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A general statistical analysis for fMRI data

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A general statistical analysis for fMRI data Keith Worsley12, Chuanhong Liao1, John Aston13, Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2, Alan Evans2 – PowerPoint PPT presentation

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Title: A general statistical analysis for fMRI data


1
A general statistical analysis for fMRI data
  • Keith Worsley12, Chuanhong Liao1, John Aston13,
  • 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

2
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?

3
More importantly ...
  • Fast execution / slow execution?
  • Matlab / C?
  • Script (batch) / GUI?
  • Lazy / hard working ?
  • Why not just use SPM?
  • Develop new ideas ...

4
Aim Simple, general, valid, robust, fast
analysis of fMRI data
Linear model, AR(p) errors
? ? Yt
(stimulust HRF) b driftt c errort
unknown parameters ?
? ?
errort a1 errort-1 ap errort-p s WNt
5
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
    scans/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.

6
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7
Example Pain perception
8
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9
First step estimate the autocorrelation
?
  • AR(1) model errort a1 errort-1 s WNt
  • Fit the linear model using least squares
  • êrrort Yt fitted Yt
  • â1 Correlation ( êrrort , êrrort-1)
  • Estimating the errors êrrort changes their
    correlation structure slightly, so â1 is
    slightly biased
  • Raw autocorrelation Smoothed 15mm
    Bias corrected

  • -0.05 0

10
Second step refit the linear model
Pre-whiten Yt Yt â1 Yt-1, then fit using
least squares Effect hot warm
Sd of effect
T statistic Effect / Sd
T gt 4.86 (P lt 0.05, corrected)
11
Higher order AR model? Try AR(4)
â1
â2
â3
â4
â2, â3, â4 0, so AR(1) seems to be adequate
12
has no effect on the T statistics
AR(1) AR(2)
AR(4) But using zero
correlation
biases T up 12 ? more false positives
13
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14
Results from 4 scans on the same subject
Scan 1 Scan 2 Scan 3 Scan 4
Effect Ei
Sd Si
T stat Ei / Si
15
MULTISTAT combines effects from different
scans/sessions/subjects
  • Ei effect for scan/session/subject i
  • Si standard error of effect
  • Mixed effects model
  • Ei covariatesi c fi ri

from FMRILM
Usually 1, but could add group, treatment,
age, sex, ...
Random effect, due to variability from scan to
scan, unknown sd ?
Fixed effects error, due to variability within
the same scan, known sd Si
16
Fitted using the EM algorithm
  • Slow to converge (10 iterations by default).
  • Stable (maintains positive variances).
  • ?2 biased if random effect is small, so
  • Sj2 ? Sj2 - minjSj2
  • ?2 ? ?2 minjSj2
  • Fit the model
  • ?2 ? ?2 - minjSj2




17
Scan 1 Scan 2 Scan 3 Scan 4
MULTISTAT
Effect Ei
very noisy sd
Sd Si
and no response is detected
T stat Ei / Si
18
Solution Spatial regularization of the sd
  • Basic idea increase df by spatial smoothing
    (local pooling) of the sd.
  • Cant smooth the random effects sd directly, -
    too much anatomical structure.
  • Instead,
  • random effects sd
  • fixed effects sd
  • which removes the anatomical structure before
    smoothing.

? )
sd smooth ?
fixed effects sd
19
Random effects sd (3 df)
Fixed effects sd (448 df)
Regularized sd (112 df)
Random effects sd Fixed effects sd
? Fixed effects sd
Over scans Over subjects
Smooth 15mm
20
Effective df
  • dfratio dfrandom ( 2 ( FWHMratio / FWHMdata )2
    1 )3/2
  • dfeff 1 / ( 1 / dfratio 1 / dffixed )
  • e.g. dfrandom 3, dffixed 112, FWHMdata 6mm
  • FWHMratio (mm) 0 5 10 15 20 infinite
  • dfeff 3 11 45 112 192
    448

variability bias
compromise!
21
Final result 15mm smoothing, 112 effective df
Scan 1 Scan 2 Scan 3 Scan 4
MULTISTAT
Effect Ei
less noisy sd
Sd Si
and now we can detect a response
T stat Ei / Si
22
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23
T gt 4.86 (P lt 0.05, corrected)
Tgt4.86
24
Conclusion
  • Largest portion of variance comes from the last
    stage i.e. combining over subjects
  • sdscan2 sdsess2
    sdsubj2
  • nscan nsess nsubj nsess nsubj
    nsubj
  • If you want to optimize total scanner time, take
    more subjects.
  • What you do at early stages doesnt matter very
    much!

25
P.S. Estimating the delay of the response
  • 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/(11/T2),
  • T is the T statistic for the magnitude.

26
Delay of the hot stimulus
T stat for magnitude 0 T stat for
delay 5.4 secs
Delay (secs) Sd of
delay (secs)
27
gt4.86 0 5.4s ?0.6s
28
T gt 4.86 (P lt 0.05, corrected)
Delay (secs)
6.5 5 5.5 4 4.5
29
T gt 4.86 (P lt 0.05, corrected)
Delay (secs)
6.5 5 5.5 4 4.5
30
References
  • http/www.math.mcgill.ca/keith/fmristat
  • Worsley et al. (2000). A general statistical
    analysis for fMRI data. NeuroImage, 11S648, and
    submitted.
  • Liao et al. (2001). Estimating the delay of the
    fMRI response. NeuroImage, 13S185 (Poster 185,
    Tuesday morning), and submitted.
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