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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 Aston123, Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2, – 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 Aston123,
  • Jean-Baptiste Poline4, Gary Duncan5, Vali Petre2,
  • Frank Morales6, 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,
  • 6Cuban Neuroscience Centre

2
fMRI data 120 scans, 3 scans each of hot, rest,
warm, rest, hot, rest,
(a) Highly significant correlation
960
hot
fMRI data
rest
940
warm
920
0
50
100
150
200
250
300
350
400
940
(b) No significant correlation
hot
fMRI data
920
rest
warm
900
0
50
100
150
200
250
300
350
400
850
fMRI data
840
(c) Drift
830
0
50
100
150
200
250
300
350
400
Time, t (seconds)
Z (effect hot warm) / S.d. N(0,1) if no
effect
3
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4
FMRISTAT 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
5
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6
FMRIDESIGN example Pain perception
(c) Response, x(t) sampled at the slice
acquisition times every 3 seconds
Time, t (seconds)
7
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8
FMRILM step 1 estimate temporal correlation
?
  • AR(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 errorts changes their correlation
    structure slightly, so â1 is slightly biased.
  • Bias correction is very quick and effective
  • Raw autocorrelation Smoothed 15mm
    Bias corrected â1

  • -0.05 0

9
FMRILM step 2 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
10
Higher order AR model? Try AR(4)
â1
â2
0
â3
â4
0
0
AR(1) seems to be adequate
11
has no effect on the T statistics
AR(1) AR(2)
AR(4)
12
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13
Results from 4 runs on the same subject
Run 1 Run 2 Run 3 Run 4
Effect Ei
Sd Si
T stat Ei / Si
14
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 ? WNiR

from FMRILM
?
?
Usually 1, but could add group, treatment,
age, sex, ...
Random effect, due to variability from run to run
Fixed effects error, due to variability within
the same run
15
REML estimation of the mixed effects model using
the EM algorithm
  • Slow to converge (10 iterations by default).
  • Stable (maintains estimate ?2 gt 0 ), but
  • ?2 biased if ?2 (random effect) is small, so
  • Re-parametrise the variance model
  • Var(Ei) Si2 ?2
  • (Si2 minj Sj2) (?2
    minj Sj2)
  • Si2
    ?2
  • ?2 ?2 minj Sj2 (less biased estimate)



?
?


16
?
Problem 4 runs, 3 df for random effects sd ? ...
Run 1 Run 2 Run 3 Run 4
MULTISTAT
Effect Ei
very noisy sd
Sd Si
and Tgt15.96 for Plt0.05 (corrected)
T stat Ei / Si
so no response is detected
17
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
18
Random effects sd (3 df)
Fixed effects sd (448 df)
19
Effective df depends on the smoothing
  • 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

FWHMratio2 3/2 FWHMdata2

Random effects Fixed effects
variability bias
compromise!
20
Final result 15mm smoothing, 112 effective df
Run 1 Run 2 Run 3 Run 4
MULTISTAT
Effect Ei
less noisy sd
Sd Si
and Tgt4.90 for Plt0.05 (corrected)
T stat Ei / Si
and now we can detect a response!
21
Conjunction All Ti gt threshold Min Ti gt
threshold
Minimum of Ti
Average of Ti
For P0.05, threshold 1.82
For P0.05, threshold 4.90
Efficiency 82
22
If the conjunction is significant, does it mean
that all effects gt 0?
  • Problem for the conjunction of 20 effects, the
    threshold can be negative!?!?!
  • Reason significance is based on the wrong null
    hypothesis, namely all effects 0
  • Correct null hypothesis is at least one effect
    0. Unfortunately the P-value depends on the
    unknown gt 0 effects
  • If the effects are random, all effects gt 0 is
    meaningless. The only parameter is the (single)
    population effect, so that the conjunction just
    tests if population effect gt 0.
  • P-values now depend on the random effects sd, not
    the fixed effects sd. But the minimum (i.e. the
    conjunction) is less efficient (sensitive) than
    the average (the usual test).

23
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24
FWHM the local smoothness of the noise
  • Used by STAT_THRESHOLD to find the P-value of
    local maxima and the
    spatial extent of clusters of voxels above a
    threshold.
  • u normalised residuals from linear model
    residuals / sd
  • u vector of spatial derivatives of u
  • ? Var(u)1/2 (mm-3)
  • FWHM (4 log 2)1/2 ?-1/3 (mm)

    (If residuals are modeled as white noise
    smoothed with a Gaussian kernel, this would
    be its FWHM).
  • ? and FWHM are corrected for low df and large
    voxel size so they are approximately unbiased.
  • For a search region S, the number of resolution
    elements is Resels(S)
    Vol(S) AvgS(FWHM-3) Vol(S) AvgS(?) (4 log
    2)-3/2
  • For local maxima in S, P_value Resels(S) x
    (function of threshold).
  • For a cluster C, P-value depends on Resels(C)
    instead of Vol(C), so that clusters in smooth
    regions are less significant.
  • Need a correction for the randomness of ? and
    FWHM - depends on df .
  • Correction is more important for small clusters C
    than for large search regions S.



