Basics of fMRI Inference

1. Basics of fMRI Inference Douglas N. Greve

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

3. Group Population (All members) Hundreds? Thousands? Billions? Sample 18 Subjects Statistical Inference • 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

4. Truth Table Conclusion Reality

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

6. How Do You Draw Conclusions? • Protocol: reduce all your data to one number (the “test statistic” T). • If T is greater than some threshold (q) 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!

7. Noise Causes Uncertainty Voxel 1 Voxel 2

8. GLM Inference T=8 T=1

9. 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-ratio • Threshold (q) is 3.41 • What is the FPR (a) and FNR (b) for this protocol?

10. FPR=area under curve to the right of line (p-value) Student’s t Distribution 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

11. Example Protocol: False Negative Rate • Need to know what the effect size is • Previous data • Guess • Power Analysis • Grants require a power analysis!

12. 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, does not affect a • Which Error is more important? Depends .. • Science? FPR=.05ish, TNR<0.2 • Pre-operative surgery?

13. What conclusions to draw from this? • Brain is activated? • Visual Cortex? • Auditory Cortex? • False Positive Rate? • Need a protocol!

14. 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: q=3.4 • What is the False Positive Rate for this protocol?

15. What does p<.01 mean? Rand(0,1) 100x100 10,000 vox p < 0.1 1000 vox p < 0.01 100 vox p < 0.001 10 vox • 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?

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

17. 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)

18. 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

19. 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

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

21. 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

22. 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.

23. Cluster-wise Correction • Cluster – set of spatially contiguous voxels that are above a given threshold. • Protocol • Perform 1st level analysis. • Threshold volume at aVox • Find clusters. • If Cluster Size > Threshold (q), Declare a Positive • Test Statistic: Cluster Size • What is the FPR (aFWE) for this protocol?

24. Random Field Theory aFWE = f(aVox,N,FWHM,ClusterSize) p=.05 p=.05

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

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

27. Cluster Table R L Radiological Orientation ROI Atlas

28. 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”)

29. 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)

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 = .0070 FDR = .01 aVox = .0070

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 = .0070

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 = .0070

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 • Can your conclusions be extended to data you have not seen? • Truth Table: False Positives (a) and False Negatives (b) • 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