Data Analysis framework in Brain Imaging

STAT 992: Image AnalysisMarch 23, 2004

Moo K. ChungDepartment of Statistics

Department of Biostatistics and Medical InformaticsW.M. Keck Brain Laboratory

University of Wisconsin-Madison

http://www.stat.wisc.edu/~mchung

Acknowledgement:Acknowledgement:

Presentation based on Will Penny, Jason LerchPresentation based on Will Penny, Jason Lerch

and Thomas Nichols’s PowerPoint slidesand Thomas Nichols’s PowerPoint slides

Some images based on Thomas Hoffman’s researchSome images based on Thomas Hoffman’s research

Brain Image AnalysisBrain Image AnalysisBrain Image AnalysisBrain 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 Before Images are transformed into a proper format for

data analysis, it will go through a bunch of image data analysis, it will go through a bunch of image processing procedures. processing procedures.

• If we can get extract proper data out of images, half of If we can get extract proper data out of images, half of the problems are already solved.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 Before Images are transformed into a proper format for

data analysis, it will go through a bunch of image data analysis, it will go through a bunch of image processing procedures. processing procedures.

• If we can get extract proper data out of images, half of If we can get extract proper data out of images, half of the problems are already solved.the problems are already solved.

realignment &motion

correction

smoothing

normalisation

General Linear Modelmodel fittingstatistic image

corrected p-values

image data parameterestimatesdesign

matrix

anatomicalreference

kernel

StatisticalParametric Map

Random Field Theory

fMRI processingsteps

MRI processing stepsMRI processing steps MRI processing stepsMRI processing steps

Montreal Neurological Institute image processing pipeline

Non-uniformity correctionNon-uniformity correctionNon-uniformity correctionNon-uniformity correction

Native Corrected

nu_correct native.mnc corrected.mnc

Classification/segmentationClassification/segmentationClassification/segmentationClassification/segmentation

classify_clean final.mnc classified.mnc

Tissue classification/segmentationClustering algorithm based onmaximum likelihood mixture model (Hartigan, 1975)

Automatic skull stripping

3 different tissue types

Binary masks: 0 or 1

Gray matter White matter

Gaussian kernel smoothing

MaskingMaskingMaskingMasking

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

Image registration

Corpus callosum (CC) is thewhite matter brain substructurethat connects hemispheres.

We have 16 autistic subjects and 12 normal subjects. Quantify the CC shape difference between two groups?

Group 1 Group 2

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.

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

He introduced deformable grid and deformation of homologous biological structures

Thompson, D. W. (1917) On growth and form. Cambridge University Press, Cambridge.

Nonlinear image registration based on Nonlinear image registration based on basis function expansionbasis function expansion

Nonlinear image registration based on Nonlinear image registration based on basis function expansionbasis function expansion

300 subjects average template

Warping into average blurred template reduce the probability of complete mismatch.

warping

Warped brain

Large scale automatic image analysis

Subject 1 Subject 2 Subject 3 Subject 498Subject 499

Subject 500

template

. . . .

500 MRIs will be warped into a template and anatomical differences can be compared at a common reference frame.

Estimating Nonlinear Image Registration

•Elastic deformation based

•Fluid dynamics based

•Intensity correlation based

•Bayesian approach

Image Registration

Similarity measure

Variational approach

PDE approach

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

3rd order polynomial warping

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

Image smoothing

Kernel smoothingKernel smoothingKernel smoothingKernel smoothing

Anisotropic Gaussian kernel smoothing

It will smooth out signals along the eigenvectors. The amount ofsmoothing is proportional to the eigenvalues. So it will basicallysmooth out along the principal eigenvector.

Isotropic kernel Anisotropic kernel

Isotropic Gaussian kernel smoothing

Principal eigenvalues > 0.6

10mm FWHM 20mm FWHM

Anisotropic Smoothing= Edge Enhancement

Before

After diffusion smoothing

initial mean curvature diffusion smoothing estimate

Inner surface = gray/white matter interface

Flattened map showing smoothing

initial mean curvature 20 iterations 100 iterations

0.00

0.01

WHY WE SMOOTH? See next slide

Smooth T random fields

6.5

-6.5

-2.0

2.0

Autocorrelation:Autocorrelation:PrecoloringPrecoloring

Autocorrelation:Autocorrelation:PrecoloringPrecoloring

• Temporally blur, smooth your dataTemporally blur, smooth your data– This induces This induces more more dependence!dependence!

