Random Field Theory
Mkael Symmonds, Bahador Bahrami
Random Field Theory
Mkael Symmonds, Bahador Bahrami
Overview

Spatial smoothing

Statistical inference

The multiple comparison problem

…and what to do about it
Overview

Spatial smoothing

Statistical inference

The multiple comparison problem

…and what to do about it
Statistical inference

Aim
– to decide if the data represents convincing
evidence of the effect we are interested in.

How
– perform a statistical test across the whole brain
volume to tell us how likely our data are to have
come about by chance (the null distribution).
Inference at a single voxel
NULL hypothesis, H0: activation is zero
t-distribution
t-value = 2.42
p-value: probability of getting
a value of t at least as extreme
as 2.42 from the t distribution (= 0.01).
a = p(t>t-value|H0)
t-value = 2.02
t-value = 2.42
alpha = 0.025
As p < α , we reject the null hypothesis
Sensitivity and Specificity
ACTION
Don’t
Reject
Reject
Chance
H0 True
TN
FP – type I error
Not by chance
H0 False
FN
TP
Specificity = TN/(# H True) = TN/(TN+FP) = 1 - a
Sensitivity = TP/(# H False) = TP/(TP+FN) = b = power
Many statistical tests




In functional imaging, there are many voxels, therefore many
statistical tests
If we do not know where in the brain our effect will occur, the
hypothesis relates to the whole volume of statistics in the
brain
We would reject H0 if the entire family of statistical values is
unlikely to have arisen from a null distribution – a familywise hypothesis
The risk of error we are prepared to accept is called the
Family-Wise Error (FWE) rate – what is the likelihood that the
family of voxel values could have arisen by chance
How to test a family-wise
hypothesis?
Height thresholding
This can localise significant test results
How to set the threshold?

Should we use the same alpha as
when we perform inference at a single
voxel?
Overview

Spatial smoothing

Statistical inference

The multiple comparison problem

…and what to do about it
How to set the threshold?
Signal + Noise
Use of ‘uncorrected’ alpha, 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%
LOTS OF SIGNIFICANT ACTIVATIONS OUTSIDE OF OUR SIGNAL BLOB!
9.5%
How to set the threshold?

So, if we see 1 t-value above our
uncorrected threshold in the family of tests,
this is not good evidence against the family-
wise null hypothesis

If we are prepared to accept a false positive
rate of 5%, we need a threshold such that,
for the entire family of statistical tests, there
is a 5% chance of there being one or more t
values above that threshold.
Bonferroni Correction

For one voxel (all values from a null distribution)
– Probability of a result greater than the threshold = α
– Probability of a result less than the threshold = 1-α

For n voxels (all values from a null distribution)
– Probability of all n results being less than the threshold
= (1-α)n
– Probability of one (or more) tests being greater than the
threshold:
= 1-(1-α)n
~= n.α (as alpha is small)
FAMILY WISE
ERROR RATE
Bonferroni Correction




So,
Set the PFWE < n.α
Gives a threshold α = PFWE / n
Should we use the Bonferroni
correction for imaging data?
100 x 100 voxels – normally
distributed independent random
numbers
10,000 tests  5% FWE rate
Apply Bonferroni correction to give
threshold of 0.05/10000 = 0.000005
This corresponds to a z-score of 4.42
We expect only 5 out of 100 such
images to have one or more z-scores
> 4.42
NULL HYPOTHESIS TRUE
100 x 100 voxels averaged
Now only 10 x 10 independent numbers
in our image
The appropriate Bonferroni correction is
0.05/100= 0.0005
This corresponds to z-score = 3.29
Only 5/100 such images will have one
or more z-scores > 3.29 by chance
Spatial correlation
Physiological Correlation
Spatial pre-processing
Smoothing
Assumes Independent Voxels
Independent Voxels
Spatially Correlated Voxels
Bonferroni is too conservative for brain images, but how to tell how many
independent observations there are?
Overview

Spatial smoothing

Statistical inference

The multiple comparison problem

…and what to do about it
Spatial smoothing
Why do you want to do it?

