Random field theory

1. Random field theory Rumana Chowdhury and Nagako Murase Methods for Dummies November 2010

3. Overview Part 1 • Multiple comparisons • Family-wise error • Bonferroni correction • Spatial correlation Part 2 • Solution = Random Field Theory • Example in SPM

4. Image time-series Statistical Parametric Map Design matrix Spatial filter Realignment Smoothing General Linear Model StatisticalInference RFT Normalisation p <0.05 Anatomicalreference Parameter estimates

5. Voxel • Raw data collected as group of voxels • 3D, volumetric pixel • Location • Value • Calculate a test statistic for each voxel • Many many many voxels…

6. 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

7. Single voxel level statistics u • 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  = p(t>u|H)

8. Use of ‘uncorrected’ p-value, a=0.1 11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5% Percentage of Null Pixels that are False Positives Multiple comparisons problem u u u u u • fMRI – lots of voxels, lots of t-tests • If use same threshold, higher probability of obtaining at least 1 false +ve      t t t t t e.g. for alpha=0.05, 10000 voxels: expect 500 false positives

9. 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 Use of ‘uncorrected’ p-value, α =0.1 Use of ‘corrected’ p-value, α =0.1 FWE

10. 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

11. Thresholding • Height thresholding • This gives us localizing power

12. 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…

13. Spatial Correlation • Dependence between voxels : physiological signal • data acquisition • spatial preprocessing Averaging over one voxel and its neighbours (independent observations) Usually weighted average using a (Gaussian) smoothing kernel

14. The problem with Bonferroni 100 x 100 voxels – normally distributed independent random numbers Averaged 10x10 Appropriate correction: 0.05/100 = 0.0005 Z score 3.29 0.05/10000 = 0.000005 Z score 4.42 Fewer independent observations than there are voxels Bonferroni is too conservative (high false negative)

15. Not making inferences on single voxels • Take into account spatial relationships • Topology

16. “the problem has no solution!” Euler • Leonhard Euler (1797-1783), Swiss mathematician • Seven bridges of Kӧnisberg

17. 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 – 12 + 6 = 2 16 – 28 + 16 – 3 =1 0 (product of 2 circles) EC is a topological measure…

18. (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

19. 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

20. 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

21. Random Field Theory Part II Nagako Murase Methods for Dummies 2010 17/11/2010 RFT for dummies - Part II 21 21

22. Overview A large volume of imaging data Multiple comparison problem <smoothing> Bonferroni correction α=PFWE/n Corrected p value Random field theory (RFT) α=PFWE ≒ E[EC] Corrected p value Treat as a single voxel by simply averaging Uncorrected p value Too false positive FWE rate is too low to 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.

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

24. designmatrix parameter estimates image data kernel Thresholding &Random Field Theory • General Linear Model • model fitting • statistic image realignment &motioncorrection smoothing normalisation StatisticalParametric Map anatomicalreference Corrected thresholds & p-values

25. 1st Smoothing →Estimation of smoothness Gaussian curves FWHM Standard Normal Distribution (Probability density function) Mean = 0 Standard Deviation = 1 • 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. • 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)

26. 1st Smoothing →Estimation of smoothness After smoothing with a Gaussian smoothing kernel FWHM in x=10, in y=10 so this FWHM=100 pixels) Original data: an image using independent random numbers from the normal distribution

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

28. 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.

29. 2nd step Apply RFT After smoothing thresholding Set the threshold as z core 2.5 Below 2.5..0..black Above 2.5..1..white EC=3 Euler characteristics (EC)= the number of blobs (minus number of holes) in an image after thresholding

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

31. Thresholding No of blobs ≒ EC

32. 3rd step Obtain PFWE Expected EC: E[EC] = the probability of finding a blob PFWE≒ E[EC] α=E[EC]= R (4 loge 2) (2π) -3/2zt exp(-zt2/2) • E[EC] depends on: • R the number of resels • Zt Z score threshold

33. 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]=0.05Z=3.8 Bonferroni correction α =0.05/10,000=0.00005Z=4.42 If the assumption of RFT are met, then the RFT threshold is more accurate than the Bonferroni correction.

34. RFT in 3D 27 Voxels 1 RESEL 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 voxels

35. RFT Note 1:When FWHM is less than 3.2 voxels, the Bonferroni correction is better than the RFT for a Gaussian statistic.

36. RFT Note2:EC depends on volume shape and size. V: volume of search region R0(V): ressel single boxel count R1(V): ressel radius R2(V): ressel surface area R3(V): ressel volume • 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. Worsley KJ, et al. , Human Brain Mapping 1996

37. Correction in case of a small shaped region T thoreshold giving a FWE rate of 0.05. Restricting the search region to a small volume within a statistical map can reduce thresholds for given FWE rates.

38. Volume of Interest: Surface Area Diameter EC Volume FWHM=20mm Threshold depends on Search Volume

39. 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

40. Peak (voxel), cluster and set level inference Regional specificity Sensitivity 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   : significant at the set level L1 > spatial extent threshold L2 < spatial extent threshold : significant at the cluster level : significant at the peak level

41. 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.

42. SPM8 and RFT: Example Data SPM manual, http://www.fil.ion.ucl.ac.uk/spm/doc/

44. 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.

45. A case where the RFT assumption is not met. The error fields are not very smooth. • Increase the subject number • Use Bonferroni correction Small number of subjects

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

47. 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.

48. Acknowledgement The topic expert: Guillaume Flandin The organisers: Christian Lambert Suz Prejawa Maria Joao Thank you for your attention!