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Data Analysis framework in Brain Imaging STAT 992: Image Analysis March 23, 2004 Moo K. Chung Department of Statistics Department of Biostatistics and Medical Informatics W.M. Keck Brain Laboratory University of Wisconsin-Madison http://www.stat.wisc.edu/~mchung

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Acknowledgement: Presentation based on Will Penny, Jason Lerch and Thomas Nichols’s PowerPoint slides Some images based on Thomas Hoffman’s research

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Brain Image Analysis Brain Image Analysis is a total science.Brain Image Analysis is a total science. Before Images are transformed into a proper format for data analysis, it will go through a bunch of image processing procedures.Before Images are transformed into a proper format for data analysis, it will go through a bunch of image processing procedures. If we can get extract proper data out of images, half of the problems are already solved.If we can get extract proper data out of images, half of the problems are already solved. Brain Image Analysis is a total science.Brain Image Analysis is a total science. Before Images are transformed into a proper format for data analysis, it will go through a bunch of image processing procedures.Before Images are transformed into a proper format for data analysis, it will go through a bunch of image processing procedures. If we can get extract proper data out of images, half of the problems are already solved.If we can get extract proper data out of images, half of the problems are already solved.

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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 fMRI processing steps

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MRI processing steps MRI processing steps Montreal Neurological Institute image processing pipeline

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Non-uniformity correction NativeCorrected nu_correct native.mnc corrected.mnc

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Classification/segmentationClassification/segmentation classify_clean final.mnc classified.mnc

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Tissue classification/segmentation Clustering algorithm based on maximum likelihood mixture model (Hartigan, 1975) Automatic skull stripping

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3 different tissue types Binary masks: 0 or 1 Gray matter White matter Gaussian kernel smoothing

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MaskingMasking cortical_surface classified.mnc mask.obj 1.5 surface_mask2 classified.mnc mask.obj masked.mnc

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Image registration

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Corpus callosum (CC) is the white matter brain substructure that connects hemispheres. We have 16 autistic subjects and 12 normal subjects. Quantify the CC shape difference between two groups? Group 1Group 2

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How do we compare shapes? Pixel by pixel comparison causes anatomical mismatching. Solution: image registration. The aim of image registration is to find a smooth one-to-one mapping that matches homologous anatomies together.

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Nonlinear image registration Estimate a continuous 3D map that matches two brain images.

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He introduced deformable grid and deformation of homologous biological structures Thompson, D. W. (1917) On growth and form. Cambridge University Press, Cambridge.

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Nonlinear image registration based on basis function expansion 300 subjects average template Warping into average blurred template reduce the probability of complete mismatch. warping Warped brain

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Large scale automatic image analysis Subject 1 Subject 2 Subject 3 Subject 498 Subject 499 Subject 500 template.. 500 MRIs will be warped into a template and anatomical differences can be compared at a common reference frame.

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Estimating Nonlinear Image Registration Elastic deformation based Fluid dynamics based Intensity correlation based Bayesian approach

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Image Registration Similarity measure Variational approach PDE approach

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x'= y'= z'= -1.212161e+02 -1.692117e+02 1.239336e+00 + 2.846339e+00 9.860215e-01 -4.670216e-03 x + 4.541216e-01 3.344188e+00 -1.022118e-02 y + 2.277959e+00 1.849708e+00 9.958860e-01 z + -9.744798e-03 -4.951447e-03 1.253383e-05 x^2 + -4.519879e-03 -4.248561e-03 -2.655254e-05 x*y + -9.122374e-04 -9.371881e-03 4.040382e-05 y^2 + -1.624103e-02 -3.371953e-03 2.356452e-06 x*z + -3.519974e-03 -2.799626e-02 9.228041e-05 y*z + -1.572948e-02 -1.688950e-03 5.386545e-05 z^2 + 2.495023e-05 4.120123e-06 3.604820e-08 x^3 + 3.232645e-06 1.739698e-05 1.044795e-07 x^2*y + 1.074305e-05 4.357408e-06 -9.302004e-10 x*y^2 + -1.059526e-06 1.699618e-05 1.166377e-07 y^3 + 5.512034e-06 9.330769e-06 -2.219099e-08 x^2*z + 1.275631e-05 -9.233413e-06 1.236940e-07 x*y*z + -5.236010e-07 3.234824e-05 -6.819396e-07 y^2*z + 9.506628e-05 1.214112e-05 -1.238024e-07 x*z^2 + 2.016546e-05 1.475354e-04 -1.693465e-08 y*z^2 + 3.377913e-06 -7.093638e-05 -2.757074e-07 z^3 3 rd order polynomial warping

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Curve registration by dynamic time warping algorithm -Thomas Hoffmann, Honors B.Sc. thesis.

