1. Group analyses of fMRI data Methods & models for fMRI data analysis November 2012 With many thanks for slides & images to: FIL Methods group, particularly Will Penny & Tom Nichols 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 Wellcome Trust Centre for Neuroimaging, University College London

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. two sources of variation (over subjects) –measurement error –Response magnitude: parameters are probabilistically distributed in the population effect (response magnitude) in each subject is randomly distributed Random/Mixed Effects

11. two sources of variation (over subjects) –measurement error –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", it makes different assumptions and addresses a different question than RFX For some questions, FFX may be appropriate (e.g., low-level physiological processes). For other questions, RFX is much more plausible (e.g., cognitive tasks, disease processes in heterogeneous populations).

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

16. 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).

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

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

19. 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

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. Estimation by Expectation Maximisation (EM) EM-algorithm E-step M-step Friston et al. 2002, NeuroImage Gauss-Newton gradient ascent 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

22. Practical problems 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 statistics approach: Holmes & Friston 1998 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

24. 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

25. Mixed effects analysis: spm_mfx 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

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

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

28. 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

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

30. Example of 2nd level non-sphericity 30 y = X  + e N  1 N  p p  1 N  1 N N error covariance 12 subjects, 4 conditions Measurements between subjects uncorrelated Measurements within subjects correlated Errors can have different variances across subjects Cor(ε) =Σ k λ k Q k

31. 2nd level non-sphericity modeling in SPM8: assumptions and limitations Cor(  ) assumed to be globally homogeneous k ’s only estimated from voxels with large F intrasubject variance assumed homogeneous

32. 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.

33. Recommended reading Linear hierarchical models Mixed effect models

34. Thank you