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 Anatomical reference Spatial filter Statistical Inference RFT p <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, H 0 (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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u t  uu t  uu t  uu t  uu e.g. for alpha=0.05, voxels: expect 500 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 imaging 1 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 correction p =  /n Corrected p-value α = acceptable Type 1 error rate n = 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 / N But …

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 = Z score 4.42 Fewer independent observations than there are voxels Bonferroni is too conservative (high false negative) Appropriate correction: 0.05/100 = Z score 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 ( ), Swiss mathematician Seven bridges of K ӧ nisberg “the problem has no solution!”

Euler characteristic: the beginnings EC V-E+F = 2 Number of polyhedra (P) V-E+F-P=1 Holes & handles reduce by 1 Topology… 0d - 1d + 2d - 3d + 4d…etc 8 – = 2 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: Guillaume Flandin’s slides: Will Penny’s slides: Random Field Theory R. Adler’s website: CBU imaging wiki: cbu.cam.ac.uk/imaging/PrinciplesRandomFields

RFT for dummies - Part II21 Random Field Theory Part II Nagako Murase 17/11/ Methods for Dummies 2010

Overview A large volume of imaging data Multiple comparison problem Bonferroni correction α=P FWE /n Corrected p value FWE rate is too low to reject the null hypothesis Too false negative Treat as a single voxel by simply averaging Uncorrected p value Too false positive Never use this. It is because Bonferroni correction is based on the assuption that all the voxels are independent. Random field theory (RFT) α = P FWE ≒ E[EC] Corrected p value

Process of RFT application: 3 steps 1 st Smoothing →Estimation of smoothness (spatial correlation) 2 nd Applying RFT 3 rd Obtaining P FWE

realignment & motion correction smoothing normalisation General Linear Model Ümodel fitting Üstatistic image Corrected thresholds & p-values image data parameter estimates design matrix anatomical reference kernel Statistical Parametric 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 = 0 Standard Deviation = 1 1 st 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) 1 st Smoothing →Estimation of smoothness

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

2 nd 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 2.5 Below black Above white EC=3 thresholding

z=2.5Z=2.75 Different Z score threshold generates different EC. EC=3 EC=1

Thresholding No of blobs ≒ EC

Expected EC: E[EC] = the probability of finding a blob P FWE ≒ E[EC] α = E[EC] = R (4 log e 2) (2π) -3/2 z t exp(-z t 2 /2) E[EC] depends on: Rthe number of resels Z t Z score threshold 3 rd step Obtain P FWE

E[EC]=0.05 RFT Using this Z score, we can conclude that any blots have a probability of ≦ 0.05 when they have occured by chance. α=E[EC]=0.05 Z=3.8 Bonferroni correction α =0.05/10,000= Z=4.42 If 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 χ 2 random fields are used to calculate threshold for height. RFT requires FWHM > 3 voxelsRFT requires FWHM > 3 voxels 27 Voxels1 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 region R 0 (V): ressel single boxel count R 1 (V): ressel radius R 2 (V): ressel surface area R 3 (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 Area Volume 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 require Height threshold spatial 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  : significant at the set level : significant at the cluster level : significant at the peak level L 1 > spatial extent threshold L 2 < spatial extent threshold

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 Data SPM manual,

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 number Use Bonferroni correction

Conclusion A large volume of imaging data Multiple comparison problem Bonferroni correction α=P FWE /n Corrected p value FWE rate is too low to reject the null hypothesis Too false negative Treat as a single voxel by simply averaging Uncorrected p value Too false positive Never use this. <smoothing with a Gaussian kernel, FWHM > Random field theory (RFT) α = P FWE ≒ E[EC] Corrected p value

Conclusion By thoresholding, expected EC is calculated by RFT, where P FWE ≒ 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 Lambert  Suz Prejawa  Maria Joao Thank you for your attention!