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

Normalisation

Statistical Parametric Map

Image time-series

Parameter estimates

General Linear ModelRealignment Smoothing

Design matrix

Anatomicalreference

Spatial filter

StatisticalInference

RFT

p <0.05p <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

•Decision rule (threshold) u, determines false +ve rate

•Choose u to give acceptable α under H0

= P(type I error) i.e. chance we are wrong when rejecting the null hypothesis

t

u

= p(t>u|H)

Multiple comparisons problem

• fMRI – lots of voxels, lots of t-tests

• If use same threshold, higher probability of obtaining at least 1 false +ve

t

u

t

u

t

u

t

u

t

u

e.g. for alpha=0.05, 10000 voxels: expect 500 false positives

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

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

FWE

Use of ‘corrected’ p-value, α =0.1

Use of ‘uncorrected’ p-value, α =0.1

Definitions

Univariate statisticsFunctional imaging1 observed data many voxels

1 statistical value family of statistical values

type I error rate family-wise error rate (FWE)

null hypothesis family-wise null hypothesis

Thresholding

•Height thresholding

•This gives us localizing power

Bonferroni correctionp = /n

Corrected p-valueα = acceptable Type 1 error raten = 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 / NBut…

Spatial Correlation

Averaging over one voxel and its neighbours (independent

observations)Usually weighted average using a

(Gaussian) smoothing kernel

•Dependence between voxels : physiological signal

data acquisition

spatial preprocessing

The problem with Bonferroni

0.05/10000 = 0.000005Z score 4.42

Fewer independent observations than there are voxels

Bonferroni is too conservative (high false negative)

Appropriate correction:0.05/100 = 0.0005

Z score 3.29

100 x 100 voxels – normally distributed independent random numbers

Averaged 10x10

• 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

• ECV-E+F = 2

• Number of polyhedra (P)V-E+F-P=1

• Holes & handlesreduce by 1

• Topology…0d - 1d + 2d - 3d + 4d…etc

8 – 12 + 6 = 2

16 – 28 + 16 – 3 = 1

EC is a topological measure…

0 (product of 2 circles)

(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

• Takes into account smoothness and shape of the data as well as number of voxels to apply an appropriate threshold

lattice representation

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.mrc-cbu.cam.ac.uk/imaging/PrinciplesRandomFields

RFT for dummies - Part II 21

Random Field Theory

Part II

Nagako Murase

17/11/2010

21

Methods for Dummies 2010

Overview A large volume of imaging data

Multiple comparison problem

Bonferroni correction α=PFWE/nCorrected p value

FWE rate is too lowto reject the null hypothesis

Too false negative

Never use this.

It is because Bonferroni correction is based on the assuption that all the voxels are independent.

Random field theory (RFT)

α = PFWE ≒ E[EC] Corrected p value

<smoothing >

Process of RFT application: 3 steps

1st

Smoothing →Estimation of smoothness (spatial

correlation)

2nd

Applying RFT

3rd

Obtaining PFWE

realignment &motion

correctionsmoothing

normalisation

General Linear Modelmodel fittingstatistic image

Corrected thresholds & p-values

image dataparameterestimates

designmatrix

anatomicalreference

kernel

StatisticalParametric Map

Thresholding &Random Field

Theory

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)

FWHM

Gaussian curves

Standard Normal Distribution (Probability density function)Mean = 0Standard Deviation = 1

1st

Smoothing →Estimation of smoothness

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.

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)

1st

Smoothing →Estimation of smoothness

The number of ressels depend on the FWHM the number of boxels (pixels).

<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.

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

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

Euler characteristics (EC)= the number of blobs (minus number of holes) in an image after thresholding

After smoothing

Set the threshold as z core 2.5Below 2.5..0..blackAbove 2.5..1..white

EC=3

thresholding

z=2.5 Z=2.75

Different Z score thresholdgenerates different EC.

EC=3 EC=1

Thresholding

No of blobs≒ EC

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

3rd step Obtain PFWE

E[EC]=0.05

RFTUsing this Z score, we can conclude that any blots have a probabilityof ≦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=4.42If 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 voxelsRFT 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 regionR0(V): ressel single boxel countR1(V): ressel radiusR2(V): ressel surface areaR3(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.

EC Diameter Surface AreaVolume

FWHM=20mm

Threshold depends on Search Volume

Volume of Interest:

Note 3:voxel-level inference → a larger framework inference: different thresholding method

• Cluster-level inference

• Set-level inference

• These framework requireHeight thresholdspatial extent threshold

Peak (voxel), cluster and set level inference

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)

Set level inference:the height and volume threshold (spatial extent above u)→ number of clusters above u

Sensitivity

Regional specificity

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 DataSPM 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 numberUse Bonferroni correction

Conclusion A large volume of imaging data

Multiple comparison problem

Bonferroni correction α=PFWE/nCorrected p value

FWE rate is too lowto reject the null hypothesis

Too false negative

Never use this.

<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 LambertSuz PrejawaMaria Joao

Thank you for your attention!