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Statistical Inference and Random Field Theory Will Penny SPM short course, Kyoto, Japan, 2002 Will Penny SPM short course, Kyoto, Japan, 2002

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realignment & motion correction smoothing normalisation General Linear Model Ümodel fitting Üstatistic image corrected p-values image data parameter estimates design matrix anatomical reference kernel Statistical Parametric Map Random Field Theory

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…a voxel by voxel hypothesis testing approach reliably identify regions showing a significant experimental effect Assessment of statistic imagesAssessment of statistic images multiple comparisonsmultiple comparisons random field theoryrandom field theory smoothnesssmoothness spatial levels of inference & powerspatial levels of inference & power …a voxel by voxel hypothesis testing approach reliably identify regions showing a significant experimental effect Assessment of statistic imagesAssessment of statistic images multiple comparisonsmultiple comparisons random field theoryrandom field theory smoothnesssmoothness spatial levels of inference & powerspatial levels of inference & power OverviewOverview

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Classical hypothesis testing Null hypothesis HNull hypothesis H –test statistic –null distributions Hypothesis testHypothesis test –control Type I error incorrectly reject Hincorrectly reject H –test level or error rate Pr(“reject” H | H) Pr(“reject” H | H) p –valuep –value –min at which H rejected –Pr(T t | H) –characterising “surprise” Null hypothesis HNull hypothesis H –test statistic –null distributions Hypothesis testHypothesis test –control Type I error incorrectly reject Hincorrectly reject H –test level or error rate Pr(“reject” H | H) Pr(“reject” H | H) p –valuep –value –min at which H rejected –Pr(T t | H) –characterising “surprise” t –distribution, 32 df. F –distribution, 10,32 df.

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Family of hypothesesFamily of hypotheses –H k k = {1,…,K} –H = H 1 H 2 … H k H K Familywise Type I errorFamilywise Type I error –weak control – omnibus test Pr(“reject” H H ) Pr(“reject” H H ) “anything, anywhere” ?“anything, anywhere” ? –strong control – localising test Pr(“reject” H W H W ) Pr(“reject” H W H W ) W: W & H W “anything, & where” ?“anything, & where” ? Family of hypothesesFamily of hypotheses –H k k = {1,…,K} –H = H 1 H 2 … H k H K Familywise Type I errorFamilywise Type I error –weak control – omnibus test Pr(“reject” H H ) Pr(“reject” H H ) “anything, anywhere” ?“anything, anywhere” ? –strong control – localising test Pr(“reject” H W H W ) Pr(“reject” H W H W ) W: W & H W “anything, & where” ?“anything, & where” ? Multiple comparisons terminology Activation is zero everywhere eg. Look at average activation over volume eg. Look at maxima of statistical field

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The Bonferroni correction Given a family of N independent voxels and a voxel-wise error rate v the Family-Wise Error rate (FWE) or ‘corrected’ error rate is α = 1 – (1-v) N α = 1 – (1-v) N ~ Nv Therefore, to ensure a particular FWE we choose v = α / N A Bonferroni correction is appropriate for independent tests If v=0.05 then over 100 voxels we’ll get 5 voxel-wise type I errors. But we’ll get a much higher α. To ensure α=0.05 we need v=0.0005 !

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The Bonferroni correction Independent VoxelsSpatially Correlated Voxels Bonferroni is too conservative for brain images

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Random Field Theory Consider a statistic image as a lattice representation of a continuous random fieldConsider a statistic image as a lattice representation of a continuous random field Use results from continuous random field theoryUse results from continuous random field theory Consider a statistic image as a lattice representation of a continuous random fieldConsider a statistic image as a lattice representation of a continuous random field Use results from continuous random field theoryUse results from continuous random field theory Lattice representation

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Euler Characteristic (EC) Topological measure –threshold an image at u –excursion set u –excursion set u u = # blobs - # holes u ) = # blobs - # holes -At high u, u = # blobs -At high u, u ) = # blobs Reject H Ω if Euler char non-zero Reject H Ω if Euler char non-zero α Pr( u > 0 ) α Pr( u ) > 0 ) Expected Euler char p–value (at high u) (at high u) α E [ u ] α E [ u )] Topological measure –threshold an image at u –excursion set u –excursion set u u = # blobs - # holes u ) = # blobs - # holes -At high u, u = # blobs -At high u, u ) = # blobs Reject H Ω if Euler char non-zero Reject H Ω if Euler char non-zero α Pr( u > 0 ) α Pr( u ) > 0 ) Expected Euler char p–value (at high u) (at high u) α E [ u ] α E [ u )]

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Example – 2D Gaussian images α = R (4 ln 2) (2π) -3/2 u exp (-u 2 /2) Voxel-wise threshold, u Number of Resolution Elements (RESELS), R N=100x100 voxels, Smoothness FWHM=10, gives R=10x10=100

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Example – 2D Gaussian images α = R (4 ln 2) (2π) -3/2 u exp (-u 2 /2) For R=100 and α=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.

