1. 1 Data Modeling General Linear Model & Statistical Inference Thomas Nichols, Ph.D. Assistant Professor Department of Biostatistics http://www.sph.umich.edu/~nichols Brain Function and fMRI ISMRM Educational Course July 11, 2002

2. 2 Motivations Data Modeling –Characterize Signal –Characterize Noise Statistical Inference –Detect signal –Localization (Where’s the blob?)

3. 3 Outline Data Modeling –General Linear Model –Linear Model Predictors –Temporal Autocorrelation –Random Effects Models Statistical Inference –Statistic Images & Hypothesis Testing –Multiple Testing Problem

4. 4 Basic fMRI Example Data at one voxel –Rest vs. passive word listening Is there an effect?

5. 5 A Linear Model Intensity Time = 11 22 + + error x1x1 x2x2  “Linear” in parameters  1 &  2

6. 6 Linear model, in image form… =++

7. 7 =++ Estimated

8. 8 … in image matrix form… =  + 

9. 9 … in matrix form. = + Y Y N: Number of scans, p: Number of regressors

10. 10 Linear Model Predictors Signal Predictors –Block designs –Event-related responses Nuisance Predictors –Drift –Regression parameters

11. 11 Signal Predictors Linear Time-Invariant system LTI specified solely by –Stimulus function of experiment –Hemodynamic Response Function (HRF) Response to instantaneous impulse Blocks Events

12. 12 Convolution Examples Event-Related Hemodynamic Response Function Predicted Response Block Design Experimental Stimulus Function

13. 13 HRF Models Canonical HRF –Most sensitive if it is correct –If wrong, leads to bias and/or poor fit E.g. True response may be faster/slower E.g. True response may have smaller/ bigger undershoot SPM’s HRF

14. 14 HRF Models Smooth Basis HRFs –More flexible –Less interpretable No one parameter explains the response –Less sensitive relative to canonical (only if canonical is correct) Gamma Basis Fourier Basis

15. 15 HRF Models Deconvolution –Most flexible Allows any shape Even bizarre, non-sensical ones –Least sensitive relative to canonical (again, if canonical is correct) Deconvolution Basis

16. 16 Drift Models Drift –Slowly varying –Nuisance variability Models –Linear, quadratic –Discrete Cosine Transform Discrete Cosine Transform Basis

17. 17 General Linear Model Recap Fits data Y as linear combination of predictor columns of X Very “General” –Correlation, ANOVA, ANCOVA, … Only as good as your X matrix

18. 18 Temporal Autocorrelation Standard statistical methods assume independent errors –Error  i tells you nothing about  j i  j fMRI errors not independent –Autocorrelation due to –Physiological effects –Scanner instability

19. 19 Temporal Autocorrelation In Brief Independence Precoloring Prewhitening

20. 20 Autocorrelation: Independence Model Ignore autocorrelation Leads to –Under-estimation of variance –Over-estimation of significance –Too many false positives

21. 21 Autocorrelation: Precoloring Temporally blur, smooth your data –This induces more dependence! –But we exactly know the form of the dependence induced –Assume that intrinsic autocorrelation is negligible relative to smoothing Then we know autocorrelation exactly Correct GLM inferences based on “known” autocorrelation [Friston, et al., “To smooth or not to smooth…” NI 12:196-208 2000]

22. 22 Autocorrelation: Prewhitening Statistically optimal solution If know true autocorrelation exactly, can undo the dependence –De-correlate your data, your model –Then proceed as with independent data Problem is obtaining accurate estimates of autocorrelation –Some sort of regularization is required Spatial smoothing of some sort

23. 23 Autocorrelation Redux AdvantageDisadvantageSoftware Indep.SimpleInflated significance All PrecoloringAvoids autocorr. est. Statistically inefficient SPM99 WhiteningStatistically optimal Requires precise autocorr. est. FSL, SPM2

24. 24 Autocorrelation: Models Autoregressive –Error is fraction of previous error plus “new” error –AR(1):  i =  i-1 +  I Software: fmristat, SPM99 AR + White Noise or ARMA(1,1) –AR plus an independent WN series Software: SPM2 Arbitrary autocorrelation function –  k = corr(  i,  i-k ) Software: FSL’s FEAT

25. 25 Statistic Images & Hypothesis Testing For each voxel –Fit GLM, estimate betas Write b for estimate of  –But usually not interested in all betas Recall  is a length-p vector

26. 26 Building Statistic Images =  +  =  + YX  Predictor of interest

27. 27 Building Statistic Images Contrast –A linear combination of parameters –c’  T = contrast of estimated parameters variance estimate T = s 2 c’(X’X) + c c’b c’ = 1 0 0 0 0 0 0 0 b  b  b  b  b ....