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Image smoothing

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Kernel smoothing

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Anisotropic Gaussian kernel smoothing It will smooth out signals along the eigenvectors. The amount of smoothing is proportional to the eigenvalues. So it will basically smooth out along the principal eigenvector. Isotropic kernel Anisotropic kernel

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Isotropic Gaussian kernel smoothing Principal eigenvalues > 0.6 10mm FWHM20mm FWHM

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Anisotropic Smoothing= Edge Enhancement

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Before After diffusion smoothing

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initial mean curvature diffusion smoothing estimate Inner surface = gray/white matter interface

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Flattened map showing smoothing initial mean curvature20 iterations100 iterations 0.00 0.01

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WHY WE SMOOTH? See next slide Smooth T random fields 6.5 -6.5 -2.0 2.0

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Autocorrelation: Precoloring Temporally blur, smooth your dataTemporally blur, smooth your data –This induces more dependence! –But we exactly know the form of the dependence induced –Assume that intrinsic autocorrelation is negligible relative to smoothing Then we know autocorrelation exactlyThen we know autocorrelation exactly Correct GLM inferences based on “known” autocorrelationCorrect GLM inferences based on “known” autocorrelation Temporally blur, smooth your dataTemporally blur, smooth your data –This induces more dependence! –But we exactly know the form of the dependence induced –Assume that intrinsic autocorrelation is negligible relative to smoothing Then we know autocorrelation exactlyThen we know autocorrelation exactly Correct GLM inferences based on “known” autocorrelationCorrect GLM inferences based on “known” autocorrelation [Friston, et al., “To smooth or not to smooth…” NI 12:196-208 2000]

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Autocorrelation: Prewhitening Statistically optimal solutionStatistically optimal solution If know true autocorrelation exactly, can undo the dependenceIf know true autocorrelation exactly, can undo the dependence –De-correlate your data, your model –Then proceed as with independent data Problem is obtaining accurate estimates of autocorrelationProblem is obtaining accurate estimates of autocorrelation –Some sort of regularization is required Spatial smoothing of some sortSpatial smoothing of some sort Statistically optimal solutionStatistically optimal solution If know true autocorrelation exactly, can undo the dependenceIf know true autocorrelation exactly, can undo the dependence –De-correlate your data, your model –Then proceed as with independent data Problem is obtaining accurate estimates of autocorrelationProblem is obtaining accurate estimates of autocorrelation –Some sort of regularization is required Spatial smoothing of some sortSpatial smoothing of some sort

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fMRI example

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Basic fMRI Example Time series at each voxel.Time series at each voxel.

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one stimulus two stimuli fMRI time series modeling

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A Linear Model Intensity Time = 11 22 + + error x1x1 x2x2 “Linear” in parameters 1 & 2“Linear” in parameters 1 & 2

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… in matrix form. = + Y Y N: Number of scans, p: Number of regressors

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leftrightleftright subject 20

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subject 41 leftrightleftright

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OSL fitting of subject 20 left amygdala

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OSL fitting of subject 20 right amygdala

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Generalized Least Squares (GLS) Estimation

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HRF snake attacking snake crawling fish swimming AFNI result subject 20 right amygdala

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HRF reconvolved with the initial stimuli Black: HR based on OSL Red: HR based on GSL In this particular example, GSL can get the dip OSL can not get.

