Model-driven statistical analysis of fMRI data Keith Worsley Department of Mathematics and Statistics, Brain Imaging Centre, Montreal Neurological Institute, McGill University www.math.mcgill.ca/keith

References Worsley et al. (2002). A general statistical analysis for fMRI data. NeuroImage, 15:1-15. Liao et al. (2002). Estimating the delay of the response in fMRI data. NeuroImage, 16:593-606. FMRISTAT: MATLAB package from www.math.mcgill.ca/keith/fmristat

500 1000 First scan of fMRI data -5 5 T statistic for hot - warm effect 100 200 300 870 880 890 hot rest warm Highly significant effect, T=6.59 800 820 No significant effect, T=-0.74 790 810 Drift Time, seconds fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, … T = (hot – warm effect) / S.d. ~ t110 if no effect

Exploring the data: PCA of time  space 20 40 60 80 100 120 4 3 2 1 Component Frame Temporal components (sd, % variance explained) 105.7, 77.8% 26.1, 4.8% 15.8, 1.7% 14.8, 1.5% Slice Spatial components 6 8 10 1: exclude first frames 2: drift 3: long-range correlation or anatomical effect: remove by converting to % of brain 4: signal?

Modeling the data: Choices … Time domain / frequency domain? AR / ARMA / state space models? Linear / non-linear time series model? Fixed HRF / estimated HRF? Voxel / local / global parameters? Fixed effects / random effects? Frequentist / Bayesian? Compromise: Simple, general, valid, robust, fast statistical analysis

Covariates example: pain perception 50 100 150 200 250 300 350 -1 1 2 Alternating hot and warm stimuli separated by rest (9 seconds each). hot warm -0.2 0.2 0.4 Hemodynamic response function: difference of two gamma densities Responses = stimuli * HRF, sampled every 3 seconds Time, seconds

Linear model for fMRI time series with AR(p) correlated errors ? ? Yt = (stimulust * HRF) b + driftt c + errort AR(p) errors: ? ? ? errort = a1 errort-1 + … + ap errort-p + s WNt ‘White Noise’ unknown parameters

First step: estimate the autocorrelation ? AR(1) model: errort = a1 errort-1 + s WNt Fit the linear model using least squares errort = Yt – fitted Yt â1 = Correlation ( errort , errort-1) Estimating errort’s changes their correlation structure slightly, so â1 is slightly biased: Raw autocorrelation Smoothed 15mm Bias corrected â1 ~ -0.05 ~ 0 -0.1 0.1 0.2 0.3

Pre-whiten: Yt* = Yt – â1 Yt-1, then refit using least squares: Second step: pre-whiten, refit the linear model Pre-whiten: Yt* = Yt – â1 Yt-1, then refit using least squares: -1 -0.5 0.5 1 Hot - warm effect, % 0.05 0.1 0.15 0.2 0.25 Sd of effect, % -6 -4 -2 2 4 6 T = effect / sd, 110 df T > 4.93 (P < 0.05, corrected)

Higher order AR model? Try AR(3): 1 2 3 0.3 0.2 AR(1) seems to be adequate 0.1 … has little effect on the T statistics: -0.1 No correlation biases T up ~12%  more false positives AR(1) AR(2) AR(3) 5 -5

Results from 4 runs on the same subject

Mixed effects linear model for combining effects from different runs/sessions/subjects: Ei = effect for run/session/subject i Si = standard error of effect Mixed effects model: Ei = covariatesi c + Si WNiF +  WNiR }from Lin. Mod. ? ? Usually 1, but could add group, treatment, age, sex, ... ‘Fixed effects’ error, due to variability within the same run Random effect, due to variability from run to run

REML estimation using the EM algorithm Slow to converge (10 iterations by default). Stable (maintains estimate 2 > 0 ), but 2 biased if 2 (random effect) is small, so: Re-parameterize the variance model: Var(Ei) = Si2 + 2 = (Si2 – minj Sj2) + (2 + minj Sj2) = Si*2 + *2 2 = *2 – minj Sj2 (less biased estimate) ^ ^ ? ? ^ ^