– But we exactly know the form of the dependence inducedBut we exactly know the form of the dependence induced

– Assume that intrinsic autocorrelation is negligible relative to Assume that intrinsic autocorrelation is negligible relative to smoothingsmoothing

• Then we know autocorrelation exactlyThen we know autocorrelation exactly• Correct GLM inferences based on “known” Correct GLM inferences based on “known”

autocorrelationautocorrelation

• Temporally blur, smooth your dataTemporally blur, smooth your data– This induces This induces more more dependence!dependence!

– But we exactly know the form of the dependence inducedBut we exactly know the form of the dependence induced

– Assume that intrinsic autocorrelation is negligible relative to Assume that intrinsic autocorrelation is negligible relative to smoothingsmoothing

• Then we know autocorrelation exactlyThen we know autocorrelation exactly• Correct GLM inferences based on “known” Correct GLM inferences based on “known”

autocorrelationautocorrelation

[Friston, et al., “To smooth or not to smooth…” NI 12:196-208 2000]

Autocorrelation:Autocorrelation:PrewhiteningPrewhitening

Autocorrelation:Autocorrelation:PrewhiteningPrewhitening

• Statistically optimal solutionStatistically optimal solution• If know true autocorrelation exactly, canIf know true autocorrelation exactly, can

undo the dependenceundo the dependence– DeDe-correlate your data, your model-correlate your data, your model

– Then proceed as with independent dataThen proceed as with independent data

• Problem is obtaining accurate estimates of Problem is obtaining accurate estimates of autocorrelationautocorrelation– Some sort of regularization is requiredSome sort of regularization is required

• Spatial smoothing of some sortSpatial smoothing of some sort

• Statistically optimal solutionStatistically optimal solution• If know true autocorrelation exactly, canIf know true autocorrelation exactly, can

undo the dependenceundo the dependence– DeDe-correlate your data, your model-correlate your data, your model

– Then proceed as with independent dataThen proceed as with independent data

• Problem is obtaining accurate estimates of Problem is obtaining accurate estimates of autocorrelationautocorrelation– Some sort of regularization is requiredSome sort of regularization is required

• Spatial smoothing of some sortSpatial smoothing of some sort

fMRI example

Basic fMRI ExampleBasic fMRI ExampleBasic fMRI ExampleBasic fMRI Example

• Time series at Time series at each voxel.each voxel.

• Time series at Time series at each voxel.each voxel.

one stimulus

two stimuli

fMRI time series modeling

A Linear ModelA Linear ModelA Linear ModelA Linear Model

Intensity

Tim

e = 1 2+ + erro

r

x1 x2

• ““Linear” in Linear” in parameters parameters 11

&& 22

• ““Linear” in Linear” in parameters parameters 11

&& 22

… … in matrix form.in matrix form.… … in matrix form.in matrix form.

XY

=

+YY X

N

1

N N

1 1p

p

N: Number of scans, p: Number of regressors

left right left right

subject 20

subject 41

left rightleftright

OSL fitting of subject 20 left amygdala

OSL fitting of subject 20 right amygdala

Generalized Least Squares (GLS) Estimation

HRF snake attacking snake crawling fish swimming

AFNI resultsubject 20right amygdala

HRF reconvolved with the initial stimuli

Black: HR based on OSLRed: HR based on GSL

In this particular example, GSL can get the dip OSL can not get.

Whitening by GSL correlation

Multiple Testing ProblemMultiple Testing ProblemMultiple Testing ProblemMultiple Testing Problem

• Inference on statistic imagesInference on statistic images– Fit GLM at each voxelFit GLM at each voxel

– Create statistic images of effectCreate statistic images of effect

• Which of 100,000 voxels are significant?Which of 100,000 voxels are significant? =0.05 =0.05 5,000 false positives! 5,000 false positives!

• Inference on statistic imagesInference on statistic images– Fit GLM at each voxelFit GLM at each voxel

– Create statistic images of effectCreate statistic images of effect

• Which of 100,000 voxels are significant?Which of 100,000 voxels are significant? =0.05 =0.05 5,000 false positives! 5,000 false positives!

t > 0.5 t > 1.5 t > 2.5 t > 3.5 t > 4.5 t > 5.5 t > 6.5

MCP Solutions:MCP Solutions:Measuring False PositivesMeasuring False Positives

MCP Solutions:MCP Solutions:Measuring False PositivesMeasuring False Positives

• Familywise Error Rate (FWER)Familywise Error Rate (FWER)– Familywise ErrorFamilywise Error

• Existence of one or more false positivesExistence of one or more false positives

– FWER is probability of familywise error FWER is probability of familywise error corrected corrected PP-value-value

• False Discovery Rate (FDR)False Discovery Rate (FDR)– R voxels declared active, V falsely soR voxels declared active, V falsely so

• Observed false discovery rate: V/RObserved false discovery rate: V/R

– FDR = E(V/R) FDR = E(V/R) Q-valueQ-value

– This is a relative measure.This is a relative measure.

