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

Statistical Inference, Multiple Comparisons in addition to R in addition to om Field Theory Andrew Holme www.phwiki.com

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 p–values 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 p–value 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 Mekelburg, Mike Ajo Copper News Head Reporter www.phwiki.com

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

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