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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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Presentation Transcript
slide1

Random Effects Analysis

Will Penny

Wellcome Department of Imaging Neuroscience,

University College London, UK

SPM Course, London, May 2004

slide2

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11

12

1

2

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^

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^

2

12

1

11

Summary Statistic Approach

1st Level 2nd Level

DataDesign MatrixContrast Images

SPM(t)

One-sample

t-test @2nd level

validity of approach
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

^

^

Effect size

example multi session study of auditory processing
Example: Multi-session study of auditory processing

SS results

EM results

Friston et al. (2004) Mixed effects and fMRI studies, Submitted.

two populations
Two populations

Estimated

population

means

Contrast images

Two-sample

t-test @2nd level

Patients

Controls

One or two

variance

components ?

slide6

The General Linear Model

y = X + e

N 1 N  L L  1 N  1

Error covariance

N

2 Basic Assumptions

  • Identity
  • Independence

N

slide7

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

slide8

y = X + e

N 1 N  L L  1 N  1

(

)

-

1

-

=

T

1

C

X

C

X

e

q

E-Step

y

-

h

=

T

1

C

X

C

y

e

q

q

y

y

for i and j {

=

-

h

r

y

X

q

y

M-Step

-

-

-

-

-

=

-

-

1

T

1

1

T

1

1

g

tr

{

Q

C

}

r

C

Q

C

r

tr

{

C

X

C

Q

C

X

}

e

e

e

e

e

q

i

i

i

i

y

-

-

=

}

1

1

J

tr

{

Q

C

Q

C

}

e

e

ij

j

i

-

l

=

l

-

1

J

g

å

=

+

l

C

C

Q

e

q

k

k

Estimating variances

EM algorithm

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

slide9

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

slide10

Population Differences

Controls

Blinds

1st Level

2nd Level

}

Contrast vector

for t-test

Covariance

Matrix

}

Design matrix

Difference

of the

2 group effects