• Familywise Error Rate (FWER)Familywise Error Rate (FWER)– Familywise ErrorFamilywise Error

• Existence of one or more false positivesExistence of one or more false positives

– FWER is probability of familywise error FWER is probability of familywise error corrected corrected PP-value-value

• False Discovery Rate (FDR)False Discovery Rate (FDR)– R voxels declared active, V falsely soR voxels declared active, V falsely so

• Observed false discovery rate: V/RObserved false discovery rate: V/R

– FDR = E(V/R) FDR = E(V/R) Q-valueQ-value

– This is a relative measure.This is a relative measure.

FWER MCP Solutions:FWER MCP Solutions:Random Field TheoryRandom Field Theory

FWER MCP Solutions:FWER MCP Solutions:Random Field TheoryRandom Field Theory

• Euler Characteristic Euler Characteristic uu

– Topological MeasureTopological Measure• #blobs - #holes#blobs - #holes

– At high thresholds,At high thresholds,just counts blobsjust counts blobs

– FWERFWER = P(Max voxel = P(Max voxel uu | | HHoo))

= P(One or more blobs | = P(One or more blobs | HHoo))

P( P(uu 1 | 1 | HHoo))

E( E(uu | | HHoo))

• Euler Characteristic Euler Characteristic uu

– Topological MeasureTopological Measure• #blobs - #holes#blobs - #holes

– At high thresholds,At high thresholds,just counts blobsjust counts blobs

– FWERFWER = P(Max voxel = P(Max voxel uu | | HHoo))

= P(One or more blobs | = P(One or more blobs | HHoo))

P( P(uu 1 | 1 | HHoo))

E( E(uu | | HHoo))

Random Field

Suprathreshold Sets

Threshold

Example – 2D Gaussian imagesExample – 2D Gaussian imagesExample – 2D Gaussian imagesExample – 2D Gaussian images

αα = R (4 ln 2) (2 = R (4 ln 2) (2ππ) ) -3/2-3/2 u exp (-u u exp (-u22/2)/2)

For R=100 and α=0.05RFT gives u=3.8

Using R=100 in a Bonferroni correction gives u=3.3

Friston et al. (1991) J. Cer. Bl. Fl. M.

DevelopmentsDevelopmentsDevelopmentsDevelopments

Friston et al. (1991) J. Cer. Bl. Fl. M. (Not EC Method)

2D Gaussian fields

3D Gaussian fields

3D t-fieldsWorsley et al. (1992) J. Cer. Bl. Fl. M.

Worsley et al. (1993) Quant. Brain. Func.

• General form for expected Euler characteristicGeneral form for expected Euler characteristic• 22, , FF, & , & tt fields fields •• restricted search regions restricted search regions

αα = = R Rd d (()) d d ((uu))

• General form for expected Euler characteristicGeneral form for expected Euler characteristic• 22, , FF, & , & tt fields fields •• restricted search regions restricted search regions

αα = = R Rd d (()) d d ((uu))

Unified TheoryUnified TheoryUnified TheoryUnified Theory

Rd (): RESEL count

R0() = () Euler characteristic of

R1() = resel diameter

R2() = resel surface area

R3() = resel volume

d (u): d-dimensional EC density –

E.g. Gaussian RF:

0(u) = 1- (u)

1(u) = (4 ln2)1/2 exp(-u2/2) / (2)

2(u) = (4 ln2) exp(-u2/2) / (2)3/2

3(u) = (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2)2

4(u) = (4 ln2)2 (u3 -3u) exp(-u2/2) / (2)5/2

Au

Worsley et al. (1996), HBM

Expected EC for a stationary Gaussian field

EC densityMinkowski functionals

Adler (1981) , Worsley (1996)

RFT AssumptionsRFT AssumptionsRFT AssumptionsRFT Assumptions

• Model fit & assumptionsModel fit & assumptions– valid distributional resultsvalid distributional results

