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## Statistical Inference, Multiple Comparisons in addition to R in addition to om Field Theory Andrew Holme

Mekelburg, Mike, Head Reporter has reference to this Academic Journal, PHwiki organized this Journal Statistical Inference, Multiple Comparisons in addition to R in addition to om Field Theory Andrew Holmes SPM short course, May 2002 Overview Overview a voxel by voxel hypothesis testing approach reliably identify regions showing a significant experimental effect of interest Assessment of statistic images multiple comparisons r in addition to om field theory smoothness spatial levels of inference & power false discovery rate later Generalisability, r in addition to om effects & population inference inferring to the population group comparisons Non-parametric inference later

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realignment & motion correction smoothing normalisation General Linear Model model fitting statistic image corrected p-values image data parameter estimates design matrix anatomical reference kernel Statistical Parametric Map r in addition to om field theory Statistical Parametric Mapping voxel by voxel modelling parameter estimate variance estimate = statistic image or SPM Multiple comparisons

Classical hypothesis testing Null hypothesis H test statistic null distributions Hypothesis test control Type I error incorrectly reject H test level Pr(reject H H) test size Pr(reject H H) p value min a at which H rejected Pr(T t H) characterising surprise Multiple comparisons t59 Gaussian 10mm FWHM (2mm pixels) p = 0.05 Threshold at p expect (100 p)% by chance Surprise extreme voxel values voxel level inference big suprathreshold clusters cluster level inference many suprathreshold clusters set level inference Power & localisation sensitivity spatial specificity Multiple comparisons terminology Family of hypotheses Hk k = {1, ,K} H = Hk Familywise Type I error weak control omnibus test Pr(reject H H) anything, anywhere strong control localising test Pr(reject HW HW) W: W & HW anything, & where Adjusted pvalues test level at which reject Hk

Simple threshold tests Threshold u tk > u reject Hk reject any Hk reject H reject H if tmax > u Valid test weak control Pr(Tmax > u H ) strong control since W Pr(TWmax > u HW ) Adjusted p values Pr(Tmax > tk H) u The Bonferroni correction The Bonferroni inequality Carlo Emilio Bonferroni (1936) For any set of events Ak : Bonferroni correction Ak : correctly accept Hk Tk < u & Hk Assess Hk at level ' correction ' = / K Adjusted p values min(1,K pk ) Conservative as long as correlated tests independent: K tests some dependence : tests totally dependent: 1 test 5mm 10mm 15mm R in addition to om field theory SPM approach: R in addition to om fields Consider statistic image as lattice representation of a continuous r in addition to om field Use results from continuous r in addition to om field theory lattice represtntation Euler characteristic Topological measure of excursion set Au Au = {x R3 : Z(x) > u} components – holes Single threshold test large u, near Tmax Euler char. local max Expected Euler char pvalue Pr(Zmax > u ) Pr((Au) > 0 ) E[(Au)] single threshold test u s.t. E[(Au) ] = Expected Euler characteristic E[(Au)] () (u 2 -1) exp(-u 2/2) / (2)2 large search region R3 ( volume smoothness Au excursion set Au = {x R3 : Z(x) > u} Z(x) Gaussian r in addition to om field x R3 + Multivariate Normal Finite Dimensional distributions + continuous + strictly stationary + marginal N(0,1) + continuously differentiable + twice differentiable at 0 + Gaussian ACF (at least near local maxima)

Smoothness, PRF, resels Smoothness variance-covariance matrix of partial derivatives (possibly location dependent) Point Response Function PRF Full Width at Half Maximum FWHM Gaussian PRF kernel var/cov matrix ACF 2 = (2)-1 FWHM f = (8ln(2)) fx 0 0 = 0 fy 0 1 0 0 fz 8ln(2) ignoring covariances = (4ln(2))3/2 / (fx fy fz) Resolution Element (RESEL) Resel dimensions (fx fy fz) R3() = () / (fx fy fz) if strictly stationary E[(Au)] = R3() (4ln(2))3/2 (u 2 -1) exp(-u 2/2) / (2)2 R3() (1 (u)) as long as high thresholds u Component fields data matrix design matrix parameters errors + = voxels scans estimate residuals estimated component fields parameter estimates Image regression variance s2 estimated variance = Component fields = + Component fields T statistic image

