WellcomeTrust Centre for Neuroimaging University College London Group analyses Will Penny SPM Short Course, Zurich 2008 WellcomeTrust Centre for Neuroimaging University College London
Overview Why hierarchical models? The model Estimation Expectation-Maximization Summary Statistics approach Variance components Examples
Overview Why hierarchical models? Why hierarchical models? The model Estimation Expectation-Maximization Summary Statistics approach Variance components Examples
Data Time fMRI, single subject EEG/MEG, single subject fMRI, multi-subject ERP/ERF, multi-subject Hierarchical model for all imaging data!
Reminder: voxel by voxel model specification parameter estimation Time hypothesis statistic Time Intensity single voxel time series SPM
Overview Why hierarchical models? The model The model Estimation Expectation-Maximization Summary Statistics approach Variance components Examples
General Linear Model = + Model is specified by Design matrix X Assumptions about e N: number of scans p: number of regressors
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.
Example: Two level model = + = + Second level First level
Algorithmic Equivalence Parametric Empirical Bayes (PEB) Hierarchical model EM = PEB = ReML Single-level model Restricted Maximum Likelihood (ReML)
Overview Why hierarchical models? The model Estimation Estimation Expectation-Maximization Summary Statistics approach Variance components Examples
Estimation EM-algorithm E-step M-step Assume, at voxel j: Friston et al. 2002, Neuroimage
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?
Summary Statistics approach First level Second level Data Design Matrix Contrast Images SPM(t) One-sample t-test @ 2nd level
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.
Mixed-effects Summary statistics EM approach Step 1 Step 2 Friston et al. (2004) Mixed effects and fMRI studies, Neuroimage
Overview Why hierarchical models? The model Estimation Expectation-Maximization Summary Statistics approach Variance components Variance components Examples
Sphericity Scans ‚sphericity‘ means: i.e. Scans
2nd level: Non-sphericity Error covariance Errors are independent but not identical Errors are not independent and not identical
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.
Population differences 1st level: Controls Blinds 2nd level:
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.
Repeated measures Anova 1st level: 1.Motion 2.Sound 3.Visual 4.Action ? = ? = ? = 2nd level:
Repeated measures Anova 1st level: Motion Sound Visual Action ? = ? = ? = 2nd level:
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.
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.