A general statistical analysis for fMRI data

Keith Worsley12, Chuanhong Liao1, John Aston123,

Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2, 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

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 ...

Aim: 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

MATLAB: reads MINC or analyze format (www/math.mcgill.ca/keith/fmristat)

• FMRIDESIGN: Sets up stimulus, convolves it with the HRF and its derivatives (for estimating delay).

• FMRILM: Fits model, estimates effects (contrasts in the magnitudes, b), standard errors, T and F statistics.

• MULTISTAT: Combines effects from separate runs/sessions/subjects in a hierarchical fixed / random effects analysis.

• TSTAT_THRESHOLD: Uses random field theory / Bonferroni to find thresholds for corrected P-values for peaks and clusters of T and F maps.

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

hot

warm

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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)

Example: Pain perception

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First step: estimate the autocorrelationAR(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

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?

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Second step: 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 > 4.86 (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

Run 1 Run 2 Run 3 Run 4

EffectEi

SdSi

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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

R

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

? ?

REML estimation 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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^

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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

~1.6~1.6

Over scans

~3~3

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

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>4.86 for P<0.05 (corrected):

… and now we can detect a response!

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

Conclusion

• Largest portion of variance comes from the last stage i.e. combining over subjects:

sdrun2 sdsess

2 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!

+ +

• Delays or latency in the neuronal response are modeled as a temporal scale shift in the reference HRF:

• Fast voxel-wise delay estimator is found by adding the derivative of the reference HRF with respect to the log scale shift as an extra term to the linear model.

• Bias correction using the second derivative.• Shrunk to the reference delay by a factor of 1/(1+1/T2), T is the T statistic for the magnitude.

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Delay = 5.4 seconds, log scale shift = 0 (reference hrf, h0)

Delay = 4.0 seconds, log scale shift = -0.3

Delay = 7.3 seconds, log scale shift = +0.3

t (seconds)

P.S. Estimating the delay of the response

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Delay of the hot stimulusT stat for magnitude = 0 T stat for delay = 5.4 secs

Delay (secs) Sd of delay (secs)

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Varying the delay of the reference HRF

Ref.delay= 4.0

Ref.delay= 7.3

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T stat for mag T stat for delay Delay Sd of delay

Ref.delay= 5.4

>4.86 ~0 ~5.4s >4.86 ~0 ~5.4s 0.6s0.6s

~5.4s~5.4s

~5.4s~5.4s

Delay(secs)

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T > 4.86 (P < 0.05, corrected)

Delay(secs)

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T > 4.86 (P < 0.05, corrected)

Comparison:• 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

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 accurate, higher df• Regularization, high df• Yes

References

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

• Worsley et al. (2000). A general statistical analysis for fMRI data. NeuroImage, 11:S648, and submitted.

• Liao et al. (2001). Estimating the delay of the fMRI response. NeuroImage, 13:S185 and submitted.

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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Noise

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

False +

True +Signal

• FDR depends on the ordered P-values: P1 < P2 < … < Pn. To control the FDR at find K = max {i : Pi < (i/n) }, 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 /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?