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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 Methods & models for fMRI data analysis in neuroeconomics 11 November 2009 Overview of SPM Image time-series Kernel Design matrix Statistical parametric map (SPM) Realignment Smoothing General linear model Statistical Gaussian Normalisation inference field theory p <0.05 Template Parameter estimates Why hierachical models? fMRI, single subject EEG/MEG, single subject fMRI, multi-subject ERP/ERF, multi-subject Hierarchical models for all imaging data! Reminder: voxel-wise time series analysis! model specification parameter estimation Time hypothesis statistic BOLD signal single voxel time series SPM The model: voxel-wise GLM 1 p 1 1 y X e p e ~ N (0, I ) 2 y = X + e Model is specified by 1. Design matrix X 2. Assumptions about e N: number of scans N p: number of regressors N N The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds. GLM assumes Gaussian “spherical” (i.i.d.) errors sphericity = iid: Examples for non-sphericity: error covariance is scalar multiple of 4 0 identity matrix: Cov(e) Cov(e) = 2I 0 1 non-identity 2 1 Cov(e) 1 2 1 0 Cov(e) 0 1 non-independence Multiple covariance components at 1st level V Cov(e) e ~ N (0, V ) 2 V iQi enhanced noise model error covariance components Q and hyperparameters V Q1 Q2 = 1 + 2 Estimation of hyperparameters with ReML (restricted maximum likelihood). t-statistic based on ML estimates Wy WX We c Tˆ ˆ st d (cT ) ˆ t ˆ st d ( cT ) ˆ c (WX ) (WX ) c ˆ 2 T T ˆ (WX ) Wy c=10000000000 W V 1/ 2 ˆ 2 Wy WXˆ 2 V Cov(e) 2 tr( R) X R I WX (WX ) V Q i i For brevity: ReML- (WX ) ( X TWX )1 X T estimates 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 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 be generalized to the population Linear hierarchical model Hierarchical model Multiple variance components at each level y X (1) (1) (1) (1) X ( 2) ( 2) ( 2) C Q (i) (i) (i) k k k ( n 1) X ( n ) ( n ) ( n ) At each level, distribution of parameters is given by level above. What we don’t know: distribution of parameters and variance parameters (hyperparameters). Example: Two-level model 1 1 1 yX 1 2 2 2 X X 1(1) 1 2 y = X 2(1) + 1 1 = X 2 + 2 X 3(1) Second level First level Two-level model y X (1) (1) (1) (1) X (2) (2) (2) y X (1) X (2) (2) (2) (1) X (1) X (2) (2) X (1) (2) (1) fixed effects random effects Friston et al. 2002, NeuroImage Mixed effects analysis Non-hierarchical model y X (1) X (2) (2) X (1) (2) (1) ˆ(1) X (1) y Estimating 2nd level effects X (2) (2) (2) X (1) (1) X (2) (2) (2) Cov C Variance components at 2nd (1) (1) T level (2) (2) X (1) C X within-level between-level non-sphericity non-sphericity Within-level non-sphericity at k Qk( i ) (i ) (i ) both levels: multiple C covariance components k Friston et al. 2005, NeuroImage Estimation y X EM-algorithm N 1 N p p1 N 1 C | y ( X T C1 X ) 1 E-step | y C | y X C y T 1 maximise L ln p( y | λ) dL g d d 2L M-step J 2 C k Qk d k J 1 g GN gradient ascent Assume, at voxel j: jk j k Friston et al. 2002, NeuroImage Algorithmic equivalence y X (1) (1) (1) Parametric Hierarchical (1) X ( 2) ( 2) ( 2) Empirical model Bayes (PEB) ( n 1) X ( n ) ( n ) ( n ) EM = PEB = ReML Single-level y (1) X (1) ( 2 ) Restricted model ... Maximum X (1) X ( n 1) ( n ) Likelihood (ReML) X (1) X ( n ) ( n ) 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 First level Second level ˆ cT t Data Design Matrix Contrast Images ˆ Var (cT ) ˆ ˆ 1 12 ˆ SPM(t) ˆ 2 2 ˆ2 ˆ 11 11 ˆ2 ˆ 12 One-sample 12 ˆ2 t-test @ 2nd level 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. Mixed effects analysis y data non-hierarchical model X [ X (0) X (1) ] X [ X ( 0) X (1) X ( 2) ] V I Q {Q1(1) ,, X (1) Q1( 2) X (1)T ,} Summary Step 1 statistics ˆ (1) ( X T V 1 X ) 1 X T V 1 y REML{ yy T n , X , Q} pooling over voxels ˆ Y (1) X X ( 2) V (i1) X (1)Qi(1) X (1)T (j2)Q (j 2) i j 1st level 2nd level non-sphericity non-sphericity EM Step 2 approach ˆ ( 2 ) ( X T V 1 X ) 1 X T V 1 y Friston et al. 2005, NeuroImage ˆ ( 2) Reminder: sphericity y X C Cov( ) E ( ) T „sphericity“ means: Scans Cov( ) I 2 i.e. Var ( ) 2 i 1 0 Scans Cov( ) 0 1 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.) Example 1: non-identical & independent errors Stimuli: Auditory Presentation (SOA = 4 secs) of (i) words and (ii) words spoken backwards e.g. “Book” and “Koob” Subjects: (i) 12 control subjects (ii) 11 blind subjects Scanning: fMRI, 250 scans per subject, block design Noppeney et al. 1st level: Controls Blinds 2nd level: V cT [1 1] X Example 2: non-identical & non-independent errors Stimuli: Auditory Presentation (SOA = 4 secs) of words 1. Motion 2. Sound 3. Visual 4. Action “jump” “click” “pink” “turn” Subjects: (i) 12 control subjects 1. Words referred to body motion. Subjects decided if the body movement was slow. Scanning: fMRI, 250 scans per 2. Words referred to auditory features. Subjects subject, block design decided if the sound was usually loud 3. Words referred to visual features. Subjects What regions are generally decided if the visual form was curved. affected by the semantic content of the words? 4. Words referred to hand actions. Subjects decided Question: if the hand action involved a tool. Contrast: semantic decisions > auditory decisions on reversed words (gender identification task) Noppeney et al. 2003, Brain Repeated measures ANOVA 1st level: 1.Motion 2.Sound 3.Visual 4.Action ? ? ? = = = X 2nd level: Repeated measures ANOVA 1st level: 1.Motion 2.Sound 3.Visual 4.Action ? ? ? = = = X 2nd level: 1 1 0 0 cT 0 1 1 0 0 0 1 1 V X 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 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. Recommended reading Linear hierarchical models Mixed effect models Thank you