Random Field TheoryCiaran S Hill & Christian Lambert

Methods for Dummies 2008

Overview

PART ONE

• Statistics of a Voxel

• Multiple Comparisons and Bonferroni correction

PART TWO

• Spatial Smoothing

• Random Field Theory

PART I

realignment &motion

correctionsmoothing

normalisation

General Linear Modelmodel fittingstatistic image

Corrected thresholds & p-values

image dataparameterestimates

designmatrix

anatomicalreference

kernel

StatisticalParametric Map

Thresholding &Random Field

Theory

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

= p(t>t-value|H0)

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).

t-value = 2.42

As p < α , we reject the null hypothesist-value = 2.02

t-distribution

NULL hypothesis, H0: activation is zero

alpha = 0.025

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

t > 0.5t > 3.5

Med. Threshold Low Threshold t<0.5High Threshold (t>5.5)

Good Specificity

Poor Power(risk of false negatives)

Poor Specificity(risk of false positives)

Good Power

t > 5.5

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

(bon) = /n

Thresholding

Error signal (noise)

Pure signal

What we actually get (bit of both)

Thresholding

11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5%

Use of ‘uncorrected’ p-value, =0.1

Percentage of Null Pixels that are False Positives

Too much false positive outside our blob

Bonferroni correctionFor an individual voxel:Probability of a result > threshold = Probability of a result < threshold = 1-

( = chosen probability threshold eg 0.01)

Bonferroni correctionFor whole family of voxels:Probability of all results > threshold = ()n

Probability of all results < threshold = (1- )n

FWE (the probability that 1 or more values will be greater than threshold)

= 1 - (1 - )n

and as alpha is so small:= x n or

= FWE / n

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

FWE

Use of ‘corrected’ p-value, =0.1

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

TOPOLOGY STATISTICS

PROBABILITY

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”:

ei

• 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 = 16

E = 32

F = 24

P = 8

HP = 1

V = 16

E = 28

F = 16

P = 3

V = 8

E = 12

F = 6

P = 1

8 – 12 + 6 – 1 = 1

16 – 28 + 16 – 3 = 1

16 – 32 + 24 – 8 + 1 = 1

Euler Characteristic IIi) Holes:

• Each hole through an object reduces it’s EC by 1:

EC = 0 EC = -1 EC = -2

EC = 2

ii) Hollows:

Homeomorphic

8 Torus = 0

2 ‘sphere’ = 4

2 Solid = 2

EC of Set [Donut] = 6

iii) Set of disconnected Polyhedra

• Calculate individual EC and sum:

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

•Full Width at Half its Maximum Height: Weighting Function

•Data Smoothing (Gaussian Kernels)•Brownian Motion -> Gaussian Random Field

FWHM

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

Nil

1 Pixel 2 Pixels 3 Pixels

Johann Carl Friedrich Gauss

•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 voxelsRFT requires FWHM > 3 voxels

27 Voxels 1 RESEL

•Typical applied smoothing:Typical applied smoothing:

•Single Subj Single Subj fMRI: 6mmfMRI: 6mmPET: 12mmPET: 12mm

•Multi Subj Multi Subj fMRI: 8-12mmfMRI: 8-12mmPET: 16mmPET: 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)

PLOT(Brownian Bridge)

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 = 1

EC = 4

EC = 15

• 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 1

* Or region of interest (see later results)

•Given E(EC) = 0.05

•Just solve for t………..

2

EC Diameter Surface AreaVolume

FWHM=20mm

(1) Threshold depends on Search Volume

(2) Surface area makes a large contribution

Volume of Interest:

Signal 1

Signal 2

Threshold (t) corresponding to p(local maxima) <0.05 (“Voxel level)

Threshold (t) corresponding to probability p of a ‘cluster’ of activation size n

(“Cluster level”)

n

Cluster level threshold raised – Increasing specificity at the cost of sensitivity

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

Cluster c with n voxels;

As N>n at t, significant

Images need to follow Gaussian

Constructed statistics need to be sufficiently smooth. If underlying images are smooth, constructed statistics are smooth.

Random Field Theory Assumptions

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

^

residuals

estimatedcomponent

fields

parameterestimates

estimated variance

=

Each row isan estimatedcomponent field

Estimated component fieldsEstimated component fields

data matrix

des

ign

mat

rix

parameters errors+ ?= ?voxelsvoxels

scansscans

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

p1<p2<p3<p4……pN

Take all P Values

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 NeuroimagingUsing 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.21.2007.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?