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Basics of fMRI Inference
Douglas N. Greve
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Overview• Inference• False Positives and False Negatives• Problem of Multiple Comparisons• Bonferroni Correction• Cluster Correction (voxel-wise threshold)• False Discovery Rate• Selection Bias
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Statistical Inference
Group Population(All members)Hundreds?Thousands?Billions?
Sample18 Subjects
• Can your conclusions be extended to data you have not seen?– Subjects, Time Points, Groups, Scanners
• Or are your results the product of a chance occurrence that is unlikely to be repeated?
• Generalizability, Repeatability,
Reproducibility, Predictability
• Uncertainty
• Beyond good Experimental Design
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Truth Table
Effect Is Not Present (Neg)
Effect Is Present (Pos)
Effect Is Not Present (Neg)
True Negative False Positive
Effect Is Present (Pos)
False Negative True Positive
Conclusion
Rea
lity
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Error Rate
Effect Is Not Present (Neg)
Effect Is
Present (Pos)
Effect Is Not Present (Neg)
True Negative
TNR=1-False Positive
FPREffect Is Present (Pos)
False Negative
FNR = True PositiveTPR = 1-(Power)
ConclusionR
eali
ty
False Positive Rate (FPR) – probability that you declare an effect to be present when there is no effectFalse Negative Rate (FNR) - probability that you declare no effect to be present when there is an effect
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How Do You Draw Conclusions?
Protocol: reduce all your data to one number (the “test statistic” T). • If T is greater than some threshold () then conclude that an effect is present (ie, a positive) • Otherwise conclude that an effect is not present (ie, a negative).
Every protocol has some FPR and some FNR, though it is not always easy to figure out!
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Noise Causes Uncertainty
Voxel 1
Voxel 2
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GLM Inference
T=8 T=1
OFFin N Var, Mean,,,
ONin N Var, Mean,,,
)2()1()1(
2
2
2
22
OFFOFFOFF
ONONON
OFFON
OFFOFFONON
OFFON
N
N
NNNN
T
ONOFF
2ON2
OFF
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Example Protocol• Collect data• Motion Correct• Smooth by 5mm FWHM• Extract Voxel 1 (throw away rest of data)• Compute Mean and StdDev of ON time points• Compute Mean and StdDev of OFF time points• Compute test statistic T• If T > 3.41, Conclude that the voxel is active
Test Statistic (T) is the t-ratioThreshold () is 3.41What is the FPR () and FNR () for this protocol?
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Example Protocol: False Positive Rate• “NULL” Distribution Student’s t-Distribution• p-value is area under curve to the right of T• For T = 3.4, FPR = p =.01• For T=8, FPR = p = 10-11
• For T=1, FPR = p = 0.1• Assumptions:
• Gaussian noise• Independent noise• Homoskedastic (equal variances)
• Violation of assumptions change FPR
2
)2()1()1(
2
22
OFFON
OFFON
OFFOFFONON
OFFON
NNDOF
NNNN
T
Student’s t Distribution FPR=area under curveto the right of line (p-value)
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Example Protocol: False Negative Rate
• Need to know what the effect size is• Previous data• Guess• Power Analysis• Grants require a power analysis!
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Trade Off of Error Rates
• Inverse relationship between error rates• As False Positives () are reduced, the False Negatives () increase• Increase sample size decreases , does not affect • Which Error is more important? Depends ..
• Science? FPR=.05ish, TNR<0.2• Pre-operative surgery?
FPR=.10
FPR=.01
FPR=10-7
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What conclusions to draw from this?
• Brain is activated?• Visual Cortex?• Auditory Cortex?• False Positive Rate?
Need a protocol!
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Possible Protocol• First Level Analysis• Compute t-ratio for each voxel • Compute p-value for each voxel• If any brain voxel has p < .01, declare a positive• Same as
• Test Statistic: T = max(Ti)• Threshold: =3.4
What is the False Positive Rate for this protocol?
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What does p<.01 mean?
• p<.01 means one expects 1% of voxels will be active purely by chance• Protocol gives a False Positive any time even a single voxel has p<.01• What is the probability that at least one voxel has p<.01?
Rand(0,1)100x10010,000 vox
p < 0.11000 vox
p < 0.01100 vox
p < 0.00110 vox
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The “Problem of Multiple Comparisons”
• Vox = voxel-wise threshold (p< Vox)• FWE = Protocol False Positive Rate (FWE = Family-wise Error)• N = Number of voxels (“Search Space”)
NVoxFWE )1(1
Vox FWE
0.00001 0.095
0.0001 0.632
0.001 1.000
0.01 1.000Vox =.10 Vox =.01 Vox =10-7
N = 10,000
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Bonferroni Correction
Compute Voxel-wise threshold needed to achieve a desired Family-wise FPR.
