1. WellcomeTrust Centre for Neuroimaging University College London
Group analyses
Will Penny
SPM Short Course, Zurich 2008
WellcomeTrust Centre for Neuroimaging
University College London

2. Overview Why hierarchical models? The model Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples

3. Overview Why hierarchical models? Why hierarchical models? The model
Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples

4. Data Time fMRI, single subject EEG/MEG, single subject
fMRI, multi-subject
ERP/ERF, multi-subject
Hierarchical model for all imaging data!

5. Reminder: voxel by voxel
model
specification
parameter
estimation
Time
hypothesis
statistic
Time
Intensity
single voxel
time series
SPM

6. Overview Why hierarchical models? The model The model Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples

7. General Linear Model = +
Model is specified by
Design matrix X
Assumptions about e
N: number of scans
p: number of regressors

8. Linear hierarchical model
Multiple variance components at each level
Hierarchical model
At each level, distribution of parameters is given by level above.
What we don’t know: distribution of parameters and variance parameters.

9. Example: Two level model= + = +
Second level
First level

10. Algorithmic Equivalence
Parametric
Empirical
Bayes (PEB)
Hierarchical
model
EM = PEB = ReML
Single-level
model
Restricted
Maximum
Likelihood
(ReML)

11. Overview Why hierarchical models? The model Estimation Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples

12. Estimation EM-algorithm E-step M-step Assume, at voxel j:
Friston et al. 2002, Neuroimage

13. And in practice? Most 2-level models are just too big to compute.
And even if, it takes a long time!
Moreover, sometimes we‘re only interested in one specific effect and don‘t want to model all the data.
Is there a fast approximation?

14. Summary Statistics approach
First level
Second level
Data Design Matrix Contrast Images
SPM(t)
One-sample
2nd level

15. Validity of approach
The summary stats approach is exact if for each session/subject:
Within-session covariance the same
First-level design the same
One contrast per session
All other cases: Summary stats approach seems to be robust against typical violations.

16. Mixed-effects Summary statistics EM approach Step 1 Step 2
Friston et al. (2004) Mixed effects and fMRI studies, Neuroimage

17. Overview Why hierarchical models? The model Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Variance components
Examples

18. Sphericity
Scans
‚sphericity‘ means:
i.e.
Scans

19. 2nd level: Non-sphericity
Error
covariance
Errors are independent
but not identical
Errors are not independent
and not identical

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

21. Population differences
1st level:
Controls
Blinds
2nd level:

22. Auditory Presentation (SOA = 4 secs) of words
Example II
Stimuli:
Auditory Presentation (SOA = 4 secs) of words
Motion
Sound
Visual
Action
“jump”
“click”
“pink”
“turn”
Subjects:
(i) 12 control subjects
fMRI, 250 scans per
subject, block design
Scanning:
What regions are affected
by the semantic content of
the words?
Question:
U. Noppeney et al.

23. Repeated measures Anova
1st level:
1.Motion
2.Sound
3.Visual
4.Action ? = ? = ? =
2nd level:

24. Repeated measures Anova
1st level:
Motion
Sound
Visual
Action ? = ? = ? =
2nd level:

25. Some practical points
RFX: If not using multi-dimensional contrasts at 2nd level (F-tests), use a series of 1-sample t-tests at the 2nd level.
Use mixed-effects model only, if seriously in doubt about validity of summary statistics approach.

26. Conclusion
Linear hierarchical models are general enough for typical multi-subject imaging data (PET, fMRI, EEG/MEG).
Summary statistics are robust approximation to mixed-effects analysis.
To minimize number of variance components to be estimated at 2nd level, compute relevant contrasts at 1st level and use simple test at 2nd level.