1 / 47

Thresholding and multiple comparisons

Thresholding and multiple comparisons. Information for making inferences on activation. Where? Signal location Local maximum – no inference in SPM Could extract peak coordinates and test (e.g., Woods lab, Ploghaus, 1999) How strong? Signal magnitude

edmund
Télécharger la présentation

Thresholding and multiple comparisons

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Thresholding and multiple comparisons

  2. Information for making inferences on activation • Where? Signal location • Local maximum – no inference in SPM • Could extract peak coordinates and test (e.g., Woods lab, Ploghaus, 1999) • How strong? Signal magnitude • Local contrast intensity – Main thing tested in SPM • How large? Spatial extent • Cluster volume – Can get p-values from SPM • Sensitive to blob-defining-threshold • When? Signal timing • No inference in SPM; but see Aguirre 1998; Bellgowan 2003; Miezin et al. 2000, Lindquist & Wager, 2007

  3. Three “levels” of inference • Voxel-level • Most spatially specific, least sensitive • Cluster-level • Less spatially specific, more sensitive • Set-level • No spatial specificity (no locatization) • Can be most sensitive

  4. Voxel-level Inference • Retain voxels above α-level threshold uα • Gives best spatial specificity • The null hyp. at a single voxel can be rejected uα space Significant Voxels No significant Voxels

  5. Cluster-level Inference • Two step-process • Define clusters by arbitrary threshold uclus • Retain clusters larger than α-level threshold kα uclus space Cluster not significant Cluster significant kα kα

  6. Cluster-level Inference • Typically better sensitivity • Worse spatial specificity • The null hyp. of entire cluster is rejected • Only means that one or more voxels in cluster active uclus space Cluster not significant Cluster significant kα kα

  7. Multiple comparisons…

  8. uα α Null Distribution of T t P-val Null Distribution of T Hypothesis Testing • Null Hypothesis H0 • Test statistic T • t observed realization of T • α level • Acceptable false positive rate • Level α = P( T>uα | H0) • Threshold uα controls false positive rate at level α • P-value • Assessment of t assuming H0 • P( T > t | H0 ) • Prob. of obtaining stat. as largeor larger in a new experiment • P(Data|Null) not P(Null|Data)

  9. Multiple Comparisons Problem • Which of 100,000 voxels are significant? • Expected false positives for independent tests: • αNtests • α=0.05 : 5,000 false positive voxels • Which of(e.g.)100 clusters are significant? • α=0.05 : 5 false positives clusters • Solution: determine a threshold that controls the false positive rate (α) per map (across all voxels) rather than per voxel

  10. Why correct? • Without multiple comparisons correction, false positive rate in whole-brain analysis is high. • Using typical arbitrary height and extent threshold is too liberal! (e.g., p < .001 and 10 voxels; Forman et al., 1995). • This is because data is spatially smooth, and false positives tend to be “blobs” of many voxels: =0.10 =0.001 =0.01 True signal inside red square, no signal outside. White is “activation” From Wager, Lindquist, & Hernandez, 2009. “Essentials of functional neuroimaging.” In: Handbook of Neuroscience for the Behavioral Sciences Note: All images smoothed with FWHM=12mm

  11. MCP Solutions:Measuring False Positives • Familywise Error Rate (FWER) • False Discovery Rate (FDR)

  12. False Discovery Rate (FDR): Executive summary FDR = Expected proportion of reported positives that are false positives • (We don’t know what the actual number of false positives is, just the expected number.) • Easy to calculate based on p-value map (Benjamini and Hochberg, 1995) • Always more liberal than FWER control • Adaptive threshold: somewhere between uncorrected and FWER, depending on signal • Can claim that most results are likely to be true positives, but not which ones are false

  13. Family-wise Error Rate vs. False Discovery RateIllustration: Noise Signal Signal+Noise

  14. 11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5% 6.7% 10.5% 12.2% 8.7% 10.4% 14.9% 9.3% 16.2% 13.8% 14.0% Control of Per Comparison Rate at 10% Percentage of Null Pixels that are False Positives Control of Familywise Error Rate at 10% FWE Occurrence of Familywise Error Control of False Discovery Rate at 10% Percentage of Activated Pixels that are False Positives

