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Multiple testing Justin Chumbley Laboratory for Social and Neural Systems Research University of Zurich With many thanks for slides & images to: FIL Methods group

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Overview of SPM – Random field theory RealignmentSmoothing Normalisation General linear model Statistical parametric map (SPM) Image time-series Parameter estimates Design matrix Template Kernel Gaussian field theory p <0.05 Statisticalinference

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t = contrast of estimated parameters variance estimate t Error at a single voxel

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t = contrast of estimated parameters variance estimate t Error at a single voxel

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Decision: H 0, H 1 : zero/non-zero activation t = contrast of estimated parameters variance estimate t h Error at a single voxel

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Decision: H 0, H 1 : zero/non-zero activation t = contrast of estimated parameters variance estimate t h Error at a single voxel

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Decision: H 0, H 1 : zero/non-zero activation t = contrast of estimated parameters variance estimate t h Error at a single voxel

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Decision: H 0, H 1 : zero/non-zero activation t = contrast of estimated parameters variance estimate t h Decision rule (threshold) h, determines related error rates, Convention: Penalize complexity Choose h to give acceptable under H 0 Error at a single voxel

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Types of error Reality H1H1 H0H0 H0H0 H1H1 True negative (TN) True positive (TP) False positive (FP) False negative (FN) specificity: 1- = TN / (TN + FP) = proportion of actual negatives which are correctly identified sensitivity (power): 1- = TP / (TP + FN) = proportion of actual positives which are correctly identified Decision

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Multiple tests t = contrast of estimated parameters variance estimate t h t h h t h What is the problem?

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Multiple tests t = contrast of estimated parameters variance estimate t h t h h t h Penalize each independent opportunity for error.

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Multiple tests t = contrast of estimated parameters variance estimate t h t h h t h Convention: Choose h to limit assuming family-wise H 0

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Issues 1. Voxels or regions 2. Bonferroni too harsh (insensitive) Unnecessary penalty for sampling resolution (#voxels/volume) Unnecessary penalty for independence

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intrinsic smoothness –MRI signals are aquired in k-space (Fourier space); after projection on anatomical space, signals have continuous support –diffusion of vasodilatory molecules has extended spatial support extrinsic smoothness –resampling during preprocessing –matched filter theorem deliberate additional smoothing to increase SNR –Robustness to between-subject anatomical differences

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1.Apply high threshold: identify improbably high peaks 2.Apply lower threshold: identify improbably broad peaks 3.Total number of regions? Acknowledge/estimate dependence Detect effects in smooth landscape, not voxels

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Null distribution? 1. Simulate null experiments 2. Model null experiments

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Use continuous random field theory Discretisation (“lattice approximation”) Smoothness quantified: resolution elements (‘resels’) similar, but not identical to # independent observations computed from spatial derivatives of the residuals

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Euler characteristic

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General form for expected Euler characteristic 2, F, & t fields E [ h ( )] = d R d ( ) d ( h ) Small volumes: Anatomical atlas, ‘functional localisers’, orthogonal contrasts, volume around previously reported coordinates… Unified Formula R d ( ):d-dimensional Minkowski functional of – function of dimension, space and smoothness: R 0 ( )= ( ) Euler characteristic of R 1 ( )=resel diameter R 2 ( )=resel surface area R 3 ( )=resel volume d ( ):d-dimensional EC density of Z(x) – function of dimension and threshold, specific for RF type: E.g. Gaussian RF: 0 (h)=1- (h) 1 (h)=(4 ln2) 1/2 exp(-h 2 /2) / (2 ) 2 (h)=(4 ln2) exp(-h 2 /2) / (2 ) 3/2 3 (h)=(4 ln2) 3/2 (h 2 -1) exp(-h 2 /2) / (2 ) 2 4 (h)=(4 ln2) 2 (h 3 -3h) exp(-h 2 /2) / (2 ) 5/2

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Euler characteristic (EC) for 2D images

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Spatial extent: similar

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Voxel, cluster and set level tests e u h

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ROI VoxelField FWEFDR * voxels/volume Height Extent ROI VoxelField Height Extent There is a multiple testing problem (‘voxel’ or ‘blob’ perspective). More corrections needed as.. Detect an effect of unknown extent & location

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Further reading Friston KJ, Frith CD, Liddle PF, Frackowiak RS. Comparing functional (PET) images: the assessment of significant change. J Cereb Blood Flow Metab Jul;11(4): Genovese CR, Lazar NA, Nichols T. Thresholding of statistical maps in functional neuroimaging using the false discovery rate. Neuroimage Apr;15(4): Worsley KJ Marrett S Neelin P Vandal AC Friston KJ Evans AC. A unified statistical approach for determining significant signals in images of cerebral activation. Human Brain Mapping 1996;4:58-73.

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