Hierarchical Models

1. Hierarchical Models Stefan Kiebel Wellcome Dept. of Imaging Neuroscience 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 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 DataDesign MatrixContrast Images SPM(t) One-sample t-test @ 2nd level

15. Validity of approach The summary stats approach is exact under i.i.d. assumptions at the 2nd level, if for each session: 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. 1st level fMRI: serial correlations with autoregressive process of order 1 (AR-1) autocovariance- function

21. Restricted Maximum Likelihood observed ReML estimated

22. Estimation in SPM5 ReML (pooled estimate) Ordinary least-squares Maximum Likelihood • optional in SPM5 • one pass through data • statistic using effective degrees of freedom • 2 passes (first pass for selection of voxels) • more accurate estimate of V

23. 2nd level: Non-sphericity • Errors can be • non-independent and non-identical • when… Several parameters per subject, e.g. repeated measures design Complete characterization of the hemodynamic response, e.g. taking more than 1 HRF parameter to 2nd level Example data set (R Henson) on: http://www.fil.ion.ucl.ac.uk/spm/data/

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

25. Example I 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.

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

27. 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.

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

29. Example 3: 2-level ERP model Single subject Multiple subjects Subject 1 Trial type 1 . . . . . . Subject j Trial type i . . . . . . Subject m Trial type n

30. Test for difference in N170 Example: difference between trial types ERP-specific contrasts: Average between 150 and 190 ms 2nd level contrast -1 1 . . . = = + Identity matrix second level first level

31. 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).

32. 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.