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Basics of fMRI Inference

Learn about the basics of fMRI analysis, error rates, multiple comparisons problem, and cluster correction in neuroimaging studies. Explore trade-offs between error rates and the importance of statistical corrections to ensure accurate results.

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Basics of fMRI Inference

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  1. Basics of fMRI Inference Douglas N. Greve

  2. fMRI Analysis Over-review Subject 1 Preprocessing MC, STC, B0 Smoothing Normalization Preprocessing MC, STC, B0 Smoothing Normalization Preprocessing MC, STC, B0 Smoothing Normalization Preprocessing MC, STC, B0 Smoothing Normalization First Level GLM Analysis X X X X X C C C C C Raw Data Subject 2 First Level GLM Analysis Raw Data Higher Level GLM Subject 3 First Level GLM Analysis Raw Data Subject 4 First Level GLM Analysis Raw Data Data Reduction: 1 value per subject per voxel Data Reduction: 1 value per voxel Need one more reduction to one number: “Thumbs up” or “Thumbs down”

  3. Overview • False Positives and False Negatives • Problem of Multiple Comparisons • Bonferroni Correction • Cluster Correction (voxel-wise threshold) • False Discovery Rate • Selection Bias

  4. Truth Table Conclusion Reality

  5. Error Rate Conclusion Reality False Positive Rate (FPR,a) – probability that you declare an effect to be present when there is no effect False Negative Rate (FNR,b) - probability that you declare no effect to be present when there is an effect

  6. Noise Causes Uncertainty Voxel 1 Voxel 2

  7. GLM Inference T=8 T=1

  8. FPR=area under curve to the right of line (p-value) Student’s t Distribution False Positive Rate • “NULL” Distribution Student’s t-Distribution • Noise Assumptions: • Gaussian noise • Independent noise • Homoskedastic (equal variances) • 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 • Violation of assumptions change FPR T

  9. What does False Positive Rate Mean? FPR=.10

  10. False Negative Rate • Need to know what the effect size is • How big will the signal be? • How big will the noise be? • Previous data • Guess

  11. FPR=.10 FPR=.01 FPR=10-7 Trade Off of Error Rates • Inverse relationship between error rates • As False Positives (a) are reduced, • the False Negatives (b) increase • Increase sample size decreases b • Which Error is more important? Depends .. • Science? FPR=.05ish, FNR(b)<0.2, TPR(1-b)>0.8 • Pre-operative surgery?

  12. Power Analysis • Given any 3 of the following, you can compute the 4th: • Desired False Positive Rate (a, usually .05) • Desired False Negative Rate (b, usually .20) • Number of subjects • Effect size, ie, ratio of • Signal (eg, Angry-vs-Neutral, Schizophrenia-vs-Normal) • Noise • Obtained from previous data or guess Grants require a power analysis!

  13. Voxel-wise vs “Family-wise” Error Rate Rand(0,1) 100x100 10,000 vox p < 0.1 1000 vox p < 0.01 100 vox p < 0.001 10 vox p < 0.1 • A voxel-wise p<.01 means one expects 1% of voxels will be active purely by chance • What if you say that if even a single voxel has p<.01, you declare a “thumbs up”? • What is the probability that at least one voxel has p<.01?

  14. The “Problem of Multiple Comparisons” N = 10,000 • aVox = voxel-wise threshold (p< aVox) • aFWE = “Thumbs up” False Positive Rate • (FWE = Family-wise Error) • N = Number of voxels (“Search Space”) aVox =.10 aVox =.01 aVox =10-7

  15. Bonferroni Correction Compute Voxel-wise threshold needed to achieve a desired Family-wise FPR. To achieve aFWE = 0.01 with N = 10,000 voxels Need aVox = 0.000001 (10-6)

  16. 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

  17. Spatial Smoothing • Spatially convolve image with Gaussian kernel. • Kernel sums to 1 • Full-Width/Half-max:FWHM = s/sqrt(log(256)) • s = standard deviation of the Gaussian Full-Width/Half-max 0 FWHM 5 FWHM 10 FWHM Full Max 2mm FWHM Half Max 5mm FWHM Smoothing causes irreversible loss of information/resolution 10mm FWHM

  18. Spatial Smoothing 0mm 5mm 10mm Smoothing 1mm 4mm 8mm Increased Voxel Size Smoothing causes irreversible loss of information (resolution), similar to increasing voxel size.

