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

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Errors can now have different variances and there can be correlations

K

K=2

E-Step

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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

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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.

2nd Level

Controls Blinds

1st Level

Differenceof the

2 group effects

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Contrast vector for t-testCovariance

Matrix

Population Differences