  15. Controlling FWE withBonferroni…

  16. FWE MCP Solutions: Bonferroni • For a statistic image T... • Tiith voxel of statistic image T, where i = 1…V • ...use α = αc/V • αc desired FWER level (e.g. 0.05) • α new alpha level to achieve αc corrected • V number of voxels • Example: for p < .05 and V = 1000 tests, use p < .05/1000 = .00005. • Assumes independent tests (voxels) – but voxels are generally positively correlated (images are smooth)! Conservative under correlation Independent: V tests Some dep.: ? tests Total dep.: 1 test Note: most of the time, this is better: αc = 1 - (1-α)^V, α = 1 – (1-αc) ^ (1/V) See Shaffer, Ann. Rev. Psych. 1995

  17. Controlling FWE withRandom field theory…

  18. α FWER MCP Solutions: Controlling FWER w/ Max • FWER is the probability of finding any false positive result in the entire brain map • If any voxel is a false positive, the voxel with the max statistic value (lowest p-value) will be a false positive • Can assess the probability that the max stat value under the null hypothesis exceeds the threshold • FWER is the threshold that controls the probability of the max exceeding that threshold at α Distribution of maxima uα Threshold such that the max only exceeds it α% of the time How to get the distribution of maxima? GRF!

  19. Consider statistic image as lattice representation of a continuous random field Use results from continuous random field theory SPM approach:Random fields… lattice representation

  20. FWER MCP Solutions:Random Field Theory • Euler Characteristic χu • Topological Measure • #blobs - #holes • At high thresholds,just counts blobs • FWER = P(Max voxel χu | Ho) = P(One or more blobs | Ho)= P(χu > 1 | Ho)= E(χu| Ho) Threshold Random Field IF No holes IF Never more than 1 blob How high does the threshold need to be? ~p<.001? Suprathreshold Sets

  21. Random Field Intuition • Corrected P-value for voxel value t depends on • Volume of search area • Roughness of data in search area • The t-value in any given voxel tested • Statistic value t increases • Pc decreases (but only for large t) • Search volume increases • Pc increases (more severe MCP) • Roughness increases (Smoothness decreases) • Pc increases (more severe MCP)

  22. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 Random Field TheorySmoothness Parameterization • RESELS = Resolution Elements • 1 RESEL = FWHMx x FWHMy x FWHMz • Volume of search region in units of smoothness • Eg: 10 voxels, 2.5 FWHM = 4 RESELS • Beware RESEL misinterpretation • The RESEL count isnot the number of independent tests in image • See Nichols & Hayasaka, 2003, Stat. Meth. in Med. Res. • Do not Bonferroni correct based on resel count.

  23. Estimate distribution of cluster sizes (mean and variance) Mean: Expected Cluster Size Expected supra-threshold volume / expected number of clusters = E(S) / E(L) = E(L) = E(χu), expected Euler characteristic, assuming no holes (high threshold) This provides uncorrected p-values: Chances of getting a blob of >= k voxels under the null hypothesis …which are then corrected: Pc = E(L) Puncorr 5mm FWHM 10mm FWHM 15mm FWHM Random Field TheoryCluster Size Tests

  24. Lattice ImageData Continuous Random Field Random Field Theory Assumptions and Limitations • Sufficient smoothness • FWHM smoothness 3-4× voxel size (Z) • More like ~10× for low-df T images • Smoothness estimation • Estimate is biased when images not sufficiently smooth • Multivariate normality • Virtually impossible to check • Several layers of approximations • Stationarity required for cluster size results • Cluster size results tend to be too lenient in practice!

  25. FWER correctionStatistical Nonparametric Mapping - SnPMThresholding without (m)any assumptions Almost all slides from Tom Nichols

  26. 5% Parametric Null Distribution 5% Nonparametric Null Distribution Nonparametric Inference • Parametric methods • Assume distribution ofstatistic under nullhypothesis • Needed to find P-values, u • Nonparametric methods • Use data to find distribution of statisticunder null hypothesis • Any statistic!

  27. Permutation TestToy Example • Data from V1 voxel in visual stim. experiment A: Active, flashing checkerboard B: Baseline, fixation 6 blocks, ABABAB Just consider block averages... • Null hypothesis Ho • No experimental effect, A & B labels arbitrary • Statistic • Mean difference

  28. Permutation TestToy Example • Under Ho • Consider all equivalent relabelings

  29. Permutation TestToy Example • Under Ho • Consider all equivalent relabelings • Compute all possible statistic values

  30. Permutation TestToy Example • Under Ho • Consider all equivalent relabelings • Compute all possible statistic values • Find 95%ile of permutation distribution