• Multivariate normalityMultivariate normality– of of componentcomponent images images

• Covariance function of Covariance function of componentcomponent images must be images must be

- Stationary- Stationary (pre SPM99)(pre SPM99)

- - Can be nonstationaryCan be nonstationary

(SPM99 onwards)(SPM99 onwards)

- Twice differentiable- Twice differentiable

• Model fit & assumptionsModel fit & assumptions– valid distributional resultsvalid distributional results

• Multivariate normalityMultivariate normality– of of componentcomponent images images

• Covariance function of Covariance function of componentcomponent images must be images must be

- Stationary- Stationary (pre SPM99)(pre SPM99)

- - Can be nonstationaryCan be nonstationary

(SPM99 onwards)(SPM99 onwards)

- Twice differentiable- Twice differentiable

• SmoothnessSmoothness– smoothness » voxel sizesmoothness » voxel size

• lattice approximationlattice approximation

• smoothness estimationsmoothness estimation

– practicallypractically• FWHMFWHM 3 3 VoxDimVoxDim

– otherwiseotherwise• conservativeconservative

• SmoothnessSmoothness– smoothness » voxel sizesmoothness » voxel size

• lattice approximationlattice approximation

• smoothness estimationsmoothness estimation

– practicallypractically• FWHMFWHM 3 3 VoxDimVoxDim

– otherwiseotherwise• conservativeconservative

Random Field Theory Random Field Theory Random Field Theory Random Field Theory

• Closed form results for E(Closed form results for E(uu))– Z, tZ, t, , FF, Chi-Squared Continuous RFs, Chi-Squared Continuous RFs

• Results depend only on RESELsResults depend only on RESELs– RESolution ELementsRESolution ELements

– A volume element of size A volume element of size FWHMFWHMx x FWHMFWHMy y FWHMFWHMzz

– RESEL countRESEL count• Voxel-size-independent measure of volumeVoxel-size-independent measure of volume

• Inferences Inferences dodo notnot depend on stationarity depend on stationarity– Smoothness can varySmoothness can vary

(It is cluster size results that require stationarity)(It is cluster size results that require stationarity)

• Closed form results for E(Closed form results for E(uu))– Z, tZ, t, , FF, Chi-Squared Continuous RFs, Chi-Squared Continuous RFs

• Results depend only on RESELsResults depend only on RESELs– RESolution ELementsRESolution ELements

– A volume element of size A volume element of size FWHMFWHMx x FWHMFWHMy y FWHMFWHMzz

– RESEL countRESEL count• Voxel-size-independent measure of volumeVoxel-size-independent measure of volume

• Inferences Inferences dodo notnot depend on stationarity depend on stationarity– Smoothness can varySmoothness can vary

(It is cluster size results that require stationarity)(It is cluster size results that require stationarity)

Random Field Theory LimitationsRandom Field Theory LimitationsRandom Field Theory LimitationsRandom Field Theory Limitations

• Sufficient smoothnessSufficient smoothness– FWHM smoothness 3-4 times voxel sizeFWHM smoothness 3-4 times voxel size

• Smoothness estimationSmoothness estimation– Estimate is biased when images not so smooth Estimate is biased when images not so smooth

– Estimate is an estimateEstimate is an estimate—sd’s on corr. —sd’s on corr. p-valuesp-values

• Multivariate normalityMultivariate normality– Virtually impossible to checkVirtually impossible to check

• Several layers of approximationsSeveral layers of approximations– E.g., E.g., tt field results conservative for low df field results conservative for low df

• Sufficient smoothnessSufficient smoothness– FWHM smoothness 3-4 times voxel sizeFWHM smoothness 3-4 times voxel size

• Smoothness estimationSmoothness estimation– Estimate is biased when images not so smooth Estimate is biased when images not so smooth

– Estimate is an estimateEstimate is an estimate—sd’s on corr. —sd’s on corr. p-valuesp-values

• Multivariate normalityMultivariate normality– Virtually impossible to checkVirtually impossible to check

• Several layers of approximationsSeveral layers of approximations– E.g., E.g., tt field results conservative for low df field results conservative for low df

Lattice ImageData

Continuous Random Field

ze

ze

ye

ze

xe

ze

ye

ye

ye

xe

ze

xe

ye

xe

xe

var,cov,cov

,covvar,cov

,cov,covvar

Smoothness EstimationSmoothness EstimationSmoothness EstimationSmoothness Estimation