25
. FWHM depends on the spatial correlation
between neighbours
Resels1.90 P0.007
Resels0.57 P0.387
26
T gt 4.90 (P lt 0.05, corrected)
Tgt4.86
27
Smooth the data before analysis?
  • Temporal smoothing or low-pass filtering is used
    by SPM99 to validate a global AR(1) model. For
    our local AR(p) model, it is not necessary (but
    harmless).
  • Spatial smoothing is used by SPM99 to validate
    random field theory. Can be harmful for focal
    signals. Should fix the theory! STAT_THRESHOLD
    uses the better of the Bonferroni or the random
    field theory.
  • A better reason for spatial smoothing is greater
    detectability of extensive activation choose the
    FWHM to match the activation (e.g. 10mm FWHM for
    10mm activations) or try a range of FWHMs i.e.
    scale space but thresholds are higher

28
False Discovery Rate (FDR)
  • Benjamini and Hochberg (1995), Journal of the
    Royal Statistical Society
  • Benjamini and Yekutieli (2001), Annals of
    Statistics
  • Genovese 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

29
P lt 0.05 (uncorrected), Z gt 1.64 5 of volume is
false
Signal Gaussian white noise
Signal
True
Noise
False
FDR lt 0.05, Z gt 2.82 5 of discoveries is false
P lt 0.05 (corrected), Z gt 4.22 5 probability of
any false
30
Comparison of thresholds
  • FDR depends on the ordered P-values (not
    smoothness)
  • P1 lt P2 lt lt Pn. To control the FDR at a
    0.05, find
  • K max i Pi lt (i/n) a, 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 a/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

31
P lt 0.05 (uncorrected), Z gt 1.64 5 of volume is
false
FDR lt 0.05, Z gt 2.91 5 of discoveries is false
P lt 0.05 (corrected), Z gt 4.86 5 probability of
any false
Which do you prefer?
32
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33
Estimating the delay of the response
  • Delay or latency to the peak of the HRF is
    approximated by a linear combination of two
    optimally chosen basis functions

delay
basis1
basis2
HRF
shift
  • HRF(t shift) basis1(t) w1(shift)
    basis2(t) w2(shift)
  • Convolve bases with the stimulus, then add to
    the linear model

34
  • Fit linear model, estimate w1 and w2
  • Equate w2 / w1 to estimates, then solve for
    shift (Hensen et al., 2002)
  • To reduce bias when the magnitude is small, use
  • shift / (1 1/T2)
  • where T w1 / Sd(w1) is the T statistic for the
    magnitude
  • Shrinks shift to 0 where there is little
    evidence for a response.

w2 / w1
w1
w2
35
Delay of the hot stimulus ( shift 5.4 sec)
T stat for magnitude T stat for
shift
Delay (secs) Sd of delay
(secs)
36
Varying the delay and dispersion of the reference
HRF
T stat for magnitude T stat for
shift
Delay (secs) Sd of delay
(secs)
37
T gt 4.86 (P lt 0.05, corrected)
Delay (secs)
6.5 5 5.5 4 4.5
38
T gt 4.86 (P lt 0.05, corrected)
Delay (secs)
6.5 5 5.5 4 4.5
39
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40
EFFICIENCY for optimum block design
Sd of hot stimulus
Sd of hot-warm
0.5
0.5
20
20
0.4
0.4
15
15
Optimum design
Magnitude
0.3
0.3
10
10
Optimum design
0.2
0.2
X
5
5
0.1
0.1
0
X
0
0
0
5
10
15
20
5
10
15
20
InterStimulus Interval (secs)
(secs)
(secs)
1
1
20
20
0.8
0.8
15
15
Delay
0.6
0.6
Optimum design X
Optimum design X
10
10
0.4
0.4
5
5
0.2
0.2
(Not enough signal)
(Not enough signal)
0
0
0
5
10
15
20
5
10
15
20
Stimulus Duration (secs)
41
EFFICIENCY for optimum event design
0.5
(Not enough signal)
uniform . . . . . . . . .
____ magnitudes . delays
random .. . ... .. .
0.45
concentrated
0.4
0.35
0.3
Sd of effect (secs for delays)
0.25
0.2
0.15
0.1
0.05
0
5
10
15
20
Average time between events (secs)
42
How many subjects?
  • Variance sdrun2 sdsess2
    sdsubj2
  • nrun nsess nsubj
    nsess nsubj nsubj
  • The largest portion of variance comes from the
    last stage, i.e. combining over subjects.
  • If you want to optimize total scanner time, take
    more subjects, rather than more scans per
    subject.
  • What you do at early stages doesnt matter very
    much - any reasonable design will do

43
  • Comparison
  • Different slice acquisition times
  • Drift removal
  • Temporal correlation
  • Estimation of effects
  • Rationale
  • Random effects
  • FWHM
  • Map of delay
  • SPM99
  • 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, low df
  • No regularization, low df
  • Global, OK for local maxima, but not clusters
  • 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 efficient, high df
  • Regularization, high df
  • Local, is OK for local maxima and clusters
  • Yes

44
References
  • Worsley et al. (2002). A general statistical
    analysis for fMRI data. NeuroImage, 151-15.
  • Liao et al. (2002). Estimating the delay of the
    fMRI response. NeuroImage, 16593-606.
  • http//www.math.mcgill.ca/keith/fmristat -
    200K of MATLAB code
  • - fully worked example

45
Functional connectivity
  • Measured by the correlation between residuals at
    every pair of voxels (6D data!)
  • Local maxima are larger than all 12 neighbours
  • P-value can be calculated using random field
    theory
  • Good at detecting focal connectivity, but
  • PCA of residuals x voxels is better at detecting
    large regions of co-correlated voxels

Activation only
Correlation only
Voxel 2


Voxel 2







Voxel 1
Voxel 1



46
Correlations gt 0.7, Plt10-10 (corrected)
First Principal Component gt threshold
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