Increases signal-to-noise ratio

Enables averaging across subjects

Allows use of Gaussian Random Field
Theory for thresholding
Spatial Smoothing
What does it do?

Reduces effect of high frequency variation in
functional imaging data, “blurring sharp edges”
Spatial Smoothing
How is it done?

Typically in functional
imaging, a Gaussian
smoothing kernel is used
– Shape similar to normal
distribution bell curve
– Width usually described using
“full width at half maximum”
(FWHM) measure
e.g., for kernel at 10mm
FWHM:
-5
0
5
Spatial Smoothing
How is it done?

Gaussian kernel defines shape of function used
successively to calculate weighted average of each
data point with respect to its neighbouring data
points
Raw data
x
Gaussian function
=
Smoothed data
Spatial Smoothing
How is it done?

Gaussian kernel defines shape of function used
successively to calculate weighted average of each
data point with respect to its neighbouring data
points
Raw data
x
Gaussian function
=
Smoothed data
Spatial correlation
Physiological Correlation
Spatial pre-processing
Smoothing
Assumes Independent Voxels
Independent Voxels
Spatially Correlated Voxels
Bonferroni is too conservative for brain images, but how to tell how many
independent observations there are?
Overview

Spatial smoothing

Statistical inference

The multiple comparison problem

…and what to do about it
Random Field Theory
(ii)
Methods for Dummies 2008
Mkael Symmonds
Bahador Bahrami
What is a random field?

A random field is a list of random
numbers whose values are mapped
onto a space (of n dimensions).
Values in a random field are usually
spatially correlated in one way or
another, in its most basic form this
might mean that adjacent values do
not differ as much as values that are
further apart.
Why random field?

To characterise the properties our study’s
statistical parametric map under the NULL
hypothesis
– NULL hypothesis =
if all predictions were wrong
 all activations were merely driven by chance
 each voxel value was a random number

– What would the probability of getting a
certain z-score for a voxel in this situation be?
Random Field
Thresholded
@ one
Thresholded
@ Zero
Measurement 1
Thresholded
@ three
Number of
blobs = 4
Measurement 2
Number of
blobs = 0
Measurement 3
Number of
blobs = 1
Number of
blobs = 2
Measurement
1000000000
Average number of blobs = (4 + 0 + 1 + … + 2)/1000000000
The probability of getting a z-score>3 by chance
Therefore, for every z-score,
the expected value of number of blobs
=
probability of rejecting the null
hypothesis erroneously (α)
The million-dollar
question is:


thresholding the random field at which Zscore produces average number of blobs
< 0.05?
Or, Which Z-score has a probability = 0.05
of rejecting the null hypothesis
erroneously?
– Any z-scores above that will be significant!
So, it all comes down to
estimating the average number
of blobs (that you expect by
chance) in your SPM
Random field theory does that for you!
Expected number of blobs in
a random field depends on
…



Chosen threshold z-score
Volume of search region
Roughness (i.e.,1/smoothness) of the search region:
Spatial extent of correlation among values in the field;
it is described by FWHM
– Volume and Roughness are combined into RESELs
– Where does SPM get R from: it is calculated
from the residuals (RPV.img)
– Given the R and Z, RFT calculates the expected
number of blobs for you:
E(EC) = R (4 ln 2) (2π) -3/2 z exp(-z2/2)
Probability of Family Wise
Error
PFWE = average number of blobs under null
hypothesis
α = PFWE = R (4 ln 2) (2π) -3/2 z exp(-z2/2)
Thank you

References:
– Brett, Penny & Keibel. An introduction to
Random Field Theory. Chapter from Human
Brain Mapping
– Will Penny’s slides
(http://www.fil.ion.ucl.ac.uk/spm/course/slides0
5/ppt/infer.ppt#324,1,Random Field Theory)
– Jean-Etienne Poirrier’s slides
(http://www.poirrier.be/~jeanetienne/presentations/rft/spm-rft-slidespoirrier06.pdf)
– Tom Nichol’s lecture in SPM Short Course (2006)
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%
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
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
n=82
set-level
P(c  3 | n  12, u  3.09) =
At least 3 0.019
clusters above
threshold
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