Random Effects Analysis

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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.

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Random Effects Analysis

Will Penny

Wellcome Department of Imaging Neuroscience,

University College London, UK

SPM Course, London, May 2004

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11

12

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2

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12

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Summary Statistic Approach

1st Level 2nd Level

SPM(t)

One-sample

t-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 variance

we have

VAREM = Var-between/N + Var-within/Nn

• In this case, the SS approach gives the

same results, on average:

Avg[a] = MEANEM

Avg[Var(a)] =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

Estimated

population

means

Contrast images

Two-sample

t-test @2nd level

Patients

Controls

One or two

variance

components ?

The General Linear Model

y = X + e

N 1 N  L L  1 N  1

Error covariance

N

2 Basic Assumptions

• Identity
• Independence

N

y = X + e

N 1 N  L L  1 N  1

Multiple variance components

K

=1

Error covariance

N

Errors can now have

different variances and

there can be correlations

N

K=2

y = X + e

N 1 N  L L  1 N  1

(

)

-

1

-

=

T

1

C

X

C

X

e

q

E-Step

y

-

h

=

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X

C

y

e

q

q

y

y

for i and j {

=

-

h

r

y

X

q

y

M-Step

-

-

-

-

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=

-

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1

T

1

1

T

1

1

g

tr

{

Q

C

}

r

C

Q

C

r

tr

{

C

X

C

Q

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}

e

e

e

e

e

q

i

i

i

i

y

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1

1

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tr

{

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e

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ij

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l

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å

=

+

l

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Q

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k

k

Estimating variances

EM algorithm

Friston, K. et al. (2002), Neuroimage

Example I

U. Noppeney et al.

Stimuli:Auditory Presentation (SOA = 4 secs) of

(i) words and (ii) words spoken backwards

Subjects: (i) 12 control subjects

(ii) 11 blind subjects

jump

Eg. “Book” and “Koob”

touch

“click”

Scanning: fMRI, 250 scans per subject, block design

Population Differences

Controls

Blinds

1st Level

2nd Level

}

Contrast vector

for t-test

Covariance

Matrix

}

Design matrix

Difference

of the

2 group effects