Smoothness estimation Smoothness from st in addition to ardised residuals empirical derivatives at each voxel Resels per voxel (RPV) an image of smoothness correction as long as estimation of variance field 2 function of degrees of freedom covariances often ignored Euler Characteristics using discrete methods Unified p-values General as long as m as long as expected Euler characteristic 2, F, & t fields restricted search regions D dimensions E[(WAu)] = S Rd (W) rd (u) Rd (W): d-dimensional Minkowski functional of W function of dimension, space W in addition to smoothness: R0(W) = (W) Euler characteristic of W R1(W) = resel diameter R2(W) = resel surface area R3(W) = resel volume rd (W): d-dimensional EC density of Z(x) function of dimension in addition to threshold, specific as long as RF type: E.g. Gaussian RF: (strictly stationary &c ) r0(u) = 1- (u) r1(u) = (4 ln2)1/2 exp(-u2/2) / (2p) r2(u) = (4 ln2) exp(-u2/2) / (2p)3/2 r3(u) = (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2p)2 r4(u) = (4 ln2)2 (u3 -3u) exp(-u2/2) / (2p)5/2 Suprathreshold cluster tests Primary threshold u examine connected components of excursion set Suprathreshold clusters Reject HW as long as clusters of voxels W of size S > s Localisation (Strong control) at cluster level increased power esp. high resolutions (f MRI) Thresholds, p values Pr(Smax > s H ) Nosko, Friston, (Worsley) Poisson occurrence (Adler) Assumme as long as m as long as Pr(S=sS>0) 5mm FWHM 10mm FWHM 15mm FWHM (2mm2 pixels)

Poisson Clumping Heuristic Expected number of clusters p{cluster volume > k} Expected cluster volume EC density ( Search volume (R) Smoothness Levels of inference Parameters u – 3.09 k – 12 voxels S – 323 voxels FWHM – 4.7 voxels D – 3 set-level P(c 3 n 12, u 3.09) = 0.019 cluster-level P(c 1 n 82, t 3.09) = 0.029 (corrected) P(n 82 t 3.09) = 0.019 (uncorrected) voxel-level P(c 1 n 0, t 4.37) = 0.048 (corrected) P(t 4.37) = 1 – {4.37} < 0.001 (uncorrected) omnibus P(c7 n 0, u 3.09) = 0.031 Summary: Levels of inference & power

SPM results SPM results SPM results

SPM results SPM results

Multiple Comparisons, & R in addition to om Field Theory Worsley KJ, Marrett S, Neelin P, Evans AC (1992) A three-dimensional statistical analysis as long as CBF activation studies in human brain Journal of Cerebral Blood Flow in addition to Metabolism 12:900-918 Worsley KJ, Marrett S, Neelin P, V in addition to al AC, Friston KJ, Evans AC (1995) A unified statistical approach as long as determining significant signals in images of cerebral activation Human Brain Mapping 4:58-73 Friston KJ, Worsley KJ, Frackowiak RSJ, Mazziotta JC, Evans AC (1994) Assessing the Significance of Focal Activations Using their Spatial Extent Human Brain Mapping 1:214-220 Cao J (1999) The size of the connected components of excursion sets of 2, t in addition to F fields Advances in Applied Probability (in press) Worsley KJ, Marrett S, Neelin P, Evans AC (1995) Searching scale space as long as activation in PET images Human Brain Mapping 4:74-90 Worsley KJ, Poline J-B, V in addition to al AC, Friston KJ (1995) Tests as long as distributed, non-focal brain activations NeuroImage 2:183-194 Friston KJ, Holmes AP, Poline J-B, Price CJ, Frith CD (1996) Detecting Activations in PET in addition to fMRI: Levels of Inference in addition to Power Neuroimage 4:223-235 Ch5 Ch4 index overview multiple comparisons r in addition to om field theory r in addition to om effects hypothesis testing fallacy

## Mekelburg, Mike Head Reporter

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