Download Random Field Theory - Wellcome Trust Centre for Neuroimaging

Methods for Dummies 2008
Random Field Theory
Ciaran S Hill & Christian Lambert
Overview
PART ONE
• Statistics of a Voxel
• Multiple Comparisons and Bonferroni
correction
PART TWO
• Spatial Smoothing
• Random Field Theory
PART I
image data
design
matrix
parameter
estimates
kernel
realignment &
motion
correction
General Linear Model
smoothing
model fitting
statistic image
Thresholding &
Random Field
Theory
normalisation
anatomical
reference
Statistical
Parametric Map
Corrected thresholds & p-values
A Voxel
“A volume element: A unit of graphical
information that defines a point in 3D
space”
Consists of
1. Location
2. Value
Usually we divide the brain into 20,000+
Statistics of a voxel
• Determine if the value of a single specified voxel is
significant
• Create a null hypothesis
• Compare our voxel’s value to a null distribution:
“the distribution we would expect if there is no effect”
Statistics of a 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
Statistics of a voxel
• Compare each voxel’s value to whole brain volume to see
if significantly different
• A statistically different voxel should give us “localizing
power”
• Because we have so many voxels we will have a lot of
errors:
1000 voxels: 10 false positives (using 0.01)
This creates the multiple comparison problem
Statistics of a voxel
• If we do not know where the effect we are searching for
occurs in the brain we look at all the voxels
• Family-Wise Hypothesis
• “ The whole family of voxels arose by chance”
• If we think that the voxels in the brain as a whole is
unlikely to have arisen from a null distribution then we
would reject the null hypothesis
Statistics of a voxel
• Choose a Family Wise Error rate
• Equivalent to alpha threshold
• The risk of error we are prepared to accept
• This is the likelihood that all the voxels we see have
arisen by chance from the null distribution
• If >10 in 1000 then there is probably a statistical
difference somewhere in the brain but we can’t trust our
localization
• How do we test our Family Wise Hypothesis/choose our
FWE rate?
Thresholding
• Height thresholding
• This gives us localizing power
Thresholding
High Threshold (t>5.5)
t > 5.5
Good Specificity
Poor Power
(risk of false negatives)
Med. Threshold
t > 3.5
Low Threshold t<0.5
t > 0.5
Poor Specificity
(risk of false positives)
Good Power
Carlo E. Bonferroni
Bonferroni correction
•A method of setting a threshold above which
results are unlikely to have arisen by chance
•If we would trust a p value of 0.05 for one
hypothesis then for multiple hypotheses we should
use 1/n
•For example if one hypothesis requires 0.05 then
two should require 0.05/2 = 0.025
More conservative to ensure we maintain probability
Bonferroni correction
a(bon) = a /n
Thresholding
Error signal (noise)
Pure signal
What we actually get (bit of both)
Thresholding
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
Too much false positive outside our blob
10.2%
9.5%
Bonferroni correction
For an individual voxel:
Probability of a result > threshold = a
Probability of a result < threshold = 1- a
(a = chosen probability threshold eg 0.01)
Bonferroni correction
For whole family of voxels:
Probability of all results > threshold = (a)n
Probability of all results < threshold = (1- a)n
FWE (the probability that 1 or
more values will be greater than
threshold)
= 1 - (1 - a)n
and as alpha is so small:
=axn
a = FWE / n
or
Bonferroni correction
• Is this useful for imaging data?
• 100,000 voxels = 100,000 t values
• If we choose FWE 0.05 then using Bonferroni correction
= 0.05/100,000 = 0.0000005
• Corresponding t value is 5.77 so any statistic above this threshold is
significant
• This is for a p value of 0.05 corrected for the multiple comparisons
• Controls type I error
• BUT increases type II error
Thresholding
Use of ‘corrected’ p-value, a=0.1
FWE
Too conservative for functional imaging
How can we make our threshold less
conservative without creating too many
false positives?
Spatial Correlation
• The Bonferroni correction doesn’t consider spatial
correlation
• Voxels are not independent
– The process of data acquisition
– Physiological signal
– Spatial preprocessing applied before analysis
– Corrections for movement
• Fewer independent observations than voxels
Spatial Correlation
•This is a 100 by 100 square full of voxels
•There are 10,000 tests (voxels) with a 5% family wise error rate
•Bonferroni correction gives us a threshold of:
• 0.05/10000 = 0.000005
•This corresponds to a z score of 4.42
Spatial Correlation
•If we average contents of the boxes we get 10 x10
•Simple smoothing
•Our correction falls to 0.0005
•This corresponds to a z score of 3.29
•We still have 10,000 z scores but only 100
independent variables. Still too conservative.
The problem is we are
considering variable that are
spatially correlated to be
independent?
How do we know how many
independent variable there
really are?
