random effects analysis
DESCRIPTION
Random Effects Analysis. Will Penny. Wellcome Department of Imaging Neuroscience, University College London, UK. SPM Course, London, May 2004. ^. ^. ^. ^. ^. 11. 12. . 1. 2. ^. ^. ^. ^. 2. 12. 1. 11. Summary Statistic Approach. - PowerPoint PPT PresentationTRANSCRIPT
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Random Effects AnalysisRandom Effects Analysis
Will Penny
Wellcome Department of Imaging Neuroscience, University College London, UK
SPM Course, London, May 2004
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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
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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^
^
Effect size
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Example: Multi-session study of auditory processing
SS results EM results
Friston et al. (2004) Mixed effects and fMRI studies, Submitted.
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Two populations
Contrast images
Estimatedpopulation means
Two-samplet-test @2nd level
Patients
ControlsOne or twovariancecomponents ?
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y = X + N 1 N L L 1 N 1
2 Basic AssumptionsIdentityIndependence
The General Linear Model
IC
N
N
Error covariance
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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
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E-Step
yCXC
XCXC
T
yy
T
y
1
11
M-Stepy
Xyr
for i and j {
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11
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CQCQtrJ
XCQCXCtrrCQCrCQtrg
ijij
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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
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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.
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2nd Level
Controls Blinds
1st Level
Differenceof the
2 group effects
}
} Design matrix
Contrast vector for t-testCovariance
Matrix
Population Differences