Statistical Inference and Random
Field Theory
Will Penny
SPM short course, London, May 2003
M.Brett et al. Introduction to Random Field
Theory, To appear in HBF, 2nd Edition.
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. Theory
3. Imaging Data
4. Levels of Inference
5. SPM Results
+FDR ?
Overview
1. Terminology
2. Theory
3. Imaging Data
4. Levels of Inference
5. SPM Results
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.
Sensitivity and Specificity
ACTION
TRUTH
H True (o)
H False (x)
At u1
Don’t
Reject
Reject
TN
FP
At u2
TP
Sens=7/10=70%
Spec=9/10=90%
FN
Sens=10/10=100%
Spec=7/10=70%
Eg. t-scores
from regions
that truly do and
do not activate
Sensitivity = TP/(TP+FN) = b
Specificity = TN/(TN+FP) = 1 - a
FP = Type I error or ‘error’
oooooooxxxooxxxoxxxx
FN = Type II error
a = p-value/FP rate/error rate/significance level
b = power
u1
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
• Family of hypotheses
– Hk k   = {1,…,K}
– H = H1  H2 …  Hk  HK
If we reject a voxel null hypothesis
at any voxel, we reject the family-wise
Null hypothesis
A FP anywhere gives a Family
Wise Error (FWE)
Family-wise error rate = ‘corrected’ p-value
Use of ‘uncorrected’ p-value, a=0.1
Use of ‘corrected’ p-value, a=0.1
FWE
The Bonferroni correction
Given a family of N independent voxels and a voxel-wise error rate v
the Family-Wise Error rate (FWE) or ‘corrected’ error rate is
α = 1 – (1-v)N
~ Nv
Therefore, to ensure a particular FWE we choose
If v=0.05 then over
100 voxels we’ll get
5 voxel-wise type I
errors. But we’ll
get a much higher
α. To ensure α=0.05
we need v=0.0005 !
v=α/N
A Bonferroni correction is appropriate for independent tests
A correction for multiple comparisons
The Bonferroni correction
Independent Voxels
Spatially Correlated Voxels
Bonferroni is too conservative for brain images
Overview
1. Terminology
2. Theory
3. Imaging Data
4. Levels of Inference
5. SPM Results
Random Field Theory
• Consider a statistic image as a lattice representation of a
continuous random field
• Use results from continuous random field theory
Lattice
representation
Euler Characteristic (EC)
Topological measure
–
–

-
threshold an image at u
excursion set Au
(Au) = # blobs - # holes
At high u, (Au) = # blobs
Reject HΩ if Euler char non-zero
α  Pr((Au) > 0 )
Expected Euler char  p–value
(at high u)
α  E[(Au)]
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
Using R=100 in a
Bonferroni correction
gives u=3.3
Friston et al. (1991) J. Cer. Bl. Fl. M.
Developments
2D Gaussian fields
Friston et al. (1991) J. Cer. Bl. Fl. M. (Not EC Method)
3D Gaussian fields
Worsley et al. (1992) J. Cer. Bl. Fl. M.
3D t-fields
Worsley et al. (1993) Quant. Brain. Func.
Restricted search regions
Box and
frame have
same
number of
voxels
Box has
16 markers
Frame has
32 markers
Unified Theory
• General form for expected Euler characteristic
• 2, F, & t fields • restricted search regions
Au
α = S Rd () rd (u)
Rd (): RESEL count; depends on
the search region – how big, how
smooth, what shape ?
Worsley et al. (1996), HBM

rd (u): EC density; depends on
type of field (eg. Gaussian, t) and the
threshold, u.
Unified Theory
• General form for expected Euler characteristic
• 2, F, & t fields • restricted search regions
Au
α = S Rd () rd (u)
Rd (): RESEL count

rd (u): d-dimensional EC density
–
E.g. Gaussian RF:
R0()
R1()
R2()
R3()
=
=
=
=
() Euler characteristic of 
resel diameter
resel surface area
resel volume
Worsley et al. (1996), HBM
r0(u)
r1(u)
r2(u)
r3(u)
r4(u)
= 1- (u)
= (4 ln2)1/2 exp(-u2/2) / (2p)
= (4 ln2)
exp(-u2/2) / (2p)3/2
= (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2p)2
= (4 ln2)2 (u3 -3u) exp(-u2/2) / (2p)5/2
Resel Counts for Brain Structures
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
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
Component fields
voxels
data matrix
scans
b
= X
=
design matrix
Y

+
?
parameters
+


errors
?
variance
component
fields
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
Smoothness Estimation
• Roughness ||
• Gaussian PRF
S

 var xe  cov xe , ye

   cov xe , ye
var ye

e e
e e
 covx , z  cov y , z






covxe , ez 

cov ye , ez 

var ez  


fx 0 0
0 fy 0
0 0 fz
 || = (4ln(2))3/2 / (fx  fy  fz)
• Point Response Function PRF
• RESEL COUNT
R3() = () / (fx  fy  fz)
Approximate the peak of the
Covariance function with a Gaussian
α = R3() (4ln(2))3/2 (u 2 -1) exp(-u 2/2) / (2p)2
RFT Assumptions
• Model fit & assumptions
– valid distributional results
• Multivariate normality
– of component images
• Covariance function of
component images must be
- Stationary (pre SPM99)
- Can be nonstationary
(SPM99 onwards)
- Twice differentiable
Smoothness
smoothness » voxel size
lattice approximation
smoothness estimation
practically
FWHM  3  VoxDim
otherwise
conservative
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 and Set-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 n or more connected
components in the excursion set ie. 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
Weak vs Strong
control over FWE
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
SPM99 results I
Activations
Significant at
Cluster level
But not at
Voxel Level
SPM99 results II
Activations
Significant at
Voxel and
Cluster level
SPM results...
False Discovery Rate
ACTION
TRUTH
H True (o)
H False (x)
At u1
Don’t
Reject
Reject
TN
FP
At u2
TP
FDR=1/8=13%
a=1/10=10%
FN
FDR=3/13=23%
a=3/10=30%
Eg. t-scores
from regions
that truly do and
do not activate
FDR = FP/(FP+TP)
a = FP/(FP+TN)
oooooooxxxooxxxoxxxx
u1
u2
False Discovery Rate
Illustration:
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 correct for multiple comparisons
• We can use Random Field Theory (RFT)
• RFT requires (i) a good lattice approximation to underlying
multivariate Gaussian fields, (ii) that these fields are continuous
with a twice differentiable correlation function
• To a first approximation, RFT is a Bonferroni correction using
RESELS.
• We only need to correct for the volume of interest.
• Depending on nature of signal we can trade-off anatomical
specificity for signal sensitivity with the use of cluster-level
inference.