Group analyses of fMRI data

1. Group analyses of fMRI data Klaas Enno Stephan Translational Neuromodeling Unit (TNU)Institute for Biomedical Engineering, University of Zurich & ETH Zurich Laboratory for Social & Neural Systems Research (SNS), University of Zurich WellcomeTrust Centre for Neuroimaging, University College London With many thanks for slides & images to: FIL Methods group, particularly Will Penny & Tom Nichols Methods & models for fMRI data analysisNovember 2012

2. Overview of SPM Statistical parametric map (SPM) Design matrix Image time-series Kernel Realignment Smoothing General linear model Gaussian field theory Statistical inference Normalisation p <0.05 Template Parameter estimates

3. model specification parameter estimation hypothesis statistic Reminder: voxel-wise time series analysis! Time Time BOLD signal single voxel time series SPM

4. The model: voxel-wise GLM X + y = • Model is specified by • Design matrix X • Assumptions about e N: number of scans p: number of regressors The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.

5. GLM assumes Gaussian “spherical” (i.i.d.) errors Examples for non-sphericity: sphericity = iid:error covariance is scalar multiple of identity matrix: Cov(e) = 2I non-identity non-independence

6. Multiple covariance components at 1st level enhanced noise model error covariance components Q and hyperparameters V Q2 Q1 1 + 2 = Estimation of hyperparameters with ReML (restricted maximum likelihood).

7. t-statisticbased on ML estimates c = 1 0 0 0 0 0 0 0 0 0 0 For brevity: ReML-estimates

8. Distribution of each subject’s estimated effect Fixed vs.random effects analysis 2FFX Subj. 1 Subj. 2 • Fixed Effects • Intra-subject variation suggests most subjects different from zero • Random Effects • Inter-subject variation suggests population is not very different from zero Subj. 3 Subj. 4 Subj. 5 Subj. 6 0 2RFX Distribution of population effect

9. Fixed Effects • Assumption: variation (over subjects) is only due to measurement error • parameters are fixed properties of the population (i.e., they are the same in each subject)

10. Random/Mixed Effects • twosources of variation (over subjects) • measurementerror • Response magnitude: parameters are probabilistically distributed in the population • effect (response magnitude) in each subject is randomly distributed

11. Random/Mixed Effects • twosources of variation (over subjects) • measurementerror • Response magnitude: parameters are probabilistically distributed in the population • effect (response magnitude) in each subject is randomly distributed • variation around population mean

12. Group level inference: fixed effects (FFX) • assumes that parameters are “fixed properties of the population” • all variability is only intra-subject variability, e.g. due to measurement errors • Laird & Ware (1982): the probability distribution of the data has the same form for each individual and the same parameters • In SPM: simply concatenate the data and the design matrices  lots of power (proportional to number of scans), but results are only valid for the group studied and cannot be generalized to the population

13. Group level inference: random effects (RFX) • assumes that model parameters are probabilistically distributed in the population • variance is due to inter-subject variability • Laird & Ware (1982): the probability distribution of the data has the same form for each individual, but the parameters vary across individuals • hierarchical model much less power (proportional to number of subjects), but results can be generalized to the population

14. FFX vs. RFX • FFX is not "wrong", itmakes different assumptionsandaddresses a different questionthan RFX • Forsomequestions, FFX maybeappropriate (e.g., low-level physiologicalprocesses). • Forotherquestions, RFX ismuchmore plausible (e.g., cognitivetasks, diseaseprocesses in heterogeneouspopulations).

15. Hierachical models fMRI, single subject EEG/MEG, single subject time fMRI, multi-subject ERP/ERF, multi-subject Hierarchical models for all imaging data!

16. Linear hierarchicalmodel 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 (hyperparameters).

17. Example: Two-level model = + + = Second level First level

18. Two-level model random effects fixed effects Friston et al. 2002, NeuroImage

19. Mixed effectsanalysis Non-hierarchical model Estimating 2nd level effects Variance components at 2nd level within-level non-sphericity between-level non-sphericity Within-level non-sphericity at both levels: multiple covariance components Friston et al. 2005, NeuroImage

20. Algorithmicequivalence Parametric Empirical Bayes (PEB) Hierarchical model EM = PEB = ReML Single-level model Restricted Maximum Likelihood (ReML)

21. EstimationbyExpectationMaximisation (EM) EM-algorithm E-step M-step • E-step: finds the (conditional) expectation of the parameters, holding the hyperparameters fixed • M-step: updates the maximum likelihood estimate of the hyperparameters, keeping the parameters fixed Gauss-Newton gradientascent Friston et al. 2002, NeuroImage

22. Practicalproblems • Full MFX inference using REML or EM for a whole-brain 2-level model has enormous computational costs • for many subjects and scans, covariance matrices become extremely large • nonlinear optimisation problem for each voxel • Moreover, sometimes we are only interested in one specific effect and do not want to model all the data. • Is there a fast approximation?

23. Summary statisticsapproach: Holmes & Friston 1998 First level Second level DataDesign MatrixContrast Images SPM(t) One-sample t-test @ 2nd level

24. Validityofthesummarystatisticsapproach 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 But: Summary stats approach is fairly robust against violations of these conditions.

25. Mixed effectsanalysis: spm_mfx non-hierarchical model Summary statistics Step 1 pooling over voxels 1st level non-sphericity 2nd level non-sphericity EM approach Step 2 Friston et al. 2005, NeuroImage

26. 2nd level non-sphericitymodeling in SPM8 • 1 effect per subject • use summary statistics approach • >1 effect per subject • model sphericity at 2nd level using variance basis functions

27. Reminder: sphericity Scans „sphericity“ means: i.e. Scans

28. 2nd level: non-sphericity Error covariance • Errors are independent • but not identical: • e.g. different groups (patients, controls) • Errors are not independent • and not identical: • e.g. repeated measures for each subject (multiple basis functions, multiple conditions etc.)

29. Exampleof 2nd level non-sphericity Error Covariance Qk: ... ...

30. Exampleof 2nd level non-sphericity Cor(ε) =ΣkλkQk y= X + e N 1 N  pp  1 N  1 error covariance • 12 subjects, 4 conditions • Measurements between subjects uncorrelated • Measurements within subjects correlated • Errors can have different variances across subjects N N

31. 2nd level non-sphericitymodeling in SPM8:assumptionsandlimitations • Cor() assumed to be globally homogeneous • lk’s only estimated from voxels with large F • intrasubjectvariance assumed homogeneous

32. Practicalconclusions • Linear hierarchical models are used for group analyses of multi-subject imaging data. • The main challenge is to model non-sphericity (i.e. non-identity and non-independence of errors) within and between levels of the hierarchy. • This is done by estimating hyperparameters using EM or ReML (which are equivalent for linear models). • The summary statistics approach is robust approximation to a full mixed-effects analysis. • Use mixed-effects model only, if seriously in doubt about validity of summary statistics approach.

33. Recommended reading Linear hierarchical models Mixed effect models

34. Thank you