28. 28 Hypothesis Test So now have a value T for our statistic How big is big –Is T=2 big? T=20?

29. 29 Hypothesis Testing Assume Null Hypothesis of no signal Given that there is no signal, how likely is our measured T? P-value measures this –Probability of obtaining T as large or larger  level –Acceptable false positive rate P-val T

30. 30 Random Effects Models GLM has only one source of randomness –Residual error But people are another source of error –Everyone activates somewhat differently…

31. 31 Subj. 1 Subj. 2 Subj. 3 Subj. 4 Subj. 5 Subj. 6 0 Fixed vs. Random Effects Fixed Effects –Intra-subject variation suggests all these subjects different from zero Random Effects –Intersubject variation suggests population not very different from zero Distribution of each subject’s effect

32. 32 Random Effects for fMRI Summary Statistic Approach –Easy Create contrast images for each subject Analyze contrast images with one-sample t –Limited Only allows one scan per subject Assumes balanced designs and homogeneous meas. error. Full Mixed Effects Analysis –Hard Requires iterative fitting REML to estimate inter- and intra subject variance –SPM2 & FSL implement this, very differently –Very flexible

33. 33 Random Effects for fMRI Random vs. Fixed Fixed isn’t “wrong”, just usually isn’t of interest If it is sufficient to say “I can see this effect in this cohort” then fixed effects are OK If need to say “If I were to sample a new cohort from the population I would get the same result” then random effects are needed

34. 34 Multiple Testing Problem Inference on statistic images –Fit GLM at each voxel –Create statistic images of effect Which of 100,000 voxels are significant? –  =0.05  5,000 false positives! t > 0.5 t > 1.5 t > 2.5 t > 3.5 t > 4.5 t > 5.5t > 6.5

35. 35 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) –R voxels declared active, V falsely so Observed false discovery rate: V/R –FDR = E(V/R)

36. 36 FWER MCP Solutions Bonferroni Maximum Distribution Methods –Random Field Theory –Permutation

37. 37 FWER MCP Solutions Bonferroni Maximum Distribution Methods –Random Field Theory –Permutation

38. 38 FWER MCP Solutions: Controlling FWER w/ Max FWER & distribution of maximum FWER= P(FWE) = P(One or more voxels  u | H o ) = P(Max voxel  u | H o ) 100(1-  )%ile of max dist n controls FWER FWER = P(Max voxel  u  | H o )   uu 

39. 39 FWER MCP Solutions: Random Field Theory Euler Characteristic  u –Topological Measure #blobs - #holes –At high thresholds, just counts blobs –FWER= P(Max voxel  u | H o ) = P(One or more blobs | H o )  P(  u  1 | H o )  E(  u | H o ) Random Field Suprathreshold Sets Threshold

40. 40 Controlling FWER: Permutation Test Parametric methods –Assume distribution of max statistic under null hypothesis Nonparametric methods –Use data to find distribution of max statistic under null hypothesis –Any max statistic! 5% Parametric Null Max Distribution 5% Nonparametric Null Max Distribution

41. 41 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) –R voxels declared active, V falsely so Observed false discovery rate: V/R –FDR = E(V/R)

42. 42 Measuring False Positives FWER vs FDR Signal+Noise Noise

43. 43 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

44. 44 Controlling FDR: Benjamini & Hochberg 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 p(i)p(i) i/Vi/V i/V  q p-value 01 0 1

45. 45 Conclusions Analyzing fMRI Data –Need linear regression basics –Lots of disk space, and time –Watch for MTP (no fishing!)

46. 46 Thanks Slide help –Stefan Keibel, Rik Henson, JB Poline, Andrew Holmes