Smoothing
• We can improve spatial correlation with
smoothing
• Increases signal-to-noise ratio
• Enables averaging across subjects
• Allows use of Gaussian Random Field
Theory for thresholding
PART II
PROBABILITY
TOPOLOGY
STATISTICS
Leonhard Euler (1707-1783)
•
Leonhard Paul Euler - Swiss Mathematician
who worked in Germany & Russia
•
Prolific mathematician - 80 volumes of work
covering almost every area of mathematics
(geometry, calculus, number theory, physics)
•
Of interest wrote “most beautiful formula
ever”:
eip + 1 = 0
•
Laid the foundations of topology (Euler's
1736 paper on Seven Bridges of Königsberg)
•
EULER CHARACTERISTIC
Euler Characteristic I
•Polyhedron e.g. Cube:
•Can subdivide object into the number of
verticies, edges, faces:
•Euler observed for all solid polyhedra:
V–E+F = 2
•Can generalise this formula by including P
(number of polyhedra):
V – E + F – P = EC
•Property of topological space: 0d - 1d + 2d - 3d + 4d…etc.,= EC
EC is 1 for ALL SOLID POLYHEDRA
V=8
E = 12
8 – 12 + 6 – 1 = 1
F=6
P=1
V = 16
E = 28
16 – 28 + 16 – 3 = 1
F = 16
P=3
V = 16
E = 32
F = 24
P=8
HP = 1
16 – 32 + 24 – 8 + 1 = 1
Euler Characteristic II
i) Holes:
• Each hole through an object reduces it’s EC by 1:
EC = 0
EC = -1
EC = -2
ii) Hollows:
Homeomorphic
EC = 2
iii) Set of disconnected Polyhedra
• Calculate individual EC and sum:
8 Torus = 0
2 ‘sphere’ = 4
2 Solid = 2
EC of Set [Donut] = 6
iv) EC of a Three Dimensionsal set:
•Often look at EXCURSION SETS
•Define a fixed threshold
•Define the objects that exceed that density and calculate EC for them
•Simplified EC = Maxima – Saddles + Minima
•VERY dependent on threshold
– RANDOM FIELD THEORY
Gaussian Curves
•
•
•
Standard Normal Distribution (Probability density function)
Mean = 0
Standard Deviation = 1
FWHM
•Full Width at Half its Maximum Height: Weighting Function
•Data Smoothing (Gaussian Kernels)
•Brownian Motion -> Gaussian Random Field
Data Smoothing
•
•
•
•
•
•
Smoothing: The process by which data points are averaged with their
neighbours in a series
Attempting to maximise the signal to noise ratio
Kernel: Defines the shape of the function that is used to take the average of
the neighbouring points
Each pixel's new value is set to a weighted average of that pixel's
neighbourhood.
The original pixel's value receives the heaviest weight and neighbouring
pixels receive smaller weights as their distance to the original pixel
increases.
This results in a blur that preserves boundaries and edges better than other,
more uniform blurring filters
Visualisation of Gaussian Kernel:
Value 1 in the centre (c), 0’s everywhere else
Effect of Kernel seen in (d)
Gaussian Smoothing
the
Gaussian Kernel
Johann Carl Friedrich
Gauss
Nil
1 Pixel
2 Pixels
3 Pixels
•Number of Resolution Elements (RESELS), R
•A block of values (e.g pixels) that is the same size as the FWHM.
•In 3D a cube of voxels that is of size (FWHM in x) by (FWHM in y) by (FWHM in z)
•RFT requires FWHM > 3 voxels
27 Voxels
1 RESEL
•Typical applied smoothing:
•Single Subj fMRI: 6mm
PET: 12mm
•Multi Subj
fMRI: 8-12mm
PET: 16mm
Stationary Gaussian Random Field
•
•
•
•
Studied by Robert Adler as his PhD thesis 1976
Published the book 1981, The Geometry of Random Fields
There are several types (Gaussian, non-Gaussian, Markov, Poisson…)
Deals with the behaviour of a stochastic process over a specified dimension
D (often D=3, but can be higher)
•
To create a stationary Gaussian field, create a lattice of independent
Gaussian observations (i.e. White Noise).
Each will have a mean = 0, SD = 1 (standard normal distribution)
Take the weighted average (using Gaussian Kernel)
•
•
(Brownian Bridge)
PLOT
Mean = 0
SD = 1
Mean = 0
SD = 1
•The link between the topology of an excursion set and the local maxima was
published by Hasofer (Adler’s Supervisor) in 1978:
•As threshold increases the holes in the excursion set disappears until each
component of the excursion set contains just one local maximum
•EC = Number of local maxima (at high thresholds)
•Just below the global maximum the EC = 1; Just above = 0
•At high thresholds the expected EC approximates the probability that global
maximum exceeds threshold
EC = 15
EC = 4
EC = 1
• Mathematical method for generating threshold (t)
• 2 Formulas
• Required values (whole brain*):
–
–
–
–
Volume: 1,064cc
Surface Area: 1,077cm2
Caliper Diameter= 0.1cm
EC = 2 (Ventricles)
• FWHM Value to calculate , a measure of the roughness of the field
 = 4.loge2/FWHM
* Or region of interest (see later results)
1
2
•Given E(EC) = 0.05
•Just solve for t………..