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DevelopmentsDevelopments 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.

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Restricted search regions Box has 16 markers Frame has 32 markers Box and frame have same number of voxels

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

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General form for expected Euler characteristicGeneral form for expected Euler characteristic 2, F, & t fields restricted search regions 2, F, & t fields restricted search regions α = R d ( ) d (u) General form for expected Euler characteristicGeneral form for expected Euler characteristic 2, F, & t fields restricted search regions 2, F, & t fields restricted search regions α = R d ( ) d (u) 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 (u):d-dimensional EC density – E.g. Gaussian RF: 0 (u)=1- (u) 1 (u)=(4 ln2) 1/2 exp(-u 2 /2) / (2 ) 2 (u)=(4 ln2) exp(-u 2 /2) / (2 ) 3/2 3 (u)=(4 ln2) 3/2 (u 2 -1) exp(-u 2 /2) / (2 ) 2 4 (u)=(4 ln2) 2 (u 3 -3u) exp(-u 2 /2) / (2 ) 5/2 AuAu Worsley et al. (1996), HBM

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Resel Counts for Brain Structures FWHM=20mm

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Functional Imaging Data The Random Fields are the component fields,The Random Fields are the component fields, Y = Xw +E, e=E/σ Y = Xw +E, e=E/σ We can only estimate the component fields, usingWe can only estimate the component fields, using estimates of w and σ estimates of w and σ To apply RFT we need the RESEL count which requires smoothness estimatesTo apply RFT we need the RESEL count which requires smoothness estimates The Random Fields are the component fields,The Random Fields are the component fields, Y = Xw +E, e=E/σ Y = Xw +E, e=E/σ We can only estimate the component fields, usingWe can only estimate the component fields, using estimates of w and σ estimates of w and σ To apply RFT we need the RESEL count which requires smoothness estimatesTo apply RFT we need the RESEL count which requires smoothness estimates

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Component fields = +YX component fields data matrix design matrix parameters += ? voxels scans errors ? variance

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Estimated component fields data matrix design matrix parameters errors + ? = ? voxels scans Üestimate ^ residuals estimated component fields parameter estimates estimated variance = Each row is an estimated component field

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Smoothness Estimation Roughness | |Roughness | | Point Response Function PRFPoint Response Function PRF Roughness | |Roughness | | Point Response Function PRFPoint Response Function PRF Gaussian PRFGaussian PRF f x f x 00 f y f z f y 0 00 f z | | = (4ln(2)) 3/2 / (f x f y f z ) RESEL COUNTRESEL COUNT R 3 ( ) = ( ) / (f x f y f z ) α = R 3 ( ) (4ln(2)) 3/2 (u 2 -1) exp(-u 2 /2) / (2 ) 2 Approximate the peak of the Covariance function with a Gaussian

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Family of hypothesesFamily of hypotheses –H k k = {1,…,K} –H = H 1 H 2 … H k H K Familywise Type I errorFamilywise Type I error –weak control – omnibus test Pr(“reject” H H ) Pr(“reject” H H ) “anything, anywhere” ?“anything, anywhere” ? –strong control – localising test Pr(“reject” H W H W ) Pr(“reject” H W H W ) W: W & H W “anything, & where” ?“anything, & where” ? Family of hypothesesFamily of hypotheses –H k k = {1,…,K} –H = H 1 H 2 … H k H K Familywise Type I errorFamilywise Type I error –weak control – omnibus test Pr(“reject” H H ) Pr(“reject” H H ) “anything, anywhere” ?“anything, anywhere” ? –strong control – localising test Pr(“reject” H W H W ) Pr(“reject” H W H W ) W: W & H W “anything, & where” ?“anything, & where” ? Multiple comparisons terminology Activation is zero everywhere eg. Look at average activation over volume eg. Look at maxima of statistical field

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Cluster and Set-level Inference We can increase sensitivity by trading off anatomical specificityWe can increase sensitivity by trading off anatomical specificity Given a voxel level threshold u, we can computeGiven 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 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 CLUSTER-LEVEL INFERENCE Similarly, we can compute the likelihood of getting cSimilarly, we can compute the likelihood of getting c clusters each having at least n voxels clusters each having at least n voxels SET-LEVEL INFERENCE SET-LEVEL INFERENCE We can increase sensitivity by trading off anatomical specificityWe can increase sensitivity by trading off anatomical specificity Given a voxel level threshold u, we can computeGiven 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 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 CLUSTER-LEVEL INFERENCE Similarly, we can compute the likelihood of getting cSimilarly, we can compute the likelihood of getting c clusters each having at least n voxels clusters each having at least n voxels SET-LEVEL INFERENCE SET-LEVEL INFERENCE