Problem: 4 runs, 3 df for random effects sd  ...  Problem: 4 runs, 3 df for random effects sd  ... Run 1 Run 2 Run 3 Run 4 Effect, E i Sd, S T stat, / S -1 1 MULTISTAT 0.1 0.2 -5 5 … very noisy sd: … and T>15.96 for P<0.05 (corrected): … so no response is detected …

Solution: Spatial regularization of the sd Basic idea: increase df by spatial smoothing (local pooling) of the sd. Can’t smooth the random effects sd directly, - too much anatomical structure. Instead, random effects sd fixed effects sd which removes the anatomical structure before smoothing.  ) sd = smooth  fixed effects sd

^ Average Si  divide multiply random effect, sd ratio ~1.3 Random effects sd, 3 df Fixed effects sd, 440 df Mixed effects sd, ~100 df 0.2 0.15 0.1 0.05 divide multiply Random sd / fixed sd Smoothed sd ratio 1.5 random effect, sd ratio ~1.3 1 0.5

Effective df depends on smoothing dfratio = dfrandom(2 + 1) 1 1 1 dfeff dfratio dffixed FWHMratio2 3/2 FWHMdata2 e.g. dfrandom = 3, dffixed = 4  110 = 440, FWHMdata = 8mm: = + 20 40 Infinity 100 200 300 400 fixed effects analysis, dfeff = 440 dfeff FWHM = 19mm Why 100? If out by 50%, dbn of T not much affected Target = 100 df random effects analysis, dfeff = 3 FWHMratio

Final result: 19mm smoothing, 100 effective df … Run 1 Run 2 Run 3 Run 4 Effect, E i Sd, S T stat, / S -1 1 MULTISTAT 0.1 0.2 -5 5 Final result: 19mm smoothing, 100 effective df … … less noisy sd: … and T>4.93 for P<0.05 (corrected): … and now we can detect a response!

P-values assessed for: Peaks or local maxima Spatial extent of clusters of neighbouring voxels above a pre-chosen threshold (~3) Correct for searching over a pre-specified region (usually the whole brain), which depends on: number of voxels in the search region (Bonferroni) or number of resels = volume / FWHM3 in the search region (random field theory) in practice, take the minimum of the two!

FWHM is spatially varying (non-isotropic) fMRI data is smoother in GM than WM VBM data is highly non-isotropic Has little effect on P-values for local maxima (use ‘average’ FWHM inside search region), but Has a big effect on P-values for spatial extents: smooth regions → big clusters, rough regions → small clusters, so Replace cluster volume by cluster resels = volume / FWHM3

FWHM – the local smoothness of the noise voxel size (1 – correlation)1/2 FWHM = (2 log 2)1/2 (If the noise is modeled as white noise smoothed with a Gaussian kernel, this would be its FWHM) P-values depend on resels: Volume FWHM3 resels = Local maximum T = 4.5 Clusters above t = 3.0, search volume resels = 500 0.1 0.1 0.08 0.08 0.06 0.06 P value of local max P value of cluster 0.04 0.04 0.02 0.02 500 1000 0.5 1 1.5 2 Resels of search volume Resels of cluster

Resels=1.90 P=0.007 Resels=0.57 P=0.387

Statistical summary: clusters clus vol resel p-val (one) 1 33992 54.22 0 ( 0) 2 14150 25.03 0 ( 0) 3 12382 20.29 0 ( 0) 4 2538 3.12 0.011 (0.001) 5 2538 2.77 0.016 (0.001) 6 1577 2.15 0.035 (0.002) 7 1000 1.43 0.098 (0.006) 8 500 1.31 0.119 (0.007) 9 1000 1.07 0.179 (0.011) 10 385 0.99 0.208 (0.013)

