Random Effects Analysis
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Transcript of Random Effects Analysis
Random Effects AnalysisRandom Effects Analysis
Will Penny
Wellcome Department of Imaging Neuroscience, University College London, UK
SPM Course, London, May 2004
1st Level 2nd Level
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Data Design Matrix Contrast Images
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Summary Statistic Approach
SPM(t)
One-samplet-test @2nd level
Validity of approach Gold Standard approach is EM – see later –
estimates population mean effect as MEANEM
the variance of this estimate as VAREM
For N subjects, n scans per subject and equal within-subject variancewe have
VAREM = Var-between/N + Var-within/Nn
In this case, the SS approach gives the same results, on average:
Avg[MEANEM
Avg[Var()] =VAREM^
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Effect size
Example: Multi-session study of auditory processing
SS results EM results
Friston et al. (2004) Mixed effects and fMRI studies, Submitted.
Two populations
Contrast images
Estimatedpopulation means
Two-samplet-test @2nd level
Patients
ControlsOne or twovariancecomponents ?
y = X + N 1 N L L 1 N 1
2 Basic AssumptionsIdentityIndependence
The General Linear Model
IC
N
N
Error covariance
y = X + N 1 N L L 1 N 1
Multiple variance components
N
N
Error covariance
QC kk
k
Errors can now have different variances and there can be correlations
K
K=2
E-Step
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11
M-Stepy
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for i and j {
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Friston, K. et al. (2002), Neuroimage
EM algorithmEstimating variances
y = X + N 1 N L L 1 N 1
QC kk
k
jump touch
Eg. “Book” and “Koob”
Stimuli: Auditory Presentation (SOA = 4 secs) of
(i) words and (ii) words spoken backwards
Subjects: (i) 12 control subjects(ii) 11 blind subjects
Scanning: fMRI, 250 scans per subject, block design
Example I
“click”
U. Noppeney et al.
2nd Level
Controls Blinds
1st Level
Differenceof the
2 group effects
}
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Contrast vector for t-testCovariance
Matrix
Population Differences