Statistical Inference and Random Field Theory Will Penny SPM short course, London, May 2003 Will Penny SPM short course, London, May 2003 M.Brett et al. Introduction to Random Field Theory, To appear in HBF, 2 nd Edition.

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

OverviewOverview 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results +FDR ? 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results +FDR ?

OverviewOverview 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results

Inference at a single voxel  = p(t>u|H) NULL hypothesis, H: activation is zero u=2 t-distribution p-value: probability of getting a value of t at least as extreme as u. If  is small we reject the null hypothesis.

Sensitivity and Specificity H True (o)TNFP H False (x)FNTP Don’t Reject ACTION TRUTH o o o o o o o x x x o o x x x o x x x x u1u2 Sens=10/10=100% Spec=7/10=70% At u1 Eg. t-scores from regions that truly do and do not activate Sens=7/10=70% Spec=9/10=90% At u2 Sensitivity = TP/(TP+FN)=  Specificity = TN/(TN+FP)= 1 -  FP = Type I error or ‘error’ FN = Type II error  = p-value/FP rate/error rate/significance level  = power

Inference at a single voxel  = p(t>u|H) NULL hypothesis, H: activation is zero u=2 t-distribution We can choose u to ensure a voxel-wise significance level of   his is called an ‘uncorrected’ p-value, for reasons we’ll see later. We can then plot a map of above threshold voxels.

Inference for Images Signal+Noise Noise

Using an ‘uncorrected’ p-value of 0.1 will lead us to conclude on average that 10% of voxels are active when they are not. This is clearly undesirable. To correct for this we can define a null hypothesis for images of statistics.

Family of hypothesesFamily of hypotheses –H k k   = {1,…,K} –H  = H 1  H 2 …  H k  H K Family of hypothesesFamily of hypotheses –H k k   = {1,…,K} –H  = H 1  H 2 …  H k  H K Family-wise Null Hypothesis FAMILY-WISE NULL HYPOTHESIS: Activation is zero everywhere If we reject a voxel null hypothesis at any voxel, we reject the family-wise Null hypothesis A FP anywhere gives a Family Wise Error (FWE) Family-wise error rate = ‘corrected’ p-value

Use of ‘uncorrected’ p-value,  =0.1 FWE Use of ‘corrected’ p-value,  =0.1

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= ! A correction for multiple comparisons

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

OverviewOverview 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results

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

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 )]

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

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.

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.

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

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

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

Resel Counts for Brain Structures FWHM=20mm (1) Threshold depends on Search Volume (2) Surface area makes a large contribution

OverviewOverview 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results

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

Component fields  =  +YX component fields data matrix design matrix parameters +=  ? voxels scans errors ? variance 

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

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

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 differentiableSmoothness smoothness » voxel size lattice approximation smoothness estimation practically FWHM  3  VoxDim otherwiseconservative Typical applied smoothing: Single Subj fMRI: 6mm PET: 12mm PET: 12mm Multi Subj fMRI: 8-12mm Multi Subj fMRI: 8-12mm PET: 16mm PET: 16mm

OverviewOverview 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results

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 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 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 SET-LEVEL INFERENCE SET-LEVEL INFERENCE Weak vs Strong control over FWE

Levels of inference set-level P(c  3 | n  12, u  3.09) = cluster-level P(c  1 | n  82, t  3.09) = (corrected) n=82 n=32 n=1 2 voxel-level P(c  1 | n > 0, t  4.37) = (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

OverviewOverview 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results 1.Terminology 2.Theory 3.Imaging Data 4.Levels of Inference 5. SPM Results

SPM99 results I Activations Significant at Cluster level But not at Voxel Level

SPM99 results II Activations Significant at Voxel and Cluster level

SPM results...

False Discovery Rate H True (o)TNFP H False (x)FNTP Don’t Reject ACTION TRUTH o o o o o o o x x x o o x x x o x x x x u1u2 FDR=3/13=23%  =3/10=30% At u1 Eg. t-scores from regions that truly do and do not activate FDR=1/8=13%  =1/10=10% At u2 FDR = FP/(FP+TP)  = FP/(FP+TN)

False Discovery Rate Illustration: Signal+Noise Noise

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.