A General Statistical Analysis for fMRI Data
K. J.
12
Worsley ,
C.
1
Liao ,
M.
1
Grabove ,
V.
2
Petre ,
B.
2
Ha , A.C.
2
Evans
1Department
of Mathematics and Statistics, McGill University
2McConnell Brain Imaging Centre, Montreal Neurological Institute, McGill University
MULTISTAT
Summary
FMRIDESIGN (~2 secs execution)
Many 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.
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:
(~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, WNif, WNir = Gaussian white noises,  = random/fixed sd:
Ei = xi´ + SiWNif +  WNir,
2 = (S2 + 2) / S2,
S2 = Averagei Si2.
Step 1: Fit model by EM algorithm:
FMRILM (~6 mins execution) Fits linear model with AR(1) errors.
Run 1
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
Run 2
Step 1: Fit model by least squares, calculate lag 1 autocorrelation a1, then smooth it:
^
Run 3
Smooth
15 mm
FMRIDESIGN
Drift removal by adding polynomial
variables 1, t, t2, …,tq to the model (q=3 by default).
Step 2: Model fitting biases correlation by ~ –0.05, so bias correction is needed:
fMRI data
FMRILM






Combining
the runs:
MULTISTAT
FMRILM
Step 2:
fMRI data
Combining
the subjects:
FMRILM
fMRI data


fMRI data
FMRILM






MULTISTAT
MULTISTAT
FMRILM
•
•
•
•
^
=
•
•
•
T = Effect / Sd
fMRI data
FMRILM
fMRI data


fMRI data
FMRILM






MULTISTAT
TSTAT_
THRESHOLD
Smooth
15 mm
^

Final
Run 1 Run 2 . . . Run m
SUBJECT n
•
•
•
Ratio
•
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:


•
•
•
•
Ignoring the correlation
fMRI data
Sd
Run 1 Run 2 . . . Run m
SUBJECT 2
^ ^ thresholded at P<0.05 (see TSTAT_THRESHOLD)
Step 4: T statistic T = c^ / Sd(c),
FMRILM
Resample to
Talairach space
after linear or
non-linear
transformations
Run 4
. Run m
^ .
^ and its standard deviation Sd(c):
^ ^
^ For a contrast c, find effect c
to get estimates ,
Run 1 Run 2 . .
SUBJECT 1
Step 3: Whiten data and design matrix with a^1, fit linear model again by least squares
fMRI data
FMRILM
FMRILM_ARP (>30 mins execution) Fits linear model with AR(p) errors for p>1.
Autoregressive coefficients a^p for AR(3): for p2, ^
ap~0, so that the AR(1) model fitted by FMRILM seems to be adequate
^
^
^
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:
T statistics Tp for AR(p) models: for p1 they are very similar, again indicating that the AR(1) model is adequate
Conclusions
• The simple AR(1) model appears to be
adequate.
• The FWHMratio parameter acts as a
convenient way of providing an analysis midway 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.
• There was little evidence of
random effects between runs on
^ ~ 1), but there
the same subject (
were substantial random effects
^ ~ 3):
between subjects (
^
=
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).