• Roughness Roughness ||||

• Point Response Function Point Response Function PRFPRF

• Roughness Roughness ||||

• Point Response Function Point Response Function PRFPRF

• Gaussian Gaussian PRFPRF

ffxx 0 0

ffyy00 0 ffzz

|||| = (4ln(2)) = (4ln(2))3/23/2 / (f / (fxx f fyy f fzz))

• RESEL COUNTRESEL COUNT

RR33(() = ) = (()) / (f / (fxx f fyy f fzz))

αα = R = R33(() (4ln(2))) (4ln(2))3/23/2 ( (u u 2 2 -1) exp(--1) exp(-u u 22/2) / (2/2) / (2))22

• Gaussian Gaussian PRFPRF

ffxx 0 0

ffyy00 0 ffzz

|||| = (4ln(2)) = (4ln(2))3/23/2 / (f / (fxx f fyy f fzz))

• RESEL COUNTRESEL COUNT

RR33(() = ) = (()) / (f / (fxx f fyy f fzz))

αα = R = R33(() (4ln(2))) (4ln(2))3/23/2 ( (u u 2 2 -1) exp(--1) exp(-u u 22/2) / (2/2) / (2))22

Approximate the peak of the Covariance function with a Gaussian

Cluster and Set-level InferenceCluster and Set-level InferenceCluster and Set-level InferenceCluster 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 the likelihood (under the null hypothesis) of getting n or more connected

components in the excursion set ie. a cluster containing at least n voxelscomponents in the excursion set ie. a cluster containing at least n voxels

CLUSTER-LEVEL INFERENCECLUSTER-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 voxelsclusters 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 the likelihood (under the null hypothesis) of getting n or more connected

components in the excursion set ie. a cluster containing at least n voxelscomponents in the excursion set ie. a cluster containing at least n voxels

CLUSTER-LEVEL INFERENCECLUSTER-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 voxelsclusters each having at least n voxels

Suprathreshold cluster testsSuprathreshold cluster testsSuprathreshold cluster testsSuprathreshold cluster tests

• Primary threshold Primary threshold uu– examine connected components examine connected components

of excursion setof excursion set

– Suprathreshold clustersSuprathreshold clusters

– Reject Reject HHWW for clusters of voxels for clusters of voxels WW of size of size SS > > ss

• Localisation Localisation (Strong control)(Strong control)

– at cluster levelat cluster level

– increased powerincreased power– esp. high resolutions esp. high resolutions ((ff MRI MRI))

• Thresholds, Thresholds, pp –values –values– Pr(Pr(SS

maxmax > > ss H H ) )

Nosko, Friston, (Worsley)Nosko, Friston, (Worsley)

– Poisson occurrence Poisson occurrence (Adler)(Adler)

– Assumme form for Pr(Assumme form for Pr(SS==ss||SS>0)>0)

• Primary threshold Primary threshold uu– examine connected components examine connected components

of excursion setof excursion set

– Suprathreshold clustersSuprathreshold clusters

– Reject Reject HHWW for clusters of voxels for clusters of voxels WW of size of size SS > > ss

• Localisation Localisation (Strong control)(Strong control)

– at cluster levelat cluster level

– increased powerincreased power– esp. high resolutions esp. high resolutions ((ff MRI MRI))

• Thresholds, Thresholds, pp –values –values– Pr(Pr(SS

maxmax > > ss H H ) )

Nosko, Friston, (Worsley)Nosko, Friston, (Worsley)

– Poisson occurrence Poisson occurrence (Adler)(Adler)

– Assumme form for Pr(Assumme form for Pr(SS==ss||SS>0)>0)

5mm FWHM

10mm FWHM

15mm FWHM

(2mm2 pixels)

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,Multiple Comparisons,& Random Field Theory& Random Field TheoryMultiple Comparisons,Multiple Comparisons,

& Random Field Theory& Random Field Theory

Ch5 Ch4

Controlling FWER: Permutation TestControlling FWER: Permutation TestControlling FWER: Permutation TestControlling FWER: Permutation Test

• Parametric methodsParametric methods– Assume distribution ofAssume distribution of

maxmax statistic under null statistic under nullhypothesishypothesis

• Nonparametric methodsNonparametric methods– Use Use datadata to find to find

distribution of distribution of maxmax statistic statisticunder null hypothesisunder null hypothesis

– Any max statistic!Any max statistic!

