Random field theory
Rumana Chowdhury and Nagako Murase
Methods for Dummies
November 2010
Overview
Part 1
• Multiple comparisons
• Family-wise error
• Bonferroni correction
• Spatial correlation
Part 2
• Solution = Random Field Theory
• Example in SPM
Image time-series
Realignment
Spatial filter
Design matrix
Smoothing
General Linear Model
Statistical Parametric Map
Statistical
Inference
Normalisation
Anatomical
reference
Parameter estimates
RFT
p <0.05
Voxel
• Raw data collected as group of
voxels
• 3D, volumetric pixel
– Location
– Value
• Calculate a test statistic for
each voxel
• Many many many voxels…
Null hypothesis
• Determine if value of single specified voxel is
significant
• Create a null hypothesis, H0 (activation is zero)
= data randomly distributed, Gaussian distribution of
noise
• Compare our voxel’s value to a null distribution
Single voxel level statistics
•Perform t-tests
u
•Decision rule (threshold) u,
determines false +ve rate 
 = p(t>u|H)
•Choose u to give acceptable
α under H0
t
• = P(type I error)
i.e. chance we are wrong when
rejecting the null hypothesis
Multiple comparisons problem
uu

• fMRI – lots of voxels, lots of ttests
u
u
u
• If use same threshold, higher
probability of obtaining at least
1 false +ve

