The statistical analysis of fMRI data Keith Worsley12, Chuanhong Liao1, John Aston123, Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2, Frank Morales6, Alan Evans2, Tom Nichols7, Satoru Hayasaki7 1Department of Mathematics and Statistics, McGill University, 2Brain Imaging Centre, Montreal Neurological Institute, 3Imperial College, London, 4Service Hospitalier Frédéric Joliot, CEA, Orsay, 5Centre de Recherche en Sciences Neurologiques, Université de Montréal, 6Cuban Neuroscience Centre 7University of Michigan

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

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?

More importantly ... Fast execution / slow execution? Matlab / C? Script (batch) / GUI? Lazy / hard working … ? Why not just use SPM? Develop new ideas ... FMRISTAT: Simple, general, valid, robust, fast analysis of fMRI data

PCA_IMAGE: PCA of time  space: 20 40 60 80 100 120 140 4 3 2 1 Component Frame Temporal components (sd, % variance explained) 0.68, 46.9% 0.29, 8.6% 0.17, 2.9% 0.15, 2.4% -1 -0.5 0.5 Slice (0 based) Spatial components 6 8 10 12 1: exclude first frames 2: drift 3: long-range correlation or anatomical effect: remove by converting to % of brain 4: signal?

FMRILM: fits a linear model for fMRI time series with AR(p) errors ? ? Yt = (stimulust * HRF) b + driftt c + errort AR(p) errors: ? ? ? errort = a1 errort-1 + … + ap errort-p + s WNt unknown parameters

FMRIDESIGN 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

FMRILM 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: which_stats = ‘_cor’ Raw autocorrelation Smoothed 12.4mm Bias corrected â1 ~ -0.05 ~ 0

Effective df depends on smoothing Variability in acor lowers df Df depends on contrast Smoothing acor brings df back up: dfacor = dfresidual(2 + 1) 1 1 2 acor(contrast of data)2 dfeff dfresidual dfacor FWHMacor2 3/2 FWHMdata2 = + Hot stimulus Hot-warm stimulus FWHMdata = 8.79 Residual df = 110 Residual df = 110 100 100 Target = 100 df Target = 100 df 50 Contrast of data, acor = 0.61 50 Contrast of data, acor = 0.79 dfeff dfeff 10 20 30 10 20 30 FWHM = 10.3mm FWHM = 12.4mm FWHMacor FWHMacor

FMRILM second step: refit the linear model Pre-whiten: Yt* = Yt – â1 Yt-1, then fit using least squares: Hot - warm effect, % ‘_mag_ef’ Sd of effect, % ‘_mag_sd’ 1 0.25 0.2 0.5 0.15 0.1 -0.5 0.05 -1 which_stats = ‘_mag_ef _mag_sd _mag_t’ T = effect / sd, 110 df ‘_mag_t’ 6 4 T > 4.93 (P < 0.05, corrected) 2 -2 -4 -6

Higher order AR model? Try AR(3): ‘_AR’ 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), df=100 AR(2), df=98 AR(3), df=98 5 -5

Results from 4 runs on the same subject 1 Effect, E i ‘_mag_ef’ -1 0.2 Sd, S i 0.1 ‘_mag_sd’ 5 T stat, E / S i i ‘_mag_t’ -5

MULTISTAT: 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 FMRILM ? ? 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 ‘_mag_ef’ … very noisy sd: ‘_mag_sd’ … and T>15.96 for P<0.05 (corrected): ‘_mag_t’ … 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 ‘_sdratio’ 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 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 … ‘_mag_ef’ ‘_ef’ … less noisy sd: ‘_sd’ ‘_mag_sd’ … and T>4.93 for P<0.05 (corrected): ‘_t’ ‘_mag_t’ … and now we can detect a response!

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

Non-isotropic data (spatially varying FWHM) 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

Resels=1.90 P=0.007 Resels=0.57 P=0.387 ‘_fwhm’ ‘_fwhm’ FWHM (mm) of scans (110 df) FWHM (mm) of effects (3 df) 20 20 Resels=1.90 P=0.007 15 15 10 10 Resels=0.57 P=0.387 5 5 FWHM of effects (smoothed) effects / scans FWHM (smoothed) 20 1.5 15 10 1 5 0.5

In between use Discrete Local Maxima (DLM) STAT_SUMMARY Low FWHM use Bonferroni In between use Discrete Local Maxima (DLM) High FWHM use Random Field Theory 4.7 Bonferroni 4.6 4.5 True T, 10 df 4.4 Random Field Theory 4.3 T, 20 df Gaussianized threshold 4.2 Discrete Local Maxima (DLM) 4.1 Gaussian 4 3.9 Bonferroni, N=Resels 3.8 3.7 1 2 3 4 5 6 7 8 9 10 FWHM of smoothing kernel (voxels)

In between use Discrete Local Maxima (DLM) STAT_SUMMARY Low FWHM use Bonferroni In between use Discrete Local Maxima (DLM) High FWHM use Random Field Theory 0.12 Gaussian T, 20 df T, 10 df 0.1 Bonferroni Random Field Theory 0.08 DLM can ½ P-value when FWHM ~3 voxels P-value 0.06 0.04 True Discrete Local Maxima 0.02 Bonferroni, N=Resels 1 2 3 4 5 6 7 8 9 10 FWHM of smoothing kernel (voxels)

STAT_SUMMARY example: single run, hot-warm Detected by BON and DLM but not by RFT Detected by DLM, but not by BON or RFT

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

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

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! +

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 ‘_mag_t’ T stat for shift ‘_del_t’ Shift (secs) ‘_del_ef’ Sd of shift (secs) ‘_del_sd’

Shift of the hot stimulus T stat for magnitude ‘_mag_t’ T stat for shift ‘_del_t’ T>4 T~2 Shift (secs) ‘_del_ef’ Sd of shift (secs) ‘_del_sd’ ~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 ‘_ef’ ‘_del_ef’ ‘_sd’ ‘_del_sd’ ‘_t’ ‘_del_t’

Shift of the hot stimulus Shift (secs) ‘_del_ef’ T stat for magnitude ‘_mag_t’ > 4.93

Comparison: SPM’99: fmristat: Different slice acquisition times: Drift removal: Temporal correlation: Estimation of effects: Rationale: Random effects: Map of the delay: SPM’99: Adds a temporal derivative Low frequency cosines (flat at the ends) AR(1), global parameter, bias reduction not necessary Band pass filter, then least-squares, then correction for temporal correlation More robust, but lower df No regularization, low df, no conjuncs No fmristat: Shifts the model Splines (free at the ends) AR(p), voxel parameters, bias reduction Pre-whiten, then least squares (no further corrections needed) More accurate, higher df Regularization, high df, conjuncs Yes

References http://www.math.mcgill.ca/keith/fmristat 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.

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

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 +

Random fields and brain mapping Keith Worsley Department of Mathematics and Statistics, McConnell Brain Imaging Centre, Montreal Neurological Institute, McGill University