Group analyses of fMRI data Methods & models for fMRI data analysis 26 November 2008 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

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

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

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

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.

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

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

c = ReML- estimates t-statistic based on ML estimates For brevity:

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, can’t be generalized to the population

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 In SPM: hierarchical model  much less power (proportional to number of subjects), but results can (in principle) be generalized to the population

Recommended reading Linear hierarchical models Mixed effect models

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.

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

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

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

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

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

Mixed effects analysis Summary statistics Summary statistics EM approach EM approach Step 1 Step 2 Friston et al. 2005, NeuroImage

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?

Summary statistics approach Data Design Matrix Contrast Images SPM(t) Second level First level One-sample 2 nd level One-sample 2 nd level

Validity of the summary statistics approach The summary stats approach is exact if for each session/subject: All other cases: Summary stats approach seems to be fairly robust against typical violations. Within-session covariance the same First-level design the same One contrast per session

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

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 (like multiple basis functions) Errors are not independent and not identical: e.g. repeated measures for each subject (like multiple basis functions) Error covariance Error covariance

Example 1: non-indentical & 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.

1 st level: 2 nd level: Controls Blinds

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 affected by the semantic content of the words? What regions are affected by the semantic content of the words? Example 2: non-indentical & non-independent errors Noppeney et al. 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.

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

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

Practical conclusions 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. –Use mixed-effects model only, if seriously in doubt about validity of summary statistics approach. RFX: If not using multi-dimensional contrasts at 2nd level (F- tests), use a series of 1-sample t-tests at the 2nd level. –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.

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