– Due to FisherDue to Fisher

• Parametric methodsParametric methods– Assume distribution ofAssume distribution of

maxmax statistic under null statistic under nullhypothesishypothesis

• Nonparametric methodsNonparametric methods– Use Use datadata to find to find

distribution of distribution of maxmax statistic statisticunder null hypothesisunder null hypothesis

– Any max statistic!Any max statistic!

– Due to FisherDue to Fisher

5%

Parametric Null Max Distribution

5%

Nonparametric Null Max Distribution

Measuring False PositivesMeasuring False PositivesFWER vs FDRFWER vs FDR

Measuring False PositivesMeasuring False PositivesFWER vs FDRFWER vs FDR

Signal

Signal+Noise

Noise

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% 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

Controlling FDR:Controlling FDR:Benjamini & Hochberg Benjamini & Hochberg

Controlling FDR:Controlling FDR:Benjamini & Hochberg Benjamini & Hochberg

• Select desired limit Select desired limit qq on E(FDR) on E(FDR)• Order p-values, Order p-values, pp(1)(1) pp(2)(2) ... ... pp((VV))

• Let Let rr be largest be largest ii such that such that

• Reject all hypotheses Reject all hypotheses corresponding tocorresponding to pp(1)(1), ... , , ... , pp((rr))..

• Select desired limit Select desired limit qq on E(FDR) on E(FDR)• Order p-values, Order p-values, pp(1)(1) pp(2)(2) ... ... pp((VV))

• Let Let rr be largest be largest ii such that such that

• Reject all hypotheses Reject all hypotheses corresponding tocorresponding to pp(1)(1), ... , , ... , pp((rr))..

p(i) i/V q

p(i)

i/V

i/V qp-

valu

e0 1

01

Benjamini & Hochberg:Benjamini & Hochberg:Varying Signal ExtentVarying Signal Extent

Benjamini & Hochberg:Benjamini & Hochberg:Varying Signal ExtentVarying Signal Extent

Signal Intensity 3.0 Signal Extent 25.0 Noise Smoothness 3.0

p = 0.019274 z = 2.07

7

FDR Software for SPMFDR Software for SPMFDR Software for SPMFDR Software for SPM

http://www.sph.umich.edu/~nichols/FDR

ReferencesReferencesReferencesReferences

• KM Petersson, TE Nichols, J-B Poline, and AP Holmes.KM Petersson, TE Nichols, J-B Poline, and AP Holmes.Statistical limitations in functional neuroimaging I. Non-inferential methods Statistical limitations in functional neuroimaging I. Non-inferential methods and statistical models. and statistical models. Statistical limitations in functional neuroimaging II. Signal detection and Statistical limitations in functional neuroimaging II. Signal detection and statistical inference. statistical inference. Philosophical Transactions of the Royal Society: Biological SciencesPhilosophical Transactions of the Royal Society: Biological Sciences , , 354:1239-1281, 1999.354:1239-1281, 1999.

• CR Genovese, N Lazar and TE Nichols.CR Genovese, N Lazar and TE Nichols.Thresholding of Statistical Maps in Functional Neuroimaging Using the Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. False Discovery Rate. NeuroImageNeuroImage, 15:870-878, 2002., 15:870-878, 2002.

• KM Petersson, TE Nichols, J-B Poline, and AP Holmes.KM Petersson, TE Nichols, J-B Poline, and AP Holmes.Statistical limitations in functional neuroimaging I. Non-inferential methods Statistical limitations in functional neuroimaging I. Non-inferential methods and statistical models. and statistical models. Statistical limitations in functional neuroimaging II. Signal detection and Statistical limitations in functional neuroimaging II. Signal detection and statistical inference. statistical inference. Philosophical Transactions of the Royal Society: Biological SciencesPhilosophical Transactions of the Royal Society: Biological Sciences , , 354:1239-1281, 1999.354:1239-1281, 1999.

• CR Genovese, N Lazar and TE Nichols.CR Genovese, N Lazar and TE Nichols.Thresholding of Statistical Maps in Functional Neuroimaging Using the Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. False Discovery Rate. NeuroImageNeuroImage, 15:870-878, 2002., 15:870-878, 2002.

http://www.sph.umich.edu/~nichols

Application: Surface data

Corrected P-value

Morphological descriptor

2D Euler characteristic density of t random field

Cortical surface area

Red: Tissue growthBlue: Tissue lossYellow: Structure displacement

Rejection regions.

Yellow = Hotelling’s T^2 field

Cortical thickness change t-map

Curvature change t-map