Group analyses of fMRI data

1. Group analyses of fMRI data Klaas Enno Stephan Laboratory for Social and Neural Systems Research Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust 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 analysis in neuroeconomicsNovember 2010

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-statistic based 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 • Two sources of variation (over subjects) • Measurement error • Response magnitude: parameters are probabilistically distributed in the population • Response magnitude is random • effect (parameter) in each subject has random magnitude

11. Random/Mixed Effects • Two sources of variation • Measurement error • Response magnitude: parameters are probabilistically distributed in the population • Response magnitude is random • effect (parameter) in each subject has random magnitude • 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. Hierachical models fMRI, single subject EEG/MEG, single subject time fMRI, multi-subject ERP/ERF, multi-subject Hierarchical models for all imaging data!

15. 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 (hyperparameters).

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

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

18. Mixed effects analysis 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

19. Estimation EM-algorithm E-step M-step GN gradient ascent Assume, at voxel j: Friston et al. 2002, NeuroImage

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

21. Practical problems Most 2-level models are just too big to compute. And even if, it takes a long time! Moreover, sometimes we are only interested in one specific effect and do not want to model all the data. Is there a fast approximation?

22. Summary statistics approach First level Second level DataDesign MatrixContrast Images SPM(t) One-sample t-test @ 2nd level

23. Validity of the summary statistics 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 But: Summary stats approach is fairly robust against violations of these conditions.

24. Mixed effects analysis 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

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

26. 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.)

27. 2nd level: non-sphericity Cor(ε) =ΣkλkQk y = X + e N 1 N  pp  1 N  1 Error covariance • 12 subjects, 4 conditions • Measurements btw subjects uncorrelated • Measurements w/in subjects correlated N N Errors can now have different variances and there can be correlations Allows for ‘nonsphericity’

28. Example 1: non-identical & independent errors 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 Noppeney et al.

29. Controls Blinds 1st level: 2nd level:

30. Example 2: non-identical & non-independent errors Stimuli: Auditory Presentation (SOA = 4 secs) of words Subjects: (i) 12 control subjects 1. Words referred to body motion. Subjects decided if the body movement was slow. 2. Words referred to auditory features. Subjects decided if the sound was usually loud 3. Words referred to visual features. Subjects decided if the visual form was curved. 4. Words referred to hand actions. Subjects decided if the hand action involved a tool. fMRI, 250 scans per subject, block design Scanning: What regions are generally affected by the semantic content of the words? Contrast: semantic decisions > auditory decisions on reversed words (gender identification task) Question: Noppeney et al. 2003, Brain

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

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

33. Practical conclusions • 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.

34. Recommended reading Linear hierarchical models Mixed effect models

35. Thank you