Classical Inference on SPMs

1. Classical Inference on SPMs Justin Chumbley http://www.fil.ion.ucl.ac.uk/~jchumb/ SPM Course Oct 23, 2008

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

3. (if h is fixed before) a long-run property of the decision-rule, i.e. all data-realisations E[ I(H>h) ] ‘p-value’(if h is observed data) a property of the this specific observation just a parameter of a distribution (like dn on numbers in the set). h Null Distribution of H Frequentist ‘exceedence probabilities’: p(H>h) h p-val Null Distribution of H

4. More exceedence probabilities… Bin(x|20, 0.3) Poi (x|20, 0.3)

5. h Null Distribution of T Spatially independent noise Independent Gaussian null Bernoulli Process N voxels How many errors?

6. Errors accumulate • Average number of errors is • t = Number of errors (independence) • Set h to ensure Bernoulli process rarely reaches height criterion anywhere in the field. Gives similar h to Bonferonni

7. WRONG APPROACH Independent Voxels Spatially Correlated Voxels • This is the WRONG model: • Noise (Binomial/Bonferonni too conservative under spatially dependent data) • There are geometric features in the noise: • Signal (under alternative distribution) • signal changes smoothly: neighbouring voxels should have similar signal • signal is everywhere/nowhere (due to smoothing, K-space, distributed neuronal responses)

8. Repeatable Space

9. Repeatable Space Unrepeatable

10. Repeatable Observation Space Unrepeatable

11. Binary decisions on signal geometry: How?! • Set a joint threshold (H>h,S>s) to define a set of regions with this geometric property. One positive region  One departure from null/flat signal-geometry. But how to calculate the number t of false-positive regions under the null!

12. Space h s unrepeatable Topological inference • As in temporal analysis… • Assume a modelfor spatial dependence • A Continuous Gaussian field vs Discrete 1st order Markov • estimate spatial dependence (under null) • Use the component residual fields • Set a joint threshold (H>h,S>s) to define a class of regions with some geometric property. 

13. Topological inference • As in temporal analysis… • Assume a modelfor spatial dependence • A Continuous Gaussian field vs Discrete 1st order Markov • estimate spatial dependence (under null) • Use the component residual fields • Set a joint threshold (H>h,S>s) to define a class of regions with some geometric property.  Count regions whose topology surpasses threshold: h R1 R0 s Space

14. h s Topological inference • As in temporal analysis… • Assume a modelfor spatial dependence • A Continuous Gaussian field vs Discrete 1st order Markov • estimate spatial dependence (under null) • Use the component residual fields • Set a joint threshold (H>h,S>s) to define a class of regions with some geometric property.  Count regions whose topology surpasses threshold: Calibrate class definition, , to control false-positive class members. What is the average number of false-positives?

15. h s Topological inference • For ‘high’ h, assuming that errors are a Gaussian Field. E(topological-false-positives per brain) =

16. Topological attributes Topological measure • threshold an image at h • excursion set Ah • (Ah)=# blobs - # holes • At high h, (Ah)=# blobs  P((Ah)> 0 )

17. Au  Unified Theory • General form for expected Euler characteristic • 2, F, & t fields • restricted search regions αh= S Rd (W)rd (h) rd (h): EC density; depends on type of field (eg. Gaussian, t) and the threshold, h. Rd (W): RESEL count; depends on the search region – how big, how smooth, what shape ? Worsley et al. (1996), HBM

18. Au  Unified Theory • General form for expected Euler characteristic • 2, F, & t fields • restricted search regions αh= S Rd (W)rd (h) rd (h):d-dimensional EC density – E.g. Gaussian RF: r0(h) = 1- (u) r1(h) = (4 ln2)1/2 exp(-u2/2) / (2p) r2(h) = (4 ln2) exp(-u2/2) / (2p)3/2 r3(h) = (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2p)2 r4(h) = (4 ln2)2 (u3 -3u) exp(-u2/2) / (2p)5/2 Rd (W): RESEL count R0(W) = (W) Euler characteristic of W R1(W) = resel diameter R2(W) = resel surface area R3(W) = resel volume Worsley et al. (1996), HBM

19. 5mm FWHM 10mm FWHM 15mm FWHM Topological attributes • Expected Cluster Size • E(S) = E(N)/E(L) • S cluster size • N suprathreshold volume • L number of clusters

20. Topological attributesunder independence 5mm FWHM 10mm FWHM 15mm FWHM (2mm2 pixels)

21. 3 related exceedence probabilities: • Set-level

22. Summary: Topological F W E • Brain images have spatially organised signal and noise. • Take this into account when compressing our 4-d data. • SPM infers the presence of departures from flat signal geometry • inversely related (for fixed ) • Exploit this for tall-thin/short-broad within one framework. • ‘Peak’ level is optimised for tall-narrow departures • ‘Cluster’ level is for short-broad departures. • ‘Set’ level tells us there is an unusually large number of regions.

23. FDR • Controls E( false-positives/total-positives ) • Doesn’t specify the subject of inference. • On voxels? • Preferably on Topological features.

24. THE END

25. Useful References • http://www.fil.ion.ucl.ac.uk/spm/doc/biblio/Keyword/RFT.html • http://www.math.mcgill.ca/keith/unified/unified.pdf • http://www.sph.umich.edu/~nichols/Docs/FWEfNI.pdf