Multiple Comparison Multiple Comparison Correction in SPMs Correction in SPMs Will Penny Will Penny SPM short course, Zurich, Feb 2008 SPM short course, Zurich, Feb 2008
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Multiple Comparison Multiple Comparison Correction in SPMsCorrection in SPMs
Multiple Comparison Multiple Comparison Correction in SPMsCorrection in SPMs
Will PennyWill PennySPM short course, Zurich, Feb 2008SPM short course, Zurich, Feb 2008
Will PennyWill PennySPM short course, Zurich, Feb 2008SPM short course, Zurich, Feb 2008
realignment &motion
correction
smoothing
normalisation
General Linear Modelmodel fittingstatistic image
corrected p-values
image data parameterestimatesdesign
matrix
anatomicalreference
kernel
StatisticalParametric Map
Random Field Theory
Inference at a single voxelInference at a single voxelInference at a single voxelInference at a single voxel
= p(t>u|H)
NULL hypothesis, H: activation is zero
u=2t-distribution
We can choose u to ensurea voxel-wise significance level of
his is called an ‘uncorrected’ p-value, forreasons we’ll see later.
We can then plot a map of above thresholdvoxels.
Inference for ImagesInference for ImagesInference for ImagesInference for Images
Signal
Signal+Noise
Noise
11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5%
Use of ‘uncorrected’ p-value, =0.1
Percentage of Null Pixels that are False Positives
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 HypothesisFamily-wise Null HypothesisFamily-wise Null HypothesisFamily-wise Null Hypothesis
FAMILY-WISE NULL HYPOTHESIS:Activation is zero everywhere
If we reject a voxel null hypothesisat any voxel, we reject the family-wiseNull hypothesis
A FP anywhere in the imagegives 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 correctionThe Bonferroni correctionThe Bonferroni correctionThe Bonferroni correction
The Family-Wise Error rate (FWE), The Family-Wise Error rate (FWE), ,, for a family of N a family of N independentindependent
voxels isvoxels is
αα = Nv = Nv
where v is the voxel-wise error rate. Therefore, to ensure a particular FWE set
v = α / N
BUT ...
The Bonferroni correctionThe Bonferroni correctionThe Bonferroni correctionThe Bonferroni correction
Independent Voxels Spatially Correlated Voxels
Bonferroni is too conservative for brain images
Random Field TheoryRandom Field TheoryRandom Field TheoryRandom Field Theory
• Consider a statistic image as a discretisation of a Consider a statistic image as a discretisation of a continuous underlying random fieldcontinuous 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 Consider a statistic image as a discretisation of a continuous underlying random fieldcontinuous underlying random field
• Use results from continuous random field theoryUse results from continuous random field theory
Discretisation
Euler Characteristic (EC)Euler Characteristic (EC)Euler Characteristic (EC)Euler Characteristic (EC)
Topological measureTopological measure– threshold an image at threshold an image at uu
- ECEC == # blobs # blobs
- at high u:at high u:
Prob blob = avg (EC)Prob blob = avg (EC)
SoSo
FWE, FWE, = avg (EC) = avg (EC)
Topological measureTopological measure– threshold an image at threshold an image at uu
- ECEC == # blobs # blobs
- at high u:at high u:
Prob blob = avg (EC)Prob blob = avg (EC)
SoSo
FWE, FWE, = avg (EC) = avg (EC)
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)
Voxel-wise threshold, u
Number of Resolution Elements (RESELS), R
N=100x100 voxels, Smoothness FWHM=10, gives R=10x10=100
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
Estimated component fieldsEstimated component fieldsEstimated component fieldsEstimated component fields
data matrix
des
ign
mat
rix
parameters errors+ ?= ?voxelsvoxels
scansscans
estimate
^
residuals
estimatedcomponent
fields
parameterestimates
estimated variance
=
Each row isan estimatedcomponent field
Applied SmoothingApplied SmoothingApplied SmoothingApplied Smoothing
SmoothnessSmoothnesssmoothness » voxel sizesmoothness » voxel size
practicallypracticallyFWHMFWHM 3 3 VoxDimVoxDim
Typical applied smoothing:Typical applied smoothing:
Single Subj fMRI: 6mmSingle Subj fMRI: 6mm
PET: 12mmPET: 12mm
Multi Subj fMRI: 8-12mmMulti Subj fMRI: 8-12mm
PET: 16mm PET: 16mm
SmoothnessSmoothnesssmoothness » voxel sizesmoothness » voxel size
practicallypracticallyFWHMFWHM 3 3 VoxDimVoxDim
Typical applied smoothing:Typical applied smoothing:
Single Subj fMRI: 6mmSingle Subj fMRI: 6mm
PET: 12mmPET: 12mm
Multi Subj fMRI: 8-12mmMulti Subj fMRI: 8-12mm
PET: 16mm PET: 16mm
SPM results ISPM results ISPM results ISPM results I
ActivationsSignificant atCluster levelBut not atVoxel Level
SPM results IISPM results IISPM results IISPM results II
Activations Significant atVoxel andCluster level
SPM results...SPM results...SPM results...SPM results...
False Discovery RateFalse Discovery RateFalse Discovery RateFalse Discovery Rate
H True (o) TN=7 FP=3
H False (x) FN=0 TP=10
Don’tReject
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-scoresfrom regionsthat truly do and do not activateFDR = FP/(# Reject)
= FP/(# H True)
False Discovery RateFalse Discovery RateFalse Discovery RateFalse Discovery Rate
H True (o) TN=9 FP=1
H False (x) FN=3 TP=7
Don’tReject
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-scoresfrom regionsthat truly do and do not activate
FDR=1/8=13%=1/10=10%
At u2
FDR = FP/(# Reject)
= FP/(# H True)
False Discovery RateFalse Discovery RateFalse Discovery RateFalse Discovery Rate
Signal
Signal+Noise
Noise
FWE
Control of Familywise Error Rate at 10%
Occurrence of Familywise Error
6.7% 10.4% 14.9% 9.3% 16.2% 13.8% 14.0% 10.5% 12.2% 8.7%
Control of False Discovery Rate at 10%
Percentage of Activated Pixels that are False Positives
SummarySummarySummarySummary
• 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
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