Statistical summary: peaks clus peak p-val (one) q-val (i j k) ( x y z ) 1 12.72 0 ( 0) 0 (59 74 1) ( 10.5 -28.7 24.1) 1 12.58 0 ( 0) 0 (60 75 1) ( 8.2 -31 23.7) 1 11.45 0 ( 0) 0 (61 73 2) ( 5.9 -25.3 17.5) 1 11.08 0 ( 0) 0 (62 66 4) ( 3.5 -6.9 6.3) 1 10.95 0 ( 0) 0 (61 70 4) ( 5.9 -16.2 4.8) 1 10.6 0 ( 0) 0 (62 69 3) ( 3.5 -15 12.1) 2 5.07 0.029 (0.004) 0 (48 69 10) ( 36.3 -7.3 -36.3) 3 5.06 0.029 (0.004) 0 (73 72 9) (-22.3 -15.3 -30.5) 3 5.03 0.033 (0.004) 0 (81 63 10) ( -41 6.6 -34.1) 13 5.02 0.035 (0.005) 0 (88 72 8) (-57.4 -16.4 -23.6) 6 4.91 0.054 (0.007) 0 (42 69 3) ( 50.4 -15 12.1) 11 4.91 0.055 (0.007) 0 (69 70 7) (-12.9 -12.9 -15.9) 9 4.91 0.055 (0.007) 0 (48 46 5) ( 36.3 40.5 6.7) 1 4.85 0.069 (0.008) 0 (52 93 2) ( 27 -71.6 10.2) 3 4.82 0.08 (0.009) 0 (79 66 8) (-36.3 -2.5 -21.4) 3 4.81 0.082 (0.009) 0 (78 65 8) ( -34 -0.2 -21) 1 4.8 0.086 ( 0.01) 0 (62 59 5) ( 3.5 10.4 1.9) 3 4.77 0.097 (0.011) 0 (82 61 10) (-43.4 11.2 -33.4) 1 4.75 0.106 (0.012) 0 (55 71 2) ( 19.9 -20.7 18.3) 5 4.73 0.114 (0.012) 0 (67 84 2) ( -8.2 -50.8 13.5)

T>4.86

T>4.86 T > 4.93 (P < 0.05, corrected)

T>4.86 T > 4.93 (P < 0.05, corrected)

T>4.86

Efficiency : optimum block design Sd of hot stimulus Sd of hot-warm 20 0.5 20 0.5 0.4 0.4 15 15 Magnitude Optimum design 0.3 0.3 10 10 0.2 Optimum design 0.2 X 5 5 0.1 0.1 X 5 10 15 20 5 10 15 20 InterStimulus Interval (secs) (secs) (secs) 20 1 20 1 0.8 0.8 15 15 Delay 0.6 0.6 Optimum design X Optimum design X 10 10 0.4 0.4 5 5 0.2 0.2 (Not enough signal) (Not enough signal) 5 10 15 20 5 10 15 20 Stimulus Duration (secs)

Efficiency : optimum event design 0.5 (Not enough signal) ____ magnitudes ……. delays uniform . . . . . . . . . 0.45 random .. . ... .. . concentrated : 0.4 0.35 0.3 Sd of effect (secs for delays) 0.25 0.2 0.15 0.1 0.05 5 10 15 20 Average time between events (secs)

How many subjects? Largest portion of variance comes from the last stage i.e. combining over subjects: sdrun2 sdsess2 sdsubj2 nrun nsess nsubj nsess nsubj nsubj If you want to optimize total scanner time, take more subjects. What you do at early stages doesn’t matter very much! +

References Worsley et al. (2002). A general statistical analysis for fMRI data. NeuroImage, 15:1-15. Liao et al. (2002). Estimating the delay of the response in fMRI data. NeuroImage, 16:593-606. FMRISTAT: MATLAB package from www.math.mcgill.ca/keith/fmristat

