1. Random Effects Analysis
Will Penny
Wellcome Department of Imaging Neuroscience,
University College London, UK
SPM Course, London, May 2004

2. Summary Statistic Approach
1st Level nd Level
Data Design Matrix Contrast Images 1 ^
SPM(t)
1 ^ 2 ^
2 ^
11 ^
11 ^  ^
One-sample
level
12 ^
12 ^

3. Validity of approach ^ ^ Effect size
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

4. Example: Multi-session study of auditory processing
SS results
EM results
Friston et al. (2004) Mixed effects and fMRI studies, Submitted.

5. Two populations Estimated population means Contrast images Two-sample
level
Patients
Controls
One or two
variance
components ?

6. The General Linear Model
y = X  + e
N  N  L L  N  1
Error covariance N
2 Basic Assumptions
Identity
Independence N

7. Multiple variance componentsK
y = X  + e
N  N  L L  N  1 =1
Error covariance N
Errors can now have
different variances and
there can be correlations N
K=2

8. Estimating variances y = X  + e EM algorithm
E-Step ( ) y C X T 1 - = e q h
M-Step r
for i and j { } { Q tr J g i j ij k å + l
y = X  + e
N  N  L L  N  1
Friston, K. et al. (2002), Neuroimage

9. Example I U. Noppeney et al.
Stimuli: Auditory Presentation (SOA = 4 secs) of
(i) words and (ii) words spoken backwards
jump
Eg. “Book” and “Koob”
touch
“click”
Subjects: (i) 12 control subjects
(ii) 11 blind subjects
Scanning: fMRI, 250 scans per subject, block design

10. } } Population Differences Covariance Matrix Design matrix Controls
Blinds
1st Level
2nd Level }
Contrast vector
for t-test
Covariance
Matrix }
Design matrix
Difference
of the
2 group effects