Random Field TheoryWill PennySPM short course, London, May 2005David CarmichaelMfD 2006

realignment & motion correctionsmoothingnormalisationGeneral Linear Modelmodel fittingstatistic imagecorrected p-valuesimage dataparameterestimatesdesign matrixanatomical referencekernelStatistical Parametric MapRandom Field Theory

Overview

1.Terminology2.Random Field TheoryCluster level inferenceSPM Results FDR

Overview

1.TerminologyRandom Field TheoryCluster level inferenceSPM Results FDR

Inference at a single voxela = p(t>u|H)NULL hypothesis, H: activation is zerou=2t-distributionp-value: probability of getting a value of t at least as extreme as u. If a is small we reject the null hypothesis.u=(effect size)/std(effect size)

Sensitivity and Specificity

Sensitivity and SpecificityEg. t-scoresfrom regionsthat truly do and do not activateo o o o o o o x x x o o x x x o x x x xu1

Sensitivity and SpecificityEg. t-scoresfrom regionsthat truly do and do not activateo o o o o o o x x x o o x x x o x x x xu2

Inference at a single voxela = p(t>u|H)NULL hypothesis, H: activation is zerou=2t-distributionWe can choose u to ensurea voxel-wise significance level of a.

This is called an uncorrected p-value, forreasons well see later.

We can then plot a map of above thresholdvoxels.

Inference for ImagesSignal+NoiseNoise

Using an uncorrected p-value of 0.1 will lead us to conclude on average that 10% of voxels are active when they are not.This is clearly undesirable. To correct for this we can define a null hypothesis for images of statistics.

Family-wise Null HypothesisFamily-Wise Error (FWE) rate = corrected p-value

Use of uncorrected p-value, a=0.1FWEUse of corrected p-value, a=0.1

The Bonferroni correctionThe Family-Wise Error rate (FWE), a, for a family of N independent voxels is = Nv

where v is the voxel-wise error rate. Therefore, to ensure a particular FWE set

v = / N

BUT ...

The Bonferroni correctionAssume Independent Voxels

Independent voxels - a good assumption??Voxel Point Spread Function (PSF)- continuous signal is sampled for a discrete period- imposes a filter that when FTd gives a PSF- Gives spread of signal through the image from point source ..worse in PETPhysiological noiseSmoothingNormalisation

The Bonferroni correctionIndependent VoxelsSpatially Correlated VoxelsBonferroni is too conservative for brain images

Random Field TheoryConsider a statistic image as a discretisation of a continuous underlying random fieldUse results from continuous random field theory

Discretisation

Overview

1.TerminologyRandom Field TheoryCluster level inferenceSPM Results FDR

Euler Characteristic (EC)Topological measurethreshold an image at uEC = # blobs at high u:

Prob blob = avg (EC)

So

FWE, a = avg (EC)

Example 2D Gaussian images = R (4 ln 2) (2) -3/2 u exp (-u2/2)Voxel-wise threshold, uNumber of Resolution Elements (RESELS), RN=100x100 voxels, Smoothness FWHM=10, gives R=10x10=100

Example 2D Gaussian images = R (4 ln 2) (2) -3/2 u exp (-u2/2)For R=100 and =0.05RFT gives u=3.8

How do we know number of resels? We can simply use the FWHM of the smoothing kernelBut processes such as normalisation mean smoothness will vary

2. Estimate the FWHM at each voxel using residuals at each voxel (worsley 1998)

Resel Counts for Brain StructuresFWHM=20mm(1) Threshold depends on Search Volume(2) Surface area makes a large contribution

Overview

1.Terminology

2.Theory

3.Imaging Data

4.Levels of Inference

5. SPM Results

Applied SmoothingSmoothnesssmoothness voxel size

practicallyFWHM 3 VoxDim

Typical applied smoothing:Single Subj fMRI: 6mm PET: 12mm Multi Subj fMRI: 8-12mm PET: 16mm

Overview

1.Terminology

2.Theory

3.Imaging Data

4.Levels of Inference

5. SPM Results

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

SET-LEVEL INFERENCE

Levels of inferenceset-levelP(c 3 | n 12, u 3.09) = 0.019cluster-levelP(c 1 | n 82, t 3.09) = 0.029 (corrected)

voxel-levelP(c 1 | n > 0, t 4.37) = 0.048 (corrected)At least onecluster withunspecifiednumber of voxels abovethresholdAt least one cluster with at least 82 voxels above thresholdAt least 3 clusters abovethreshold

Overview

1.Terminology

2.Theory

3.Imaging Data

4.Levels of Inference

5. SPM Results

SPM results IActivationsSignificant atCluster levelBut not atVoxel Level

SPM results IIActivations Significant atVoxel andCluster level

SPM results...

