Basics of fMRI Inference

Douglas N. Greve

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

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)

Spatial SmoothingSmoothing causes irreversible loss of information (resolution), similar to increasing voxel size.

5mm

Smoothing

IncreasedVoxel Size

0mm 10mm

4mm1mm 8mm

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

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

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?

Random Field Theory

FWE = f(Vox,N,FWHM,ClusterSize)

p=.05 p=.05

Smoothing increases size of random clusters

FWHM 0 FWHM 2 FWHM 4 FWHM 6

Z

Z>2.3p<.01

Cluster Images

Sig MappVox < .001

Cluster MappCluster < .05

Some small clusters do not “survive”

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

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”)

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)

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

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)

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

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?

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?

False Discovery Rate Summary

• False Discoveries• FDR does not control FPR (False Positive Rate)• Careful when interpreting• Voxel-wise threshold is Data Dependent

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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