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Suprathreshold cluster tests Primary threshold uPrimary threshold u –examine connected components of excursion set –Suprathreshold clusters –Reject H W for clusters of voxels W of size S > s Localisation (Strong control)Localisation (Strong control) –at cluster level –increased power –esp. high resolutions ( f MRI ) Thresholds, p –valuesThresholds, p –values –Pr(S max > s H ) Nosko, Friston, (Worsley) –Poisson occurrence (Adler) –Assumme form for Pr(S=s|S>0) Primary threshold uPrimary threshold u –examine connected components of excursion set –Suprathreshold clusters –Reject H W for clusters of voxels W of size S > s Localisation (Strong control)Localisation (Strong control) –at cluster level –increased power –esp. high resolutions ( f MRI ) Thresholds, p –valuesThresholds, p –values –Pr(S max > s H ) Nosko, Friston, (Worsley) –Poisson occurrence (Adler) –Assumme form for Pr(S=s|S>0) 5mm FWHM 10mm FWHM 15mm FWHM (2mm 2 pixels)

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Poisson Clumping Heuristic Expected number of clusters p{cluster volume > k} Expected cluster volume EC density ( Search volume (R) Smoothness

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Levels of inference Parameters u - 3.09 k - 12 voxels S - 32 3 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) n=82 n=32 n=1 2 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

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SPM results...

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RFT Assumptions Model fit & assumptionsModel fit & assumptions –valid distributional results Multivariate normalityMultivariate normality –of component images Covariance function of component images must beCovariance function of component images must be - Stationary (pre SPM99) - Stationary (pre SPM99) - Can be nonstationary - Can be nonstationary (SPM99 onwards) (SPM99 onwards) - Twice differentiable - Twice differentiable Model fit & assumptionsModel fit & assumptions –valid distributional results Multivariate normalityMultivariate normality –of component images Covariance function of component images must beCovariance function of component images must be - Stationary (pre SPM99) - Stationary (pre SPM99) - Can be nonstationary - Can be nonstationary (SPM99 onwards) (SPM99 onwards) - Twice differentiable - Twice differentiable SmoothnessSmoothness –smoothness » voxel size lattice approximationlattice approximation smoothness estimationsmoothness estimation –practically FWHM 3 VoxDimFWHM 3 VoxDim –otherwise conservativeconservative

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Worsley KJ, Marrett S, Neelin P, Evans AC (1992) “A three-dimensional statistical analysis for CBF activation studies in human brain” Journal of Cerebral Blood Flow and Metabolism 12:900-918 Worsley KJ, Marrett S, Neelin P, Vandal AC, Friston KJ, Evans AC (1995) “A unified statistical approach for 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 and F fields” Advances in Applied Probability (in press) Worsley KJ, Marrett S, Neelin P, Evans AC (1995) “Searching scale space for activation in PET images” Human Brain Mapping 4:74-90 Worsley KJ, Poline J-B, Vandal AC, Friston KJ (1995) “Tests for 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 and fMRI: Levels of Inference and Power” Neuroimage 4:223-235 Worsley KJ, Marrett S, Neelin P, Evans AC (1992) “A three-dimensional statistical analysis for CBF activation studies in human brain” Journal of Cerebral Blood Flow and Metabolism 12:900-918 Worsley KJ, Marrett S, Neelin P, Vandal AC, Friston KJ, Evans AC (1995) “A unified statistical approach for 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 and F fields” Advances in Applied Probability (in press) Worsley KJ, Marrett S, Neelin P, Evans AC (1995) “Searching scale space for activation in PET images” Human Brain Mapping 4:74-90 Worsley KJ, Poline J-B, Vandal AC, Friston KJ (1995) “Tests for 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 and fMRI: Levels of Inference and Power” Neuroimage 4:223-235 Multiple Comparisons, & Random Field Theory Ch5Ch4

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SummarySummary We should correct for multiple comparisonsWe should correct for multiple comparisons We can use Random Field Theory (RFT)We can use Random 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 functionRFT 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.To a first approximation, RFT is a Bonferroni correction using RESELS. We only need to correct for the volume of interest.We only need to correct for the volume of interest. Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference.Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference. We should correct for multiple comparisonsWe should correct for multiple comparisons We can use Random Field Theory (RFT)We can use Random 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 functionRFT 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.To a first approximation, RFT is a Bonferroni correction using RESELS. We only need to correct for the volume of interest.We only need to correct for the volume of interest. Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference.Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference.

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