  31. Permutation TestToy Example • Under Ho • Consider all equivalent relabelings • Compute all possible statistic values • Find 95%ile of permutation distribution

  32. Permutation TestToy Example • Under Ho • Consider all equivalent relabelings • Compute all possible statistic values • Find 95%ile of permutation distribution -8 -4 0 4 8

  33. Permutation TestStrengths • Requires only assumption of exchangeability: • Under Ho, distribution unperturbed by permutation • Subjects are exchangeable (good for group analysis) • Under Ho, each subject’s A/B labels can be flipped • fMRI scans not exchangeable under Ho (bad for time series analysis/single subject) • Due to temporal autocorrelation

  34. 5% Parametric Null Max Distribution 5% Nonparametric Null Max Distribution Controlling FWER: Permutation Test • Parametric methods • Assume distribution ofmax statistic under nullhypothesis • Nonparametric methods • Use data to find distribution of max statisticunder null hypothesis • Again, any max statistic!

  35. Results:RFT vs Bonf. vs Perm.

  36. RF & Perm adapt to smoothness Perm & Truth close Bonferroni close to truth for low smoothness FWER Correction: Performance summary Nichols and Hayasaka, 2003 19 df FamilywiseErrorThresholds more 9 df

  37. Permutation vs. Bonferroni: Power curves • Bonferroni and SPM’s GRF correction do not account accurately for spatial smoothness with the sample sizes and smoothness values typical in imaging experiments. • Nonparametric tests are more accurate and usually more sensitive. d = 2 d = 2 d = 1 d = 1 d = 0.5 d = 0.5 From Wager, Lindquist, & Hernandez, 2009. “Essentials of functional neuroimaging.” In: Handbook of Neuroscience for the Behavioral Sciences

  38. Performance Summary • Bonferroni • Not adaptive to smoothness • Not so conservative for low smoothness • Random Field • Adaptive • Conservative for low smoothness & df • Permutation • Adaptive (Exact)

  39. Controlling the False Discovery Rate…

  40. MCP Solutions:Measuring False Positives • Familywise Error Rate (FWER) • Familywise Error • Existence of one or more false positives • FWER is probability of familywise error • False Discovery Rate (FDR) • FDR = E(V/R) • R voxels declared active, V falsely so • Realized false discovery rate: V/R

  41. p(i)< (i/V)q Benjamini & HochbergProcedure • Select desired limit q on FDR (e.g., 0.05) • Order p-values, p(1)<p(2)< ... < p(V) • Let r be largest i such that • Reject all hypotheses corresponding top(1), ... , p(r). • Example: • 1000 Voxels, q = .05 • Keep those above P < .05*i/100 JRSS-B (1995)57:289-300 1 p(i) p-value (i/V)q 0 0 1 i/V • Under positive dependence; OK for fMRI

  42. Benjamini & Hochberg:Key Properties • FDR is controlled E(FDP) = q m0/V • Conservative, if large fraction of nulls false • Adaptive • Threshold depends on amount of signal • More signal, More small p-values,More results with only alpha % expected false discoveries • Hard to find signal in small areas, even with low p-values

  43. Conclusions • Must account for multiple comparisons • Otherwise have a fishing expedition • FWER • Very specific, not very sensitive • FDR • Less specific, more sensitive • Adaptive • Good: Power increases w/ amount of signal • Bad: Number of false positives increases too

  44. References • Most of this talk covered in these papers TE Nichols & S Hayasaka, Controlling the Familywise Error Rate in Functional Neuroimaging: A Comparative Review. Statistical Methods in Medical Research, 12(5): 419-446, 2003. TE Nichols & AP Holmes, Nonparametric Permutation Tests for Functional Neuroimaging: A Primer with Examples. Human Brain Mapping, 15:1-25, 2001. CR Genovese, N Lazar & TE Nichols, Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. NeuroImage, 15:870-878, 2002.

  45. End of this section.

  46. Set-level Inference • Count number of blobs c • c depends on by uclus & blob size k threshold • Worst spatial specificity • Only can reject global null hypothesis uclus space k k Here c = 1; only 1 cluster larger than k

  47. Random Field TheoryCluster Size Corrected P-Values • Previous results give uncorrected P-value • Corrected P-value If E(L) is expected number of clusters: • Bonferroni • Correct for expected number of clusters • Corrected Pc = E(L) Puncorr • Poisson Clumping Heuristic (Adler, 1980) • Corrected Pc = 1 - exp( -E(L) Puncorr )

More Related