Group analyses

1. Group analyses Will Penny 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 fMRI, single subject EEG/MEG, single subject Time 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 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 DataDesign MatrixContrast Images SPM(t) One-sample t-test @ 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 Step 1 EM approach 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 II Stimuli: Auditory Presentation (SOA = 4 secs) of words 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.

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

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

23. 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. If using variance components at 2nd level: Always check your variance components (explore design).

24. 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.