Volume of Interest:
EC
Diameter Surface Area
FWHM=20mm
(1) Threshold depends on Search Volume
(2) Surface area makes a large contribution
Volume
Threshold (t)
corresponding to
p(local maxima)
<0.05 (“Voxel level)
Threshold (t)
corresponding to
probability p of a
‘cluster’ of
activation size n
n
(“Cluster level”)
Signal 1
Signal 2
Low Threshold – Could still ask at
what threshold (t) do you see a cluster
(C) ≥ N with p<0.05
Cluster c with n voxels;
As N>n at t, significant
Cluster C
with N voxels
Cluster level threshold raised
– Increasing specificity at the
cost of sensitivity
Random Field Theory
Assumptions
Images need to follow Gaussian
Constructed statistics need to be sufficiently smooth.
If underlying images are smooth, constructed statistics are smooth.
Link to General Linear Model
•
The Random Fields are the component fields:
Y = Xw +E,
e=E/σ
•
This is because under the null hypothesis there is no contribution from Xw
to observed data Y; hence E should explain all the results in this scenario, if
it does not then there is a statistically significant contribution from the Xw
term.
•
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
?
^

parameter
estimates


=
Each row is
an estimated
component field
residuals
estimated variance
estimated
component
fields
Example Data:
Beyond Random Field Theory
False Discovery Rate (FDR)
•Whenever one performs multiple tests, the FDR is the proportion of false
positives among those tests for which the null hypothesis is rejected
•This quantity gets at the essence of what one wants to control in multiple
comparisons
•This procedure is conservative if the voxels are positively dependent
Take all P Values
p1<p2<p3<p4……pN
To control the FDR at q, find the largest value i so that:
Designate imax = R
Then Threshold at R i.e. reject all the voxels that contributed from p1…..pr
Developments in Random Field Theory.
K.J. Worsley. Chapter 15 in Human Brain Function (2nd Ed)
K Friston, J Ashburner, W Penny
Thresholding of Statistical Maps in Functional Neuroimaging
Using the False Discovery Rate. C Genovese et al.
NeuroImage 15, 870–878 (2002)
Conclusion
•
We should not use uncorrected p-values
• Uncorrected = High False Positive
• Bonferroni = High False Negative
•
We can use Random Field Theory (RFT) to ‘correct’ p-values
• Combines Probability, Topology and Extreme Statistics
•
RFT requires FWHM > 3 voxels
•
We only need to correct for the volume of interest
•
Cluster-level inference – can detect changes missed on voxel level
inference
•
Only covered basics of RFT – Broad ranging applications
Acknowledgements
•
•
•
•
•
•
•
•
•
•
•
•
Will Penny’s slides
– http://www.fil.ion.ucl.ac.uk/spm/course/slides05/ppt/infer.ppt#324,1,Random Field
Theory
Jean-Etienne’s slides
– http://www.poirrier.be/~jean-etienne/presentations/rft/spm-rft-slides-poirrier06.pdf
Previous MfD slides
– http://www.fil.ion.ucl.ac.uk/~jchumb/MfDweb.htm
Brett, Penny & Keibel. An introduction to Random Field Theory. Chapter from Human
Brain Mapping
http://www.icn.ucl.ac.uk/courses/MATLAB-Tutorials/spm_icn_2008/ParamThresh.ppt
The Geometry of Random Fields, Robert Adler (1981) at
http://www.dleex.com/read/?4204
Geometry of Random Images, Keith Worsley, Chance, 1996; 9(1):27-39
An Introduction to Random Field Theory (Chapter 14) Human Brain Mapping
A Unifed Statistical Approach for Determining Signifcant Signals in Images of Cerebral
Activation, Worsley et al. Human Brain Mapping, 4:58-73.
Medical Image Analysis:
http://www.stat.wisc.edu/~mchung/teaching/MIA/lectures/MIA.lecture10.random.field.feb.2
1.2007.pdf
Calculating the Topology of Large-Scale Structure, R. J. Massey
http://www.astro.caltech.edu/~rjm/thesis/Massey_MSci_LSSTopology.pdf
Topological Characterization of Porous Media, Hans-J¨org Vogel, Morphology of
Condensed Matter, 2002 Volume 600: 75-92
….Any Questions?