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Whitening by GSL correlation

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Multiple Testing Problem Inference on statistic imagesInference on statistic images –Fit GLM at each voxel –Create statistic images of effect Which of 100,000 voxels are significant?Which of 100,000 voxels are significant? – =0.05 5,000 false positives! Inference on statistic imagesInference on statistic images –Fit GLM at each voxel –Create statistic images of effect Which of 100,000 voxels are significant?Which of 100,000 voxels are significant? – =0.05 5,000 false positives! t > 0.5 t > 1.5 t > 2.5 t > 3.5 t > 4.5 t > 5.5t > 6.5

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MCP Solutions: Measuring False Positives Familywise Error Rate (FWER)Familywise Error Rate (FWER) –Familywise Error Existence of one or more false positivesExistence of one or more false positives –FWER is probability of familywise error corrected P-value False Discovery Rate (FDR)False Discovery Rate (FDR) –R voxels declared active, V falsely so Observed false discovery rate: V/RObserved false discovery rate: V/R –FDR = E(V/R) Q-value –This is a relative measure. Familywise Error Rate (FWER)Familywise Error Rate (FWER) –Familywise Error Existence of one or more false positivesExistence of one or more false positives –FWER is probability of familywise error corrected P-value False Discovery Rate (FDR)False Discovery Rate (FDR) –R voxels declared active, V falsely so Observed false discovery rate: V/RObserved false discovery rate: V/R –FDR = E(V/R) Q-value –This is a relative measure.

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FWER MCP Solutions: Random Field Theory Euler Characteristic uEuler Characteristic u –Topological Measure #blobs - #holes#blobs - #holes –At high thresholds, just counts blobs –FWER= P(Max voxel u | H o ) = P(One or more blobs | H o ) P( u 1 | H o ) E( u | H o ) Euler Characteristic uEuler Characteristic u –Topological Measure #blobs - #holes#blobs - #holes –At high thresholds, just counts blobs –FWER= P(Max voxel u | H o ) = P(One or more blobs | H o ) P( u 1 | H o ) E( u | H o ) Random Field Suprathreshold Sets Threshold

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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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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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Expected EC for a stationary Gaussian field EC density Minkowski functionals Adler (1981), Worsley (1996)

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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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Random Field Theory Closed form results for E( u )Closed form results for E( u ) –Z, t, F, Chi-Squared Continuous RFs Results depend only on RESELsResults depend only on RESELs –RESolution ELements –A volume element of size FWHM x FWHM y FWHM z –RESEL count Voxel-size-independent measure of volumeVoxel-size-independent measure of volume Inferences do not depend on stationarityInferences do not depend on stationarity –Smoothness can vary (It is cluster size results that require stationarity) Closed form results for E( u )Closed form results for E( u ) –Z, t, F, Chi-Squared Continuous RFs Results depend only on RESELsResults depend only on RESELs –RESolution ELements –A volume element of size FWHM x FWHM y FWHM z –RESEL count Voxel-size-independent measure of volumeVoxel-size-independent measure of volume Inferences do not depend on stationarityInferences do not depend on stationarity –Smoothness can vary (It is cluster size results that require stationarity)

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Random Field Theory Limitations Sufficient smoothnessSufficient smoothness –FWHM smoothness 3-4 times voxel size Smoothness estimationSmoothness estimation –Estimate is biased when images not so smooth –Estimate is an estimate—sd’s on corr. p-values Multivariate normalityMultivariate normality –Virtually impossible to check Several layers of approximationsSeveral layers of approximations –E.g., t field results conservative for low df Sufficient smoothnessSufficient smoothness –FWHM smoothness 3-4 times voxel size Smoothness estimationSmoothness estimation –Estimate is biased when images not so smooth –Estimate is an estimate—sd’s on corr. p-values Multivariate normalityMultivariate normality –Virtually impossible to check Several layers of approximationsSeveral layers of approximations –E.g., t field results conservative for low df Lattice Image Data Continuous Random 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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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 connected components in the excursion set ie. a cluster containing at least n voxels the likelihood (under the null hypothesis) of getting n or more connected components 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 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 connected components in the excursion set ie. a cluster containing at least n voxels the likelihood (under the null hypothesis) of getting n or more connected components 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