  19. Resel • Pixel = picture element • Voxel = volume element • Resel = resolution element (depends on smoothing level) • Resel = (FWHM)3 for volumes • Resel = (FWHM)2 for surfaces • If 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) Bonferroni

  20. Clusters aVox =.10 aVox =.01 aVox =10-7 • 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.

  21. Smoothing increases size of random clusters FWHM 0 FWHM 2 FWHM 4 FWHM 6 Z Z>2.3 p<.01

  22. Gaussian Random Field Theory aFWE = f(aVox,N,FWHM,ClusterSize) • aVox Voxel-wise, Cluster-forming Threshold • N – Search space. • FWHM – Smoothing level • ClusterSize – size of cluster to be tested • aFWE – Cluster p-value

  23. Cluster Images Sig Map pVox < .001 Cluster Map pCluster < .05 Some small clusters do not “survive”

  24. Cluster Table R L Radiological Orientation ROI Atlas

  25. 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)

  26. Cluster Data Extraction • Spatial average over cluster of each subject’s contrast • Can correlate with other measures (age, test score, etc)

  27. Selection Bias: Cluster Data Extraction • Voodoo Correlations: Running a “circular” test on extracted data • Eg, cluster represents voxel-wise test on AD-vs-Normal • Then you cannot perform an AD-vs-Normal test on the extracted data • If you do, then • p-values will be much too significant and will not reflect actual false positive rate • Correlation coefficients will be much too high • Subtle and easy to do Vul, Edward, et al. "Puzzlingly high correlations in fMRI studies of emotion, personality, and social cognition." Perspectives on psychological science 4.3 (2009)

  28. bG1 bG2 1 1 1 0 0 0 0 0 1 1 = Permutation: Recall Two Group GLM Analysis • Does Group 1 differ from Group 2? • C = [1 -1], Contrast = C*b = bG1-bG2 • Compute T from t-test • t-test assumes: Gaussian, independent, homoscedastic • If not, then p-values are not accurate

  29. Permutation bG1 bG2 1 1 1 0 0 0 0 0 1 1 = • Permute rows of design matrix • Run analysis • Compute simulation test statistic Ts • Go back to step 1 • Repeat a large (~10k) times, get 10k values of Ts • Analyze your true data • Compute test statistic T • aFWE <= How often T>Ts Permutation: under the NULL, labelings in design matrix are irrelevant

  30. False Discovery Rate (FDR) p < 0.1 1000 vox p < 0.01 100 vox p < 0.001 10 vox • 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

  31. False Discovery Rate (FDR) • Number of False Positives = N*aVox • Total Number of Positives = Count from image • aVox = f(FDR,N,Data)

  32. False Discovery Rate (FDR) • FDR = .05 means that 5% of Positives are False Positives • Which 5%, no one knows • How to interpret? FDR = .05 aVox = .00700 FDR = .01 aVox = .00080

  33. False Discovery Rate (FDR) • FDR = .05 means that 5% of Positives are False Positives • Which 5%, no one knows • How to interpret? Would you change your opinion of this blob if 50 of the voxels were False Positives? FDR = .05 aVox = .0070 FDR = .01 aVox = .00080

  34. False Discovery Rate (FDR) • FDR = .05 means that 5% of Positives are False Positives • Which 5%, no one knows • How to interpret? Would you change your opinion of this blob if 50 of the voxels were False Positives? FDR = .05 aVox = .0070 FDR = .01 aVox = .00080

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

  36. Summary • Final data reduction: thumbs up or thumbs down • Truth Table: False Positives (a) and False Negatives (b) • Trade-off in Error Rates • 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) • Permutation (combine with cluster-wise) • False Discovery Rate (FDR) • Selection Bias – VooDoo Correlations

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