A general statistical analysis for fMRI data

Keith Worsley12, Chuanhong Liao1, John Aston123, Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2,

Frank Morales6, Alan Evans2

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

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fMRI data: 120 scans, 3 scans each ofhot, rest, warm, rest, hot, rest, …

Z = (effect hot – warm) / S.d. ~ N(0,1) if no effect

FMRISTAT: Simple, general, valid, robust, fast analysis of fMRI data

• Linear model: ? ? Yt = (stimulust * HRF) b + driftt c + errort

• AR(p) errors: ? ? ? errort = a1 errort-1 + … + ap errort-p + s WNt

unknown parameters

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2(a) Stimulus, s(t): alternating hot and warm stimuli on forearm, separated by rest (9 seconds each).

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(b) Hemodynamic response function, h(t): difference of two gamma densities (Glover, 1999)

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2(c) Response, x(t): sampled at the slice acquisition times every 3 seconds

Time, t (seconds)

FMRIDESIGN example: Pain perception

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FMRILM step 1: estimate temporal correlationAR(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.

• Bias correction is very quick and effective:Raw autocorrelation Smoothed 15mm Bias corrected â1

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FMRILM step 2: refit the linear modelPre-whiten: Yt

* = Yt – â1 Yt-1, then fit using least squares:

Effect: hot – warm Sd of effect

T statistic = Effect / Sd

T > 4.90 (P < 0.05, corrected)

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Higher order AR model? Try AR(4): â1 â2

â3 â4

AR(1) seems to be adequate

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… has no effect on the T statistics:AR(1) AR(2)

AR(4)

biases T up ~12% more false positives

But ignoring correlation …

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Results from 4 runs on the same subject

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EffectEi

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MULTISTAT combines 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 + WNi

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Random effect,due to variability from run to run

‘Fixed effects’ error,due to variabilitywithin the same run

Usually 1, but could add group,treatment, age,sex, ...

}from

FMRILM

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REML estimation of the mixed effects model using the EM algorithm

• Slow to converge (10 iterations by default).• Stable (maintains estimate 2 > 0 ), but2 biased if 2 (random effect) is small, so:• Re-parametrise the variance model:

Var(Ei) = Si2 + 2

= (Si2 – minj Sj

2) + (2 + minj Sj2)

= Si*2 + *2 2 = *2 – minj Sj

2 (less biased estimate)^ ^

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Run 1 Run 2 Run 3 Run 4 MULTISTAT

EffectEi

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Problem: 4 runs, 3 df for random effects sd ...

… and T>15.96 for P<0.05 (corrected):

… very noisy sd:

… so no response is detected …

• 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.

Solution: Spatial regularization of the sd

sd = smooth fixed effects sd )

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Random effects sd(3 df)

Fixed effects sd(448 df)

Random effects sdFixed effects sd

Regularized sd(112 df)

Fixed effects sd

Smooth Smooth 15mm15mm ~1~1

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Over runs

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Over subjects

dfratio = dfrandom(2 + 1)1 1 1

dfeff dfratio dffixed

e.g. dfrandom = 3, dffixed = 112, FWHMdata = 6mm:

FWHMratio (mm) 0 5 10 15 20 infinite

dfeff 3 11 45 112 192 448

Effective df depends on the smoothing

Random effects Fixed effects variability bias compromise!

FWHMratio2 3/2

FWHMdata2

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Run 1 Run 2 Run 3 Run 4 MULTISTAT

EffectEi

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Final result: 15mm smoothing, 112 effective df …

… less noisy sd:

… and T>4.90 for P<0.05 (corrected):

… and now we can detect a response!

Conjunction: All Ti > threshold = Min Ti > threshold

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‘Minimum of Ti’ ‘Average of Ti’

For P=0.05,threshold = 1.82

For P=0.05,threshold = 4.90

Efficiency = 82%

If the conjunction is significant, does it mean that all effects > 0?

• Problem: for the conjunction of 20 effects, the threshold can be negative!?!?!

• Reason: significance is based on the wrong null hypothesis, namely: all effects = 0

• Correct null hypothesis is: at least one effect = 0. Unfortunately the P-value depends on the unknown > 0 effects …

• If the effects are random, all effects > 0 is meaningless. The only parameter is the (single) population effect, so that the conjunction just tests if population effect > 0.

• P-values now depend on the random effects sd, not the fixed effects sd. But the minimum (i.e. the conjunction) is less efficient (sensitive) than the average (the usual test).

