Multiple Comparison Correction in SPMs Will Penny SPM short course, Zurich, Feb 2008 Will Penny SPM short course, Zurich, Feb 2008

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

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-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 in the image gives a Family Wise Error (FWE) Family-Wise Error (FWE) rate = ‘corrected’ p-value

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

The Bonferroni correction The Family-Wise Error rate (FWE), ,a family of N independent The Family-Wise Error rate (FWE), , for a family of N independent voxels is α = Nv α = Nv where v is the voxel-wise error rate. Therefore, to ensure a particular FWE set v = α / N BUT...

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

Random Field Theory Consider a statistic image as a discretisation of a continuous underlying random fieldConsider a statistic image as a discretisation of a continuous underlying random field Use results from continuous random field theoryUse results from continuous random field theory Consider a statistic image as a discretisation of a continuous underlying random fieldConsider a statistic image as a discretisation of a continuous underlying random field Use results from continuous random field theoryUse results from continuous random field theory Discretisation

Euler Characteristic (EC) Topological measure –threshold an image at u -EC = # blobs -at high u: Prob blob = avg (EC) So FWE,  = avg (EC) Topological measure –threshold an image at u -EC = # blobs -at high u: Prob blob = avg (EC) So FWE,  = avg (EC)

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

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

Applied Smoothing Smoothness smoothness » voxel size practically FWHM  3  VoxDim 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

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

SPM results II Activations Significant at Voxel and Cluster level

SPM results...

False Discovery Rate H True (o)TN=7FP=3 H False (x)FN=0TP=10 Don’t Reject ACTION TRUTH u1 FDR=3/13=23%  =3/10=30% At u1 o o o o o o o x x x o o x x x o x x x x Eg. t-scores from regions that truly do and do not activate FDR = FP/(# Reject)  = FP/(# H True)

False Discovery Rate H True (o)TN=9FP=1 H False (x)FN=3TP=7 Don’t Reject ACTION TRUTH u2 o o o o o o o x x x o o x x x o x x x x Eg. t-scores from regions that truly do and do not activate FDR=1/8=13%  =1/10=10% At u2 FDR = FP/(# Reject)  = FP/(# H True)

False Discovery Rate Signal+Noise Noise

SummarySummary We should not use uncorrected p-valuesWe should not use uncorrected p-values We can use Random Field Theory (RFT) to ‘correct’ p-valuesWe can use Random Field Theory (RFT) to ‘correct’ p-values RFT requires FWHM > 3 voxelsRFT requires FWHM > 3 voxels We only need to correct for the volume of interestWe only need to correct for the volume of interest Cluster-level inferenceCluster-level inference False Discovery Rate is a viable alternativeFalse Discovery Rate is a viable alternative We should not use uncorrected p-valuesWe should not use uncorrected p-values We can use Random Field Theory (RFT) to ‘correct’ p-valuesWe can use Random Field Theory (RFT) to ‘correct’ p-values RFT requires FWHM > 3 voxelsRFT requires FWHM > 3 voxels We only need to correct for the volume of interestWe only need to correct for the volume of interest Cluster-level inferenceCluster-level inference False Discovery Rate is a viable alternativeFalse Discovery Rate is a viable alternative