = Xw +E,
e=E/σ
• We can only estimate the component fields, using
estimates of w and σ
• To apply RFT we need the RESEL count which
requires smoothness estimates
voxels
data matrix
scans
=
design matrix
Estimated component fields

?
parameters
+
errors
?
^
b
 estimate

parameter
estimates


=
Each row is
an estimated
component field
residuals
estimated variance
estimated
component
fields