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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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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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Controlling FWER: Permutation Test Parametric methodsParametric methods –Assume distribution of max statistic under null hypothesis Nonparametric methodsNonparametric methods –Use data to find distribution of max statistic under null hypothesis –Any max statistic! –Due to Fisher Parametric methodsParametric methods –Assume distribution of max statistic under null hypothesis Nonparametric methodsNonparametric methods –Use data to find distribution of max statistic under null hypothesis –Any max statistic! –Due to Fisher 5% Parametric Null Max Distribution 5% Nonparametric Null Max Distribution

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Measuring False Positives FWER vs FDR Signal+Noise Noise

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FWE 6.7% 10.4%14.9%9.3%16.2%13.8%14.0% 10.5%12.2%8.7% Control of Familywise Error Rate at 10% 11.3% 12.5%10.8%11.5%10.0%10.7%11.2%10.2%9.5% Control of Per Comparison Rate at 10% Percentage of Null Pixels that are False Positives Control of False Discovery Rate at 10% Occurrence of Familywise Error Percentage of Activated Pixels that are False Positives

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Controlling FDR: Benjamini & Hochberg Select desired limit q on E(FDR)Select desired limit q on E(FDR) Order p-values, p (1) p (2) ... p (V)Order p-values, p (1) p (2) ... p (V) Let r be largest i such thatLet r be largest i such that Reject all hypotheses corresponding to p (1),..., p (r).Reject all hypotheses corresponding to p (1),..., p (r). Select desired limit q on E(FDR)Select desired limit q on E(FDR) Order p-values, p (1) p (2) ... p (V)Order p-values, p (1) p (2) ... p (V) Let r be largest i such thatLet r be largest i such that Reject all hypotheses corresponding to p (1),..., p (r).Reject all hypotheses corresponding to p (1),..., p (r). p (i) i/V q p(i)p(i) i/Vi/V i/V q p-value 01 0 1

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Benjamini & Hochberg: Varying Signal Extent Signal Intensity3.0Signal Extent25.0Noise Smoothness3.0 p = 0.019274z = 2.07 7

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FDR Software for SPM http://www.sph.umich.edu/~nichols/FDR

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ReferencesReferences KM Petersson, TE Nichols, J-B Poline, and AP Holmes. Statistical limitations in functional neuroimaging I. Non-inferential methods and statistical models. Statistical limitations in functional neuroimaging II. Signal detection and statistical inference. Philosophical Transactions of the Royal Society: Biological Sciences, 354:1239-1281, 1999.KM Petersson, TE Nichols, J-B Poline, and AP Holmes. Statistical limitations in functional neuroimaging I. Non-inferential methods and statistical models. Statistical limitations in functional neuroimaging II. Signal detection and statistical inference. Philosophical Transactions of the Royal Society: Biological Sciences, 354:1239-1281, 1999. CR Genovese, N Lazar and TE Nichols. Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. NeuroImage, 15:870-878, 2002.CR Genovese, N Lazar and TE Nichols. Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. NeuroImage, 15:870-878, 2002. KM Petersson, TE Nichols, J-B Poline, and AP Holmes. Statistical limitations in functional neuroimaging I. Non-inferential methods and statistical models. Statistical limitations in functional neuroimaging II. Signal detection and statistical inference. Philosophical Transactions of the Royal Society: Biological Sciences, 354:1239-1281, 1999.KM Petersson, TE Nichols, J-B Poline, and AP Holmes. Statistical limitations in functional neuroimaging I. Non-inferential methods and statistical models. Statistical limitations in functional neuroimaging II. Signal detection and statistical inference. Philosophical Transactions of the Royal Society: Biological Sciences, 354:1239-1281, 1999. CR Genovese, N Lazar and TE Nichols. Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. NeuroImage, 15:870-878, 2002.CR Genovese, N Lazar and TE Nichols. Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. NeuroImage, 15:870-878, 2002. http://www.sph.umich.edu/~nichols

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Application: Surface data

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Corrected P-value Morphological descriptor 2D Euler characteristic density of t random field Cortical surface area

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Red: Tissue growth Blue: Tissue loss Yellow: Structure displacement Rejection regions.

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Yellow = Hotelling’s T^2 field

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Cortical thickness change t-map

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Curvature change t-map

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1 Data Modeling General Linear Model & Statistical Inference Thomas Nichols, Ph.D. Assistant Professor Department of Biostatistics

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