Contents

## 2D Gaussian fields 3D Gaussian fields 3D t-fields

Midey, Connie, Health & Fitness Reporter has reference to this Academic Journal, PHwiki organized this Journal Statistical Inference in addition to R in addition to om Field Theory Will Penny SPM short course, Kyoto, Japan, 2002 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 Overview a voxel by voxel hypothesis testing approach reliably identify regions showing a significant experimental effect Assessment of statistic images multiple comparisons r in addition to om field theory smoothness spatial levels of inference & power

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Classical hypothesis testing Null hypothesis H test statistic null distributions Hypothesis test control Type I error incorrectly reject H test level or error rate Pr(reject H H) p value min a at which H rejected Pr(T t H) characterising surprise Multiple comparisons terminology Family of hypotheses Hk k = {1, ,K} H = H1 H2 Hk 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 Activation is zero everywhere eg. Look at average activation over volume eg. Look at maxima of statistical field The Bonferroni correction Given a family of N independent voxels in addition to a voxel-wise error rate v the Family-Wise Error rate (FWE) or corrected error rate is = 1 (1-v)N ~ Nv There as long as e, to ensure a particular FWE we choose v = / N A Bonferroni correction is appropriate as long as independent tests If v=0.05 then over 100 voxels well get 5 voxel-wise type I errors. But well get a much higher . To ensure =0.05 we need v=0.0005 !

The Bonferroni correction Independent Voxels Spatially Correlated Voxels Bonferroni is too conservative as long as brain images R in addition to om Field Theory Consider a statistic image as a lattice representation of a continuous r in addition to om field Use results from continuous r in addition to om field theory Lattice representation Euler Characteristic (EC) Topological measure threshold an image at u excursion set Au (Au) = blobs – holes At high u, (Au) = blobs Reject H if Euler char non-zero Pr((Au) > 0 ) Expected Euler char pvalue (at high u) E[(Au)]

Example 2D Gaussian images = R (4 ln 2) (2) -3/2 u exp (-u2/2) Voxel-wise threshold, u Number of Resolution Elements (RESELS), R N=100×100 voxels, Smoothness FWHM=10, gives R=10×10=100 Example 2D Gaussian images = R (4 ln 2) (2) -3/2 u exp (-u2/2) For R=100 in addition to =0.05 RFT gives u=3.8 Using R=100 in a Bonferroni correction gives u=3.3 Friston et al. (1991) J. Cer. Bl. Fl. M. Developments Friston et al. (1991) J. Cer. Bl. Fl. M. (Not EC Method) 2D Gaussian fields 3D Gaussian fields 3D t-fields Worsley et al. (1992) J. Cer. Bl. Fl. M. Worsley et al. (1993) Quant. Brain. Func.

Restricted search regions Box has 16 markers Frame has 32 markers Box in addition to frame have same number of voxels Unified Theory General as long as m as long as expected Euler characteristic 2, F, & t fields restricted search regions = S Rd (W) rd (u) Rd (W): RESEL count; depends on the search region how big, how smooth, what shape rd (u): EC density; depends on type of field (eg. Gaussian, t) in addition to the threshold, u. Worsley et al. (1996), HBM Unified Theory General as long as m as long as expected Euler characteristic 2, F, & t fields restricted search regions = S Rd (W) rd (u) Rd (W): RESEL count R0(W) = (W) Euler characteristic of W R1(W) = resel diameter R2(W) = resel surface area R3(W) = resel volume rd (u): d-dimensional EC density E.g. Gaussian RF: 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 Worsley et al. (1996), HBM

Resel Counts as long as Brain Structures FWHM=20mm Functional Imaging Data The R in addition to om Fields are the component fields, Y = Xw +E, e=E/ We can only estimate the component fields, using estimates of w in addition to To apply RFT we need the RESEL count which requires smoothness estimates Component fields component fields data matrix design matrix parameters + = voxels scans variance

Estimated component fields data matrix design matrix parameters errors + = voxels scans estimate residuals estimated component fields parameter estimates estimated variance = Each row is an estimated component field Smoothness Estimation Roughness Point Response Function PRF Gaussian PRF fx 0 0 = 0 fy 0 0 0 fz = (4ln(2))3/2 / (fx fy fz) RESEL COUNT R3() = () / (fx fy fz) = R3() (4ln(2))3/2 (u 2 -1) exp(-u 2/2) / (2)2 Approximate the peak of the Covariance function with a Gaussian Multiple comparisons terminology Family of hypotheses Hk k = {1, ,K} H = H1 H2 Hk 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 Activation is zero everywhere eg. Look at average activation over volume eg. Look at maxima of statistical field

Cluster in addition to Set-level Inference We can increase sensitivity by trading off anatomical specificity Given a voxel level threshold u, we can compute the likelihood (under the null hypothesis) of getting n or more connectedcomponents in the excursion set ie. a cluster containing at least n voxels CLUSTER-LEVEL INFERENCE Similarly, we can compute the likelihood of getting c clusters each having at least n voxels SET-LEVEL INFERENCE 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) voxel-level P(c 1 n > 0, t 4.37) = 0.048 (corrected) At least one cluster with unspecified number of voxels above threshold At least one cluster with at least 82 voxels above threshold At least 3 clusters above threshold SPM results

SPM results SPM results RFT Assumptions Model fit & assumptions valid distributional results Multivariate normality of component images Covariance function of component images must be – Stationary (pre SPM99) – Can be nonstationary (SPM99 onwards) – Twice differentiable Smoothness smoothness » voxel size lattice approximation smoothness estimation practically FWHM 3 VoxDim otherwise conservative

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 Summary We should correct as long as multiple comparisons We can use R in addition to om Field Theory (RFT) RFT requires (i) a good lattice approximation to underlying multivariate Gaussian fields, (ii) that these fields are continuous with a twice differentiable correlation function To a first approximation, RFT is a Bonferroni correction using RESELS. We only need to correct as long as the volume of interest. Depending on nature of signal we can trade-off anatomical specificity as long as signal sensitivity with the use of cluster-level inference.

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