t

t
t
t



t
e.g. for alpha=0.05, 10000 voxels: expect 500 false positives
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%
Family-wise error
• In fMRI = volume (family) of voxel statistics
• Family-wise null hypothesis = activation is zero everywhere
• Family Wise Error (FWE) = 1 false positive anywhere
• FWE rate = ‘corrected’ p-value
Use of ‘uncorrected’ p-value, α =0.1
Use of ‘corrected’ p-value, α =0.1
FWE
Definitions
Univariate statistics
Functional imaging
1 observed data
1 statistical value
type I error rate
null hypothesis
many voxels
family of statistical values
family-wise error rate (FWE)
family-wise null hypothesis
Thresholding
•Height thresholding
•This gives us localizing power
Bonferroni correction
p =  /n
Corrected p-value
α = acceptable Type 1 error rate
n = number of tests
•The Family-Wise Error rate (FWE), α, 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…
Spatial Correlation
•Dependence between voxels :
physiological signal
data acquisition
spatial preprocessing
Averaging over one voxel and its
neighbours (independent
observations)
Usually weighted average using a
(Gaussian) smoothing kernel
The problem with Bonferroni
100 x 100 voxels – normally distributed
independent random numbers
0.05/10000 = 0.000005
Z score 4.42
Averaged 10x10
Appropriate correction:
0.05/100 = 0.0005
Z score 3.29
Fewer independent observations than there are voxels
Bonferroni is too conservative (high false negative)
• Not making inferences on single voxels
• Take into account spatial relationships
• Topology
Euler
• Leonhard Euler (1797-1783), Swiss mathematician
• Seven bridges of Kӧnisberg
“the problem has
no solution!”
Euler characteristic: the beginnings
• EC
V-E+F = 2
• Number of polyhedra (P)
V-E+F-P=1
8 – 12 + 6 = 2
16 – 28 + 16 – 3 = 1
• Holes & handles
reduce by 1
0 (product of 2 circles)
• Topology…
0d - 1d + 2d - 3d + 4d…etc
EC is a topological
measure…
(a little bit more background)
• Robert J Adler (1981): relationship between topology of random field
(local maxima) and EC
• Apply a threshold to random field; regions above = excursion sets
• EC is a topological measure of excursion set
• Expected EC is a good approximation of FWE at higher threshold
• Random field theory uses the expected EC
Random field theory: overview
•
Consider statistic image as lattice representation of a continuous
random field
lattice
representation
•
Takes into account smoothness and shape of the data as well as
number of voxels to apply an appropriate threshold
References
• An Introduction to Random Field Theory (Chapter 14) Human Brain
Mapping
• Developments in Random Field Theory (Chapter 15), KJ Worsley
• Previous MfD slides:
http://www.fil.ion.ucl.ac.uk/mfd/page2/page2.html
• Guillaume Flandin’s slides:
http://www.fil.ion.ucl.ac.uk/spm/course/slides10-meeg/
• Will Penny’s slides:
http://www.fil.ion.ucl.ac.uk/spm/course/slides05/ppt/infer.ppt#324,1,
Random Field Theory
• R. Adler’s website:
http://webee.technion.ac.il/people/adler/research.html
• CBU imaging wiki: http://imaging.mrccbu.cam.ac.uk/imaging/PrinciplesRandomFields
Methods for Dummies 2010
Random Field Theory
Part II
Nagako Murase
17/11/2010
RFT for dummies - Part II
21
Overview
A large volume of imaging data
Multiple comparison problem
<smoothing >
Treat as a single voxel
by simply averaging
Bonferroni correction
α=PFWE/n
Corrected p value
Uncorrected p value
Random field theory (RFT)
α = PFWE ≒ E[EC]
Corrected p value
Too false positive
Never use this.
FWE rate is too low
to reject the null hypothesis
Too false negative
It is because Bonferroni correction
is based on the assuption that all
the voxels are independent.
Process of RFT application:
3 steps
1st
Smoothing →Estimation of smoothness
(spatial correlation)
2nd
Applying RFT
3rd
Obtaining PFWE
image data
design
matrix
parameter
estimates
kernel
realignment &
motion
correction
General Linear Model
smoothing
Ümodel fitting
Üstatistic image
Thresholding &
Random Field
Theory
normalisation
anatomical
reference
Statistical
Parametric Map
Corrected thresholds & p-values
st
1
Smoothing →Estimation of smoothness
 By smoothing, data points are
averaged with their
neighbours.
 A smoothing kernel (shape)
such a Gaussian is used.
 Then each value in the image
is replaced with a weighted
average of itself and its
neighbours.
 Smoothness is expressed as
FWHM (full width at half
maximum)
Gaussian curves
FWHM
Standard Normal Distribution
(Probability density function)
Mean = 0
Standard Deviation = 1
For example, FWHM of 10 pixels in X axis means that at 5 pixels from
the center, the value of the kernel is half of its peak value.
1st
Smoothing →Estimation of smoothness
Original data:
an image using independent
random numbers from the
normal distribution
After smoothing with a Gaussian
smoothing kernel
FWHM in x=10, in y=10
so this FWHM=100 pixels)
Resel
a block of values, e.g. pixels,
that is the same size as the FWHM.
 a word made form ‘Resolution Elements’
one of a factor which defines p value
in RFT
<example of ressel>
The FWHMs were 10 by 10 pixels.
Thus a resel is a block of 100 pixels.
As there are 10,000 pixels in our image,
there are 100 resels.
The number of ressels depend
on
 the FWHM
 the number of boxels
(pixels).
Smoothing
• Compiles the data of spatial correlation.
• Reduce the number of independent
observations.
• Generates a blurred image.
• Increases signal-to-noise ratio.
• Enables averaging across subjects.
2nd step
Apply RFT
After smoothing
Euler characteristics
(EC)= the number of
blobs (minus number
of holes) in an image
after thresholding
thresholding
Set the threshold as z core 2.5
Below 2.5..0..black
Above 2.5..1..white
EC=3
Different Z score threshold
generates different EC.
z=2.5
Z=2.75
EC=3
EC=1
Thresholding
No of blobs
≒ EC
3rd step Obtain PFWE
Expected EC: E[EC] =
the probability of finding a blob
PFWE ≒ E[EC]
α = E[EC] = R (4 loge 2) (2π) -3/2 zt exp(-zt2/2)
E[EC] depends on:
R the number of resels
Zt Z score threshold
E[EC]=0.05
RFT
Using this Z score, we can conclude that any blots have a probability
of ≦0.05 when they have occured by chance.
α=E[EC]=0.05 Z=3.8
Bonferroni correction
α =0.05/10,000=0.00005 Z=4.42
If the assumption of RFT are met, then the RFT threshold is
more accurate than the Bonferroni correction.
RFT in 3D
• EC=the number of 3D blobs
• Resel=a cube of voxels of size (FWHM in x) by
(FWHM in y) by (FWHM in z)
• In SPM, the formulae for t, F and χ2random fields
are used to calculate threshold for height.
• RFT requires FWHM > 3 voxels
27 Voxels
1 RESEL
RFT Note 1:
When FWHM is less than 3.2 voxels, the
Bonferroni correction is better than the RFT
for a Gaussian statistic.
RFT Note2:
EC depends on volume shape and
size.
• EC depends, not only on resel numbers,
but also on the shape and size of the
volume we want to search (see table).
• The shape becomes important when we
have a small or oddly shaped regions.
V: volume of search region
R0(V): ressel single boxel count
R1(V): ressel radius
R2(V): ressel surface area
R3(V): ressel volume
Worsley KJ, et al. , Human Brain Mapping 1996
Correction in case of a small shaped
region
• Restricting the
search region to a
small volume
within a statistical
map can reduce
thresholds for
given FWE rates.
T thoreshold giving a FWE rate of 0.05.
Volume of Interest:
EC
Diameter Surface Area
FWHM=20mm
Threshold depends on Search Volume
Volume
Note 3:
voxel-level inference → a larger framework
inference:
different thresholding method
• Cluster-level inference
• Set-level inference
• These framework require
Height threshold
spatial extent threshold
Peak (voxel), cluster and set
level inference
Regional
Sensitivity
specificity
Peak level inference:
height of local maxima
(Special extent threshold
is 0)
Cluster level inference:
number of activated
voxels comprising a
particular region (spatial
extent above u)
: significant at the set level
: significant at the cluster level
: significant at the peak level
L1 > spatial extent threshold
L2 < spatial extent threshold
 Set level inference:
the height and volume
threshold (spatial extent
above u)→ number of
clusters above u

