Wellcome Dept. of Imaging Neuroscience University College London Hierarchical Models Stefan Kiebel Wellcome Dept. of Imaging Neuroscience 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 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.

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

1st level fMRI: serial correlations with autoregressive process of order 1 (AR-1) autocovariance- function COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!

Restricted Maximum Likelihood observed ReML estimated COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!

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

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/

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

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: Motion Sound Visual Action ? = ? = ? = 2nd level:

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

Test for difference in N170 ERP-specific contrasts: Average between 150 and 190 ms Example: difference between trial types 2nd level contrast -1 1 . . . = = + Identity matrix second level first 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. If using variance components at 2nd level: Always check your variance components (explore design).

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.