False Discovery Rateu1o o o o o o o x x x o o x x x o x x x xEg. t-scoresfrom regionsthat truly do and do not activateFDR = FP/(# Reject) a = FP/(# H True)

False Discovery Rateu2o o o o o o o x x x o o x x x o x x x xEg. t-scoresfrom regionsthat truly do and do not activateFDR=1/8=13%a=1/10=10% At u2FDR = FP/(# Reject) a = FP/(# H True)

False Discovery Rate

Signal+NoiseNoise

SummaryWe should not use uncorrected p-values

We can use Random Field Theory (RFT) to correct p-values

RFT requires FWHM > 3 voxels

We only need to correct for the volume of interest

Cluster-level inference

False Discovery Rate is a viable alternative

Functional Imaging DataThe Random Fields are the component fields, Y = Xw +E, e=E/ We can only estimate the component fields, using estimates of w and

To apply RFT we need the RESEL count which requires smoothness estimates

Estimated component fieldsdata matrixdesign matrixparameterserrors+?=?voxelsscansestimateresidualsestimated component fieldsparameter estimatesestimated variance=Each row isan estimatedcomponent field

So this is just a reminder that both these slides and all the relating publications are available on the spm website. SO were talking about a method to get from our spm map here, where each voxel has been considered individually, to our corrected p-values where we have taken into account the relationship between voxels and what we expect to get from the whole data considered over all. Incidentally, when I use the term voxel I mean a 3D version of a pixel so a small constituent cube and when we start to consider all the voxels then it is necessary to be aware of the some elements from earlier in the chain smoothing and properties of our images. So to start lets just recap a little terminology and a few key ideas that weve met already several times so should now be becoming familiar.So this is just a reminder of the t-distribution. We assume that our noise in the images is random white noise that has a Gaussian distributed.

So our null hypothesis is that there is no activation and our data variance is pure noise. Now we can choose a threshold u=2. We can ask the question what is the probability that we get a t-value that is at least as greater than or equal to u. This is our p-value and we can decide a level of alpha that we think is reasonable 0.05.So from our threshold we can have 4 possibilities. Null hypothesis is true and we treat it is such a true -ve, Null hypothesis is true and we reject it - think its activation and get a False positive. Alternatively, the null hypothesis is false and we incorrectly reject it, a false negative. Lastly, we correctly reject the null hypothesis and so get a true positive..

We can define two terms. Firstly , the specificity that is given by the number of true negatives given that the null hypothesis is true (no activation) this is simply 1-alpha.

Secondarily, we get the sensitivity that is; given that there is an activation, the null hypothesis is wrong how much of the time do we get it correct.

Key -> WE NEED TO DEFINE A NULL HYPOTHESIS FOR THE GROUP OF VOXELS THAT MAKE UP AN IMAGE.So the fw NULL HYPOTHESIS - for our family of voxels is that there is no activation in the image it is zero eveywhere.

If we reject the null hypothesis at any voxel it is rejected for the entire image.

A False positive anywhere in the image gives the FWE

It is this FWE rate that is termed our corrected p-valuea) 1 in 10 pixels is FP activation in each image and this is seen with activation in the noise regions. Ifc we repeat 10 times we get the same result.

b) The corrected p-value of 0.05 means we only expect an FP in 1 out of 10 images / runs of experiment. You can see it happens here one out of 10 times.So the classic way to perform this correction is the Bonferroni correction. Were the FWE is calculated for a family of N independent voxels. The alpha is just the voxel wisae error rate v times the number of independent voxels N.

We can then reset v so that we get the FWE desiredWe assume for the bonferroni that each of the voxels is independent 10x10 smoothing FWE = average number of ECs We know a formula from RFT that tells us the number of ECs we expect, and we equate with the FWE alpha desired. All we have in this equation is R the number of resels and equating with alpha we find u, the voxelwise threshold, that we want.

So how many Resels do we have, in the image we saw earlier we smoothed a 100 sq image with a 10 FWHM kernal making the data have 10 sq independent resels. So RFT takes into account geometry of search space.

Cf 4mm shell and whole brain similar volume but larger surface area.Process of discretisation for RFT need to sample RF well enough so must be below res of imageMust choose cluster level threshold. This trades anatomical specificity and sensitivity