Classical Inference on SPMs Justin Chumbley SPM Course Oct 23, 2008

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 Random Field Theory

Frequentist ‘exceedence probabilities’: p(H>h) 1. (if h is fixed before) –a long-run property of the decision-rule, i.e. all data-realisations – E[ I(H>h) ] 2.‘p-value’ (if h is observed data) –a property of the this specific observation 3.just a parameter of a distribution –(like dn on numbers in the set). h p-val Null Distribution of H h

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

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

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

Independent VoxelsSpatially Correlated Voxels This is the WRONG model: 1.Noise (Binomial/Bonferonni too conservative under spatially dependent data) There are geometric features in the noise: 2.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) WRONG APPROACH

Space Repeatable

Space Repeatable Unrepeatable

Space Repeatable Unrepeatable Observation

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!

Topological inference As in temporal analysis… –Assume a model for spatial dependence A Continuous Gaussian field vs Discrete 1 st 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.  h s Space unrepeatable

Topological inference As in temporal analysis… –Assume a model for spatial dependence A Continuous Gaussian field vs Discrete 1 st 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: Space h s R1R1 R0R0

Topological inference As in temporal analysis… –Assume a model for spatial dependence A Continuous Gaussian field vs Discrete 1 st 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? h s

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

Topological attributes Topological measure –threshold an image at h –excursion set  h  h ) = # blobs - # holes -At high h,  h ) = # blobs  P(  h ) > 0 )

General form for expected Euler characteristic  2, F, & t fields restricted search regions α h =  R d (  )  d (h) Unified Theory R d (  ): RESEL count; depends on the search region – how big, how smooth, what shape ?  d (h): EC density; depends on type of field (eg. Gaussian, t) and the threshold, h. AuAu  Worsley et al. (1996), HBM

General form for expected Euler characteristic  2, F, & t fields restricted search regions α h =  R d (  )  d (h) Unified Theory R d (  ): RESEL count R 0 (  )=  (  ) Euler characteristic of  R 1 (  )=resel diameter R 2 (  )=resel surface area R 3 (  )=resel volume  d (h):d-dimensional EC density – E.g. Gaussian RF:  0 (h)=1-  (u)  1 (h)=(4 ln2) 1/2 exp(-u 2 /2) / (2  )  2 (h)=(4 ln2) exp(-u 2 /2) / (2  ) 3/2  3 (h)=(4 ln2) 3/2 (u 2 -1) exp(-u 2 /2) / (2  ) 2  4 (h)=(4 ln2) 2 (u 3 -3u) exp(-u 2 /2) / (2  ) 5/2 AuAu  Worsley et al. (1996), HBM

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

5mm FWHM 10mm FWHM 15mm FWHM (2mm 2 pixels) Topological attributes under independence

3 related exceedence probabilities: Set-level

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.

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

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Useful References Keyword/RFT.htmlhttp:// Keyword/RFT.html ied.pdfhttp:// ied.pdf WEfNI.pdfhttp:// WEfNI.pdf