Estimating the delay of the response Delay or latency to the peak of the HRF is approximated by a linear combination of two optimally chosen basis functions: delay -5 5 10 15 20 25 -0.4 -0.2 0.2 0.4 0.6 t (seconds) basis1 basis2 HRF shift HRF(t + shift) ~ basis1(t) w1(shift) + basis2(t) w2(shift) Convolve bases with the stimulus, then add to the linear model

Fit linear model, estimate w1 and w2 -5 5 -3 -2 -1 1 2 3 shift (seconds) Fit linear model, estimate w1 and w2 Equate w2 / w1 to estimates, then solve for shift (Hensen et al., 2002) To reduce bias when the magnitude is small, use shift / (1 + 1/T2) where T = w1 / Sd(w1) is the T statistic for the magnitude Shrinks shift to 0 where there is little evidence for a response. w2 / w1 w1 w2

Shift of the hot stimulus T stat for magnitude T stat for shift Shift (secs) Sd of shift (secs)

Shift of the hot stimulus T stat for magnitude T stat for shift T>4 T~2 Shift (secs) Sd of shift (secs) ~1 sec +/- 0.5 sec

Combining shifts of the hot stimulus (Contours are T stat for magnitude > 4) Run 1 Run 2 Run 3 Run 4 Effect, E i Sd, S T stat, / S -4 -2 2 4 MULTISTAT 1 -5 5

Shift of the hot stimulus Shift (secs) T stat for magnitude > 4.93

References Worsley et al. (2002). A general statistical analysis for fMRI data. NeuroImage, 15:1-15. Liao et al. (2002). Estimating the delay of the response in fMRI data. NeuroImage, 16:593-606. FMRISTAT: MATLAB package from www.math.mcgill.ca/keith/fmristat

False Discovery Rate (FDR) Benjamini and Hochberg (1995), Journal of the Royal Statistical Society Benjamini and Yekutieli (2001), Annals of Statistics Genovese et al. (2001), NeuroImage FDR controls the expected proportion of false positives amongst the discoveries, whereas Bonferroni / random field theory controls the probability of any false positives No correction controls the proportion of false positives in the volume

Signal True + Noise False + Signal + Gaussian white noise P < 0.05 (uncorrected), T > 1.64 5% of volume is false + Signal True + Noise False + FDR < 0.05, T > 2.82 5% of discoveries is false + P < 0.05 (corrected), T > 4.22 5% probability of any false +

Comparison of thresholds FDR depends on the ordered P-values: P1 < P2 < … < Pn. To control the FDR at a = 0.05, find K = max {i : Pi < (i/n) a}, threshold the P-values at PK Proportion of true + 1 0.1 0.01 0.001 0.0001 Threshold T 1.64 2.56 3.28 3.88 4.41 Bonferroni thresholds the P-values at a/n: Number of voxels 1 10 100 1000 10000 Threshold T 1.64 2.58 3.29 3.89 4.42 Random field theory: resels = volume / FHHM3: Number of resels 0 1 10 100 1000 Threshold T 1.64 2.82 3.46 4.09 4.65

P < 0.05 (uncorrected), T > 1.64 5% of volume is false +

5% of discoveries is false + FDR < 0.05, T > 2.67 5% of discoveries is false +

P < 0.05 (corrected), T > 4.93 5% probability of any false +

Conjunction: Minimum Ti > threshold ‘Minimum of Ti’ ‘Average of Ti’ For P=0.05, threshold = 1.82 For P=0.05, threshold = 4.93 Efficiency = 82%

Functional connectivity Measured by the correlation between residuals at every pair of voxels (6D data!) Local maxima are larger than all 12 neighbours P-value can be calculated using random field theory Good at detecting focal connectivity, but PCA of residuals x voxels is better at detecting large regions of co-correlated voxels Activation only Correlation only Voxel 2 + + Voxel 2 + + + + + + + Voxel 1 + Voxel 1 + +

Component > threshold |Correlations| > 0.7, P<10-10 (corrected) First Principal Component > threshold