TSTAT_THRESHOLD

1
A General Statistical Analysis for fMRI Data K.
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
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,
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
Summary 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.m
cgill.ca/users/keith) that seek a compromise
between validity, generality, simplicity and
execution speed.
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
(hs)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 (p1) aj
autoregressive parameters WNt Gaussian white
noise Yt
(hs)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

Smooth 15 mm
FMRIDESIGN
Drift removal by adding polynomial variables 1,
t, t2, ,tq to the model (q3 by default).
Step 2 Model fitting biases correlation by
0.05, so bias correction is needed
fMRI data
Resample to Talairach space after linear or
non-linear transformations
FMRILM
Combining the runs
fMRI data
FMRILM
MULTISTAT
? ? ?
? ?
? ? ?
fMRI data
Run 1 Run 2 Run 3 Run
4 Sd Ratio
Final
FMRILM
Step 2
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??)

fMRI data
Combining the subjects





FMRILM
fMRI data
FMRILM
MULTISTAT
? ? ?
MULTISTAT
? ?
? ? ?
Run 1 Run 2 . . . Run m Run 1
Run 2 . . . Run m
Run 1 Run 2 . . . Run m
SUBJECT 1 SUBJECT 2
SUBJECT n
fMRI data
FMRILM

?

T Effect / Sd
Step 4 T statistic T c?? / Sd(c??),
thresholded at Plt0.05 (see TSTAT_THRESHOLD)

Smooth 15 mm
fMRI data
FMRILM

fMRI data
?
TSTAT_ THRESHOLD
FMRILM
MULTISTAT
? ? ?
? ?
? ? ?
fMRI data
FMRILM
FMRILM_ARP (gt30 mins execution) Fits linear model
with AR(p) errors for pgt1.
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

• 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.

TSTAT_THRESHOLD (1 secs execution) Calculates
P0.05 (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,
t4.86, and voxels where Tgtt are shown in green


Autoregressive coefficients ap for AR(3) for
p?2, ap0, so that the AR(1) model fitted by
FMRILM seems to be adequate

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

?

T statistics Tp for AR(p) models for p?1 they
are very similar, again indicating that the AR(1)
model is adequate
References Bullmore, E.T. et al. (1996).
Magnetic Resonance in Medicine,
35261-277.Glover, G.H. (1999). NeuroImage,
9416-429. Purdon, P.L. et al. (1998).
NeuroImage, 7S618.Worsley, K.J. et al. (1996).
Human Brain Mapping, 458-73.Worsley, K.J. et
al. (1999). Human Brain Mapping,
898-101. Worsley, K.J. et al. (2000). NeuroImage
(submitted).