FWHM – the local smoothness of the noise • Used by STAT_THRESHOLD to find the P-value of local maxima and

the spatial extent of clusters of voxels above a threshold. • u = normalised residuals from linear model = residuals / sd• u = vector of spatial derivatives of u• λ = |Var(u)|1/2 (mm-3) • FWHM = (4 log 2)1/2 λ-1/3 (mm)

(If residuals are modeled as white noise smoothed with a Gaussian kernel, this would be its FWHM).

• λ and FWHM are corrected for low df and large voxel size so they are approximately unbiased.

• For a search region S, the number of ‘resolution elements’ is Resels(S) = Vol(S) AvgS(FWHM-3) = Vol(S) AvgS(λ) (4 log 2)-3/2

• For local maxima in S, P_value = Resels(S) x (function of threshold).• For a cluster C, P-value depends on Resels(C) instead of Vol(C), so that

clusters in smooth regions are less significant. • Need a correction for the randomness of λ and FWHM - depends on df .• Correction is more important for small clusters C than for large search

regions S.

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Resels=1.90P=0.007

Resels=0.57P=0.387

…. FWHM depends on the spatial correlation between neighbours

T>4.86T > 4.90 (P < 0.05, corrected)

Smooth the data before analysis?• Temporal smoothing or low-pass filtering is used by

SPM’99 to validate a global AR(1) model. For our local AR(p) model, it is not necessary (but ~ harmless).

• Spatial smoothing is used by SPM’99 to validate random field theory. Can be harmful for focal signals. Should fix the theory! STAT_THRESHOLD uses the better of the Bonferroni or the random field theory.

• A better reason for spatial smoothing is greater detectability of extensive activation: choose the FWHM to match the activation (e.g. 10mm FWHM for 10mm activations) – or try a range of FWHM’s i.e. scale space – but thresholds are higher …

False Discovery Rate (FDR)Benjamini and Hochberg (1995), Journal of the Royal Statistical Society

Benjamini and Yekutieli (2001), Annals of StatisticsGenovese 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

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P < 0.05 (uncorrected), Z > 1.645% of volume is false +

FDR < 0.05, Z > 2.825% of discoveries is false +

P < 0.05 (corrected), Z > 4.225% probability of any false +

Signal + Gaussian white noise

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True +Signal

• FDR depends on the ordered P-values (not smoothness): 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 Z 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 Z 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 Z 1.64 2.82 3.46 4.09 4.65

Comparison of thresholds

FDR < 0.05, Z > 2.915% of discoveries is false +

P < 0.05 (corrected), Z > 4.865% probability of any false +

P < 0.05 (uncorrected), Z > 1.645% of volume is false +

Which do you prefer?

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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:

HRF(t + shift) ~ basis1(t) w1(shift) + basis2(t) w2(shift)

• Convolve bases with the stimulus, then add to the linear model

basis1 basis2HRF

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delay

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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.

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w2 / w1

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Delay of the hot stimulus (= shift + 5.4 sec)T stat for magnitude T stat for shift

Delay (secs) Sd of delay (secs)

Varying the delay and dispersion of the reference HRF

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EFFICIENCY for optimum block design

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Optimumdesign

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EFFICIENCY for optimum event design

____ magnitudes ……. delays

(Not enough signal)

How many subjects?• Variance = sdrun

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nrun nsess nsubj nsess nsubj nsubj

• The largest portion of variance comes from the last stage, i.e. combining over subjects.

• If you want to optimize total scanner time, take more subjects, rather than more scans per subject.

• What you do at early stages doesn’t matter very much - any reasonable design will do …

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Comparison• Different slice acquisition times:• Drift removal:

• Temporal correlation:

• Estimation of effects:

• Rationale:• Random effects:• FWHM:

• Map of 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, low df• No regularization, low df• Global, ~ OK for local maxima, but not clusters• No

FMRISTAT:•Shifts the model

• Polynomials (free at the ends)• AR(p), voxel parameters, bias reduction• Pre-whiten, then least squares (no further corrections needed)

• More efficient, high df• Regularization, high df• Local, is OK for local maxima and clusters• Yes

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 fMRI response. NeuroImage, 16:593-606.

• http://www.math.mcgill.ca/keith/fmristat - 200K of MATLAB code

- fully worked example

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

Voxel 2

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Activation onlyVoxel 2

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Correlation only

First Principal Component > threshold

|Correlations| > 0.7,P<10-10 (corrected)