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False Discovery Rate for Functional Neuroimaging Thomas Nichols Department of Biostatistics University of Michigan

False Discovery Rate for Functional Neuroimaging Thomas Nichols Department of Biostatistics University of Michigan With thanks to... Christopher Genovese & Nicole Lazar Carnegie Mellon University Keith Worlsey McGill University Outline Introduction to Functional Neuroimaging

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False Discovery Rate for Functional Neuroimaging Thomas Nichols Department of Biostatistics University of Michigan

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  1. False Discovery RateforFunctional NeuroimagingThomas NicholsDepartment of BiostatisticsUniversity of Michigan With thanks to... Christopher Genovese & Nicole Lazar Carnegie Mellon University Keith Worlsey McGill University

  2. Outline • Introduction to Functional Neuroimaging • Multiple Comparison Problem • A Multiple Comparison Solution: False Discovery Rate (FDR) • FDR Properties • FDR Example

  3. Tap fingers Rest Introduction:Functional Neuroimaging •  Neuronal Activity  Blood Flow • Many functional neuroimaging methods measurecorrelates of blood flow • Functional Magnetic Resonance Imaging (fMRI) • Based on intrinsic properties of tissue • Blood Oxygenation Level Dependent effect (BOLD) •  Blood flow  fMRI Signal

  4. fMRI Multiple Comparisons Problem 1,000 • 4-Dimensional Data • 1,000 multivariate observations,each with 100,000 elements • 100,000 time series, each with 1,000 observations • Massively UnivariateApproach • 100,000 hypothesistests • Massive MCP! . . . 3 2 1

  5. Solutions forMultiple Comparison Problem • A MCP Solution Must Control False Positives • How to measure multiple false positives? • Familywise Error Rate (FWER) • Chance of any false positives • Controlled by Bonferroni & Random Field Methods • False Discovery Rate (FDR) • Proportion of false positives among rejected tests

  6. Signal False Discovery RateIllustration: Noise Signal+Noise

  7. 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

  8. p(i) i/V q/c(V) Benjamini & Hochberg Procedure • Select desired limit q on E(FDR) • Order p-values, p(1)p(2) ...  p(V) • Let r be largest i such that • Reject all hypotheses corresponding top(1), ... , p(r). 1 p(i) p-value i/V q/c(V) 0 0 1 i/V JRSS-B (1995) 57:289-300

  9. Benjamini & Hochberg Procedure • c(V) = 1 • Positive Regression Dependency on Subsets • Technical condition, special cases include • Independent data • Multivariate Normal with all positive correlations • Result by Benjamini & Yekutieli, Annals of Statistics, in press. • c(V) = i=1,...,V 1/i log(V)+0.5772 • Arbitrary covariance structure

  10. Signal Intensity 3.0 Signal Extent 1.0 Noise Smoothness 3.0 Benjamini & Hochberg:Varying Signal Extent p = z = 1

  11. Signal Intensity 3.0 Signal Extent 2.0 Noise Smoothness 3.0 Benjamini & Hochberg:Varying Signal Extent p = z = 2

  12. Signal Intensity 3.0 Signal Extent 3.0 Noise Smoothness 3.0 Benjamini & Hochberg:Varying Signal Extent p = z = 3

  13. Signal Intensity 3.0 Signal Extent 5.0 Noise Smoothness 3.0 Benjamini & Hochberg:Varying Signal Extent p = 0.000252 z = 3.48 4

  14. Signal Intensity 3.0 Signal Extent 9.5 Noise Smoothness 3.0 Benjamini & Hochberg:Varying Signal Extent p = 0.001628 z = 2.94 5

  15. Signal Intensity 3.0 Signal Extent 16.5 Noise Smoothness 3.0 Benjamini & Hochberg:Varying Signal Extent p = 0.007157 z = 2.45 6

  16. Signal Intensity 3.0 Signal Extent 25.0 Noise Smoothness 3.0 Benjamini & Hochberg:Varying Signal Extent p = 0.019274 z = 2.07 7

  17. Benjamini & Hochberg: Properties • Adaptive • Larger the signal, the lower the threshold • Larger the signal, the more false positives • False positives constant as fraction of rejected tests • Not a problem with imaging’s sparse signals • Smoothness OK • Smoothing introduces positive correlations

  18. FDR: Example • Verbal fluency data • 14 42-second blocks • ABABAB... • A: Two syllable words presented aurally • B: Silence • Imaging parameters • 2Tesla scanner, TR = 7 sec • 84 64x64x64 images of 3 x 3 x 3 mm voxels

  19. FDR Example:Plot of FDR Inequality p(i) ( i/V ) ( q/c(V) )

  20. FDR: Example FDR  0.05Indep/PRDSt0 = 3.8119 FDR  0.05Arbitrary Cov.t0 = 5.0747 FWER  0.05Bonferronit0 = 5.485

  21. FDR Software for SPM http://www.sph.umich.edu/~nichols/FDR

  22. FDR: Conclusions • False Discovery Rate • A new false positive metric • Benjamini & Hochberg FDR Method • Straightforward solution to fNI MCP • Just one way of controlling FDR • New methods under developmente.g. C. Genovese or J. Storey • Limitations • Arbitrary dependence result less sensitive Start Ill http://www.sph.umich.edu/~nichols/FDR Prop

  23. Positive Regression Dependency • Does fMRI data exhibit total positive correlation? • Example • 160 scan experiment • Spatialautocorrelationof residuals • Single voxelwith all others • Negative correlationexists!

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