TSTAT_THRESHOLD(~1 secs execution)

Calculates P=0.05 (corrected) threshold t for the T statistic using the minimum given by a Bonferroni correction and non-isotropic random field theory (Worsley et al., 1996, 1999). For this example, t=4.86, and voxels where T>t are shown in green:

FMRILM

MULTISTAT (~3 mins execution) Combines results from separate

runs of FMRILM using REML estimation with a regularized random effects analysis.

Model: Ei = effect for run i; xi = vector of regressors (= (1, 1, …, 1)´ to average the effects); = unknown vector of regression parameters; Si = standard deviation of effect; = unknown random effects standard deviation, WNi

f, WNir = Gaussian white noises, = random/fixed sd:

Ei = xi´ + SiWNif + WNi

r, 2 = (S2 + 2) / S2, S2 = Averagei Si2.

Step 1: Fit model by EM algorithm:

SummaryMany methods are available for the statistical analysis of fMRI data that range from a simple linear model for the response and a global autoregressive model for the temporal errors (Bullmore, et al., 1996; SPM), to a more sophisticated non-linear model for the response with a local state space model for the temporal errors (Purdon, et al., 1998). We have written Matlab programs FMRIDESIGN, FMRILM, MULTISTAT and TSTAT_THRESHOLD (available at http://www.bic.mni.mcgill.ca/users/keith) that seek a compromise between validity, generality, simplicity and execution speed.

A General Statistical Analysis for fMRI DataK. J. Worsley12, C. Liao1, M. Grabove1, V. Petre2, B. Ha2, A.C. Evans2

1Department of Mathematics and Statistics, McGill University2McConnell Brain Imaging Centre, Montreal Neurological Institute, McGill University

FMRIDESIGN

FMRILM

TSTAT_THRESHOLD

FMRIDESIGN (~2 secs execution)

Sets up stimuli st and convolutes it with the hemodynamic response function ht (difference of two gamma densities, Glover, 1999) to create the response design matrix xt for the linear model:

FMRILM (~6 mins execution) Fits linear model with AR(1) errors.

Model: Yt = fMRI data at time t; (h*s)t = hemodynamic response function h convoluted with vector of stimuli s, at time t; = vector of linear model parameters; dt = polynomial drift (1, t, t2, … ,tq)’; = vector of drift parameters; = standard deviation parameter; t =

AR(p) errors (p=1); aj = autoregressive parameters; WNt = Gaussian white noise:

Yt = (h*s)t´ + dt´ + t; t = a1 t-1 + …+ ap t-p + WNt

Step 1: Fit model by least squares, calculate lag 1 autocorrelation a1, then smooth it:

fMRI data

Ru

n 1 R

un

2 . . . Ru

n m

Ru

n 1 R

un

2 . . . Ru

n m

Ru

n 1 R

un

2 . . . Ru

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BJE

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1 SU

BJE

CT

2 • • • SU

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Smooth15 mm

Step 2: Model fitting biases correlation by ~ –0.05, so bias correction is needed:

Step 3: Whiten data and design matrix with a1, fit linear model again by least squares to get estimates , . For a contrast c, find effect c and its standard deviation Sd(c):

Step 4: T statistic T = c / Sd(c), thresholded at P<0.05 (see TSTAT_THRESHOLD)

FMRILM_ARP (>30 mins execution) Fits linear model with AR(p) errors for p>1.

Run 1 R

un 2 Run 3 R

un 4 Sd R

atio Final

Combining the runs:

Conclusions

• The simple AR(1) model appears to be adequate.• The FWHMratio parameter acts as a convenient way of providing an analysis mid-way between a random effects and a fixed effects analysis; setting FWHMratio = 0 (no smoothing) produces a random effects analysis; setting FWHMratio to infinity, which smoothes the sd ratio to one everywhere, produces a fixed effects analysis. In practice, we choose FWHMratio to produce a final dffinal which is at least 100, so that errors in its estimation do not greatly affect the distribution of test statistics.

Ignoring the correlation

If the temporal correlation is ignored completely, that is, the observations are treated as independent and a least squares analysis is used, then the T statistic T0 is ~11% larger than T1, the T statistic assuming AR(1) errors. This has the effect of increasing the number of false positives:

FMRILM

fMRI data

fMRI data

FMRILM

FMRILM

FMRILM

FMRILM

FMRILM

MULTISTAT

fMRI data

FMRILM

fMRI data

fMRI data

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fMRI data

fMRI data

fMRI data

MULTISTAT

Combining the subjects:

T = Effect / Sd

T statistics Tp for AR(p) models: for p1 they are very similar, again indicating that the AR(1) model is adequate

Autoregressive coefficients ap for AR(3): for p2, ap~0, so that the AR(1) model fitted by FMRILM seems to be adequate

Drift removal by adding polynomial variables 1, t, t2, …,tq to the model (q=3 by default).

MULTISTAT

MULTISTAT

=

References Bullmore, E.T. et al. (1996). Magnetic Resonance in Medicine, 35:261-277.Glover, G.H. (1999). NeuroImage, 9:416-429.Purdon, P.L. et al. (1998). NeuroImage, 7:S618.Worsley, K.J. et al. (1996). Human Brain Mapping, 4:58-73.Worsley, K.J. et al. (1999). Human Brain Mapping, 8:98-101.Worsley, K.J. et al. (2000). NeuroImage (submitted).

=

• There was little evidence of random effects between runs on the same subject ( ~ 1), but there were substantial random effects between subjects ( ~ 3):

Smooth15 mm

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Step 2:

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Resample to Talairach space after linear or non-linear transformations