Which inference we should
use?
• It depends on what you're looking at.
• Focal activation is well detected with
greater regional specificity using voxel
(peak) – level test.
• Cluster-level inference – can detect
changes missed on voxel-level inference,
because it uses the spaticial extent
threshold as well.
SPM8 and RFT: Example Data
SPM manual, http://www.fil.ion.ucl.ac.uk/spm/doc/
Random Field Theory: two
assumptions
The error fields are a reasonable lattice approximation to
an underlying random field , with a multivariate Gaussian
distribution.
The error fields are continuous.
The data can be sufficiently smoothed.
The errors are indeed Gaussian and
General Linear Models can be correctly specified.
RFT assumption is met.
A case where the RFT assumption is not
met.
Small number of subjects
The error fields are not very smooth.
Increase the subject number
Use Bonferroni correction
Conclusion
A large volume of imaging data
Multiple comparison problem
Treat as a single voxel
by simply averaging
Bonferroni correction
α=PFWE/n
Corrected p value
Uncorrected p value
Too false positive
Never use this.
FWE rate is too low
to reject the null hypothesis
Too false negative
<smoothing with a
Gaussian kernel,
FWHM >
Random field theory (RFT)
α = PFWE ≒ E[EC]
Corrected p value
Conclusion
• By thoresholding, expected EC is calculated by RFT,
where
PFWE ≒ E[EC]
• Restricting the search region to a small volume, we can
reduce the threshold for given FWE rates.
• FWHM is less than 3.2 voxels, the Bonferroni correction
is better.
• Voxel-level and cluster-level inference are used
depending on what we are looking at.
• In case of small number of subjects, RFT is not met.
Acknowledgement
• The topic expert:
Guillaume Flandin
• The organisers:
Christian Lambert
Suz Prejawa
Maria Joao
Thank you for your attention!