To achieve FWE = 0.01 with N = 10,000 voxelsNeed Vox = 0.000001 (10-6)
),(11 NfN FWEFWEN
FWEVox
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Search Space• Set of voxels over which positives are searched• Severity of correction increases with size of search space (regardless of method)• Reduce Search Space
• Reduce the area to a ROI (eg, superior temp gyrus) • Increase voxel size (cover same volume with fewer voxels)• Spatial Smoothing
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Spatial Smoothing
Full-Width/Half-max
• Spatially convolve image with Gaussian kernel.• Kernel sums to 1• Full-Width/Half-max: FWHM = /sqrt(log(256)) = standard deviation of the Gaussian
0 FWHM 5 FWHM 10 FWHM
2mm FWHM
10mm FWHM
5mm FWHM
Full Max
Half Max
Smoothing causes irreversible loss of information (resolution)
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Spatial SmoothingSmoothing causes irreversible loss of information (resolution), similar to increasing voxel size.
5mm
Smoothing
IncreasedVoxel Size
0mm 10mm
4mm1mm 8mm
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Resel• Pixel = picture element• Voxel = volume element• Resel = resolution element (depends on smoothing level) Resel = (FWHM)3 for volumesResel = (FWHM)2 for surfacesIf FWHM>Voxel Size, fewer Resels than Voxels.
Correct based on the number of Resels instead of number of voxels (math is more complicated, need Random Field Theory)
),,( FWHMNf FWEVox
NVoxN
FWEVox
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Bonferroni
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Clusters
• True signal tends to be clustered• False Positives tend to be randomly distributed in space• Cluster – set of spatially contiguous voxels that are above a given threshold.
Vox =.10 Vox =.01 Vox =10-7
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Cluster-wise Correction• Cluster – set of spatially contiguous voxels that are above a given threshold.• Protocol
• Perform 1st level analysis.• Threshold volume at Vox
• Find clusters.• If Cluster Size > Threshold (), Declare a Positive• Test Statistic: Cluster Size
• What is the FPR (FWE) for this protocol?
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Random Field Theory
FWE = f(Vox,N,FWHM,ClusterSize)
p=.05 p=.05
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Smoothing increases size of random clusters
FWHM 0 FWHM 2 FWHM 4 FWHM 6
Z
Z>2.3p<.01
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Cluster Images
Sig MappVox < .001
Cluster MappCluster < .05
Some small clusters do not “survive”
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Cluster Table
R L
RadiologicalOrientation
Cluster
MNI305
X Y Z
Size
(mm3)
Cluster
p-value
Atlas
Location
1 40 -67 -11 41368 ~0 Right Lateral
Occipital
2 -40 -85 -13 51184 ~0 Left Lateral
Occipital
3 -6 17 45 2784 .00026 Left Superior Frontal
4 -50 7 23 3768 .00002 Left Precentral
ROI Atlas
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Cluster Data Extraction
• Spatial average over cluster of each subject’s contrast• Can correlate with other measures (age, test score, etc)• Be careful of “Selection Bias” (“Voodoo Correlations”)
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Cluster Correction Summary
• Cluster – set of supra-threshold voxels (size)• Critical Size Threshold given by Random Field Theory
• Search Space• Voxel-wise threshold (arbitrary)• FWHM (smoothing level)• Assumptions on each
• Loose small clusters (False Negatives)
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False Discovery Rate (FDR)
• Given the voxel-wise threshold, know expected number of False Positives• If there are more Positives than this, then some of them must be True Positives
p < 0.11000 vox
p < 0.01100 vox
p < 0.00110 vox
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False Discovery Rate (FDR)
PositivesNumber Total
Positives False ofNumber
Positives True ofNumber Positives False ofNumber
Positives False ofNumber
FDR
• Number of False Positives = N*Vox
• Total Number of Positives = Count from image• Vox = f(FDR,N,Data)
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False Discovery Rate (FDR)
• FDR = .05 means that 5% of Positives are False Positives• Which 5%, no one knows• How to interpret?
FDR = .05Vox = .0070
FDR = .01Vox = .0070
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False Discovery Rate (FDR)
• FDR = .05 means that 5% of Positives are False Positives• Which 5%, no one knows• How to interpret?
FDR = .05Vox = .0070
FDR = .01Vox = .0070
Would you change your opinion of this blob if 50 of the voxels were False Positives?
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False Discovery Rate (FDR)
• FDR = .05 means that 5% of Positives are False Positives• Which 5%, no one knows• How to interpret?
FDR = .05Vox = .0070
FDR = .01Vox = .0070
Would you change your opinion of this blob if 50 of the voxels were False Positives?
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False Discovery Rate Summary
• False Discoveries• FDR does not control FPR (False Positive Rate)• Careful when interpreting• Voxel-wise threshold is Data Dependent
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Summary• Can your conclusions be extended to data you have not seen?• Truth Table: False Positives () and False Negatives ()• Protocol – describes how you will draw conclusions • Problem of Multiple Comparisons (Family-wise Error)• Search Space, Search Space reduction
• Larger voxels (less resolution)• Smoothing (Resels)
• Bonferroni Correction• Cluster Correction (voxel-wise threshold)• False Discovery Rate• Selection Bias
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