Methods for Dummies Random Field Theory

1. Methods for Dummies Random Field Theory Dominic Oliver & Stephanie Alley

2. Structure • What is the multiple comparisons problem? • Why we can’t use the classical Bonferroni approach • What is Random Field Theory? • How do we implement it in SPM?

3. Processing so far

4. Hypothesis testing and the type I error u • Testing t-statistic against our null hypothesis (H0) • Probability of result arising by chance • p < 0.05 • Chosen threshold (u) determines false positive rate () • t-statistic distribution means you always run a risk of false positive or type I error t

5. Significance of type I errors in fMRI u u u u u • Since α remains constant, the more tests you do, the moretype I errors • Due to the number of tests performed in fMRI, the number of type I errors with this threshold is highly impractical      t t t t t 60,000 voxels per brain 60,000 t-tests α = 0.05 60,000 x 0.05 = 3,000 type I errors

6. Why is this a problem? • 3,000 is a large number but it’s still only 5% of our total voxels • Neural Correlates of Interspecies Perspective Taking in the Post-Mortem Atlantic Salmon • t(131) > 3.15, p(uncorrected) < 0.001 Bennett, Miller & Wolford, 2009

7. t > 2.5 t > 4.5 t > 0.5 t > 1.5 t > 3.5 t > 5.5 t > 6.5 So what can we do about it? u u • Adjust the threshold (u), taking in account the number of tests being performed • Any values above new threshold are highly unlikely to be false positives t t

8. The Bonferroni Correction Why can’t we use it?

9. Bonferroni correction example single-voxel probability threshold  Number of voxels  = PFWE / n e.g. 100,000 t values, each with 40 d.f. • When PFWE = 0.05: 0.05/100000 = 0.0000005 t = 5.77 • Voxel where t > 5.77 has only 5% chance of appearing in a volume of 100,000 t stats drawn from the null distribution Family-wise error rate

10. So what’s the issue? • Bonferroni correction is too conservative for fMRI data • Bonferroni relies on independent values • But fMRI data is not independent due to • How the scanner collects data and reconstructs the images • Anatomy and physiology • Preprocessing (e.g. normalisation, smoothing)

11. The problem • What Bonferroni needs: • A series of independent statistical tests • What we have: • A spatially dependent, continuous statistical image

12. Spatial dependence • This slice is made of 100 x 100 voxels • Random values from normal distribution • 10,000 independent values • Bonferroni accurate

13. Spatial dependence • Add spatial dependence by averaging across 10 x 10 squares • Still 10,000 numbers in image but only 100 are independent • Bonferroni correction 100 times too conservative

14. Smoothing • Convolve fMRI signal with Gaussian kernel • Improves signal to noise ratio (SNR) • Increased sensitivity • Incorporates anatomical and functional variations between subjects • Improves statistical validity by making distribution of errors more normal

15. Spatial dependence is increased by smoothing • When map is smoothed, each value in the image is replaced by a weighted average of itself and its neighbours • Image is blurred and independent values are reduced

16. If not Bonferroni, then what? • Idea behind Bonferroni correction in this context is sound • We need to adjust our threshold in a similar way but also take into account the influence on spatial dependence on our data • What does this? Random Field Theory

17. Random Field Theory

18. Random field theory provides a means of working with smooth statistical maps. • It allows us to determine the height threshold that will give the desired FWE. • Random fields have specified topological characteristics • Apply topological inference to detect activations in SPMs

19. Three stages of applying RFT: 1. Estimate smoothness 2. Calculate the number of resels 3. Get an estimate of the Euler Characteristic at different thresholds

20. 1. Estimate Smoothness • Image contains both applied and intrinsic smoothness • Estimated using the residuals (error) from GLM estimate

21. 2. Calculate the number of reselsLook at your estimated smoothness (FWHM) FWHM (Full Width at Half Maximum)

22. Express your search volume in resels • Resel - A block of values that is the same size as the FWHM • Resolution element (Ri = FWHMx x FWHMy x FWHMz) • “Restores” independence of the data

23. 3. Get an estimate of the Euler Characteristic • Euler Characteristic – a topological property • SPM  threshold  EC • EC = number of blobs (minus number of holes)* “Seven bridges of Kӧnisberg” *Not relevant: we are only interested in EC at high thresholds (when it approximates P of FEW)

24. Steps 1 and 2 yield a ‘fitted random field’ (appropriate smoothness) • Now: how likely is it to observe above threshold (‘significantly different’) local maxima (or clusters, sets)under H0? • How to find out? EC!

25. Euler Characteristic and FWE Zt = 2.5 EC = 3 • The probability of a family wise error is approximately equivalent to the expected Euler Characteristic • Number of “above threshold blobs” Zt = 2.75 EC = 1

26. How to get E [EC] at different thresholds # of resels Expected Euler Characteristic Z (or T) -score threshold

27. Given # of resels and E[EC], we can find the appropriate z/t-threshold E [EC] for an image of 100 resels, for Z score thresholds 0 – 5

28. How to get E [EC] at different thresholds # of resels Expected Euler Characteristic Z (or T) -score threshold From this equation, it looks like the threshold depends only on the number of resels in our image

29. Shape of search region matters, too! • #resels  E [EC] • not strictly accurate! • close approximation if ROI is large compared to the size of a resel • # + shape + size resel  • E [EC] • Matters if small or oddly shaped ROI • Example: • central 30x30 pixel box = max. 16 ‘resels’ (resel-width = 10pixel) • edge 2.5 pixel frame = same volume but max. 32 ‘resels’ • Multiple comparison correction for frame must be more stringent

30. intensity Different types of topological Inference tα • Topological inference can be about • Peak height (voxel level) • Regional extent (cluster level) • Number of clusters (set level) tclus space Different height and spatial extent thresholds

31. Assumptions made by RFT • Underlying fields are continuous with twice-differentiable autocorrelation function • Error fields are reasonable lattice approximation to underlying random field with multivariate Gaussian distribution lattice representation Check: FWHM must be min. 3 voxels in each dimension In general: if smooth enough + GLM correct (error = Gaussian)  RFT

32. Assumptions are not met if... • Data is not sufficiently smooth • Small number of subjects • High-resolution data Non-parametric methods to get FWE (permutation) Alternative ways of controlling for errors: FDR (false discovery rate)

33. Implementation in SPM

34. Implementation in SPM

36. Sources • Human Brain Function, Chapter 14, An Introduction to Random Field Theory (Brett, Penny, Kiebel) • http://www.fil.ion.ucl.ac.uk/spm/course/video/#RFT • http://www.fil.ion.ucl.ac.uk/spm/course/video/#MEEG_MCP • Former MfD Presentations • Expert Guillaume Flandin