1. Group analyses of fMRI data Methods & models for fMRI data analysis in neuroeconomics November 2010 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

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

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

4. The model: voxel-wise GLM = + y y X X Model is specified by 1.Design matrix X 2.Assumptions about e Model is specified by 1.Design matrix X 2.Assumptions about e N: number of scans p: number of regressors 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 sphericity = iid: error covariance is scalar multiple of identity matrix: Cov(e) =  2 I sphericity = iid: error covariance is scalar multiple of identity matrix: Cov(e) =  2 I Examples for non-sphericity: non-identity non-independence

6. Multiple covariance components at 1 st level = 1 + 2 Q1Q1 Q2Q2 Estimation of hyperparameters with ReML (restricted maximum likelihood). V enhanced noise model error covariance components Q and hyperparameters 

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

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

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. 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 Random/Mixed Effects

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

15. Linear hierarchical model Hierarchical model Multiple variance components at each level 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 Friston et al. 2002, NeuroImage fixed effects random effects

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

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

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

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 Data Design Matrix Contrast Images SPM(t) Second level First level One-sample t-test @ 2 nd level One-sample t-test @ 2 nd level

23. Validity of the summary statistics approach The summary stats approach is exact if for each session/subject: But: Summary stats approach is fairly robust against violations of these conditions. Within-session covariance the same First-level design the same One contrast per session

24. Mixed effects analysis Summary statistics Summary statistics EM approach EM approach Step 1 Step 2 Friston et al. 2005, NeuroImage non-hierarchical model 1 st level non-sphericity 2 nd level non-sphericity pooling over voxels

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

26. 2nd level: non-sphericity Errors are independent but not identical: e.g. different groups (patients, controls) 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.) Errors are not independent and not identical: e.g. repeated measures for each subject (multiple basis functions, multiple conditions etc.) Error covariance Error covariance

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

28. Example 1: non-identical & independent errors Stimuli: Auditory Presentation (SOA = 4 secs) of (i) words and (ii) words spoken backwards Auditory Presentation (SOA = 4 secs) of (i) words and (ii) words spoken backwards Subjects: e.g. “Book” and “Koob” e.g. “Book” and “Koob” fMRI, 250 scans per subject, block design Scanning: (i) 12 control subjects (ii) 11 blind subjects (i) 12 control subjects (ii) 11 blind subjects Noppeney et al.

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

30. Stimuli: Auditory Presentation (SOA = 4 secs) of words Subjects: fMRI, 250 scans per subject, block design fMRI, 250 scans per subject, block design Scanning: (i) 12 control subjects 1. Motion2. Sound3. Visual4. Action “jump”“click”“pink”“turn” Question: What regions are generally affected by the semantic content of the words? Contrast: semantic decisions > auditory decisions on reversed words (gender identification task) What regions are generally affected by the semantic content of the words? Contrast: semantic decisions > auditory decisions on reversed words (gender identification task) Example 2: non-identical & non-independent errors Noppeney et al. 2003, Brain 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.

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

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

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