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

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

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

4. fMRI Multiple Comparisons Problem 4-Dimensional Data –1,000 multivariate observations, each with 100,000 elements –100,000 time series, each with 1,000 observations Massively Univariate Approach –100,000 hypothesis tests Massive MCP! 1,000 1 2 3...

5. Solutions for Multiple 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. False Discovery Rate Illustration: Signal+Noise Noise

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

8. 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 to p (1),..., p (r). p (i)  i/V  q/c(V) p(i)p(i) i/Vi/V i/V  q/c(V) p-value 01 0 1 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. Benjamini & Hochberg: Varying Signal Extent Signal Intensity3.0Signal Extent 1.0Noise Smoothness3.0 p = z = 1

11. Benjamini & Hochberg: Varying Signal Extent Signal Intensity3.0Signal Extent 2.0Noise Smoothness3.0 p = z = 2

12. Benjamini & Hochberg: Varying Signal Extent Signal Intensity3.0Signal Extent 3.0Noise Smoothness3.0 p = z = 3

13. Benjamini & Hochberg: Varying Signal Extent Signal Intensity3.0Signal Extent 5.0Noise Smoothness3.0 p = 0.000252z = 3.48 4

14. Benjamini & Hochberg: Varying Signal Extent Signal Intensity3.0Signal Extent 9.5Noise Smoothness3.0 p = 0.001628z = 2.94 5

15. Benjamini & Hochberg: Varying Signal Extent Signal Intensity3.0Signal Extent16.5Noise Smoothness3.0 p = 0.007157z = 2.45 6

16. Benjamini & Hochberg: Varying Signal Extent Signal Intensity3.0Signal Extent25.0Noise Smoothness3.0 p = 0.019274z = 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  0.05 Indep/PRDS t 0 = 3.8119 FWER  0.05 Bonferroni t 0 = 5.485 FDR  0.05 Arbitrary Cov. t 0 = 5.0747

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 development e.g. C. Genovese or J. Storey Limitations –Arbitrary dependence result less sensitive http://www.sph.umich.edu/~nichols/FDR Prop Ill Start

23. Positive Regression Dependency Does fMRI data exhibit total positive correlation? Example –160 scan experiment –Spatial autocorrelation of residuals –Single voxel with all others Negative correlation exists!