Contents. Data. Introduction & recap Variance components Hierarchical model RFX and summary statistics Variance/covariance matrix «Take home» message

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1 SPM course, CRC, Liege,, Septembre 2009 Contents Group analysis (RF) Variance components Hierarchical model RF and summary statistics Variance/covariance matrix «Tae home» message C. Phillips, Centre de Recherches du Cyclotron, ULg, Belgium Based on slides from: T. ichols, S. Kiebel, JB. Poline Data Reminder: : voxel by voxel analysis fmri, single subject Time EEG/MEG, single subject Time model specification parameter estimation hypothesis statistic fmri, multi-subject ERP/ERF, multi-subject Hierarchical model for all imaging data! Time single voxel time series Intensity SPM

2 1 General Linear Model p 1 1 Contents y = p + ε y = Error Covariance C ε = λ Q Variance components Hierarchical model RF and summary statistics Variance/covariance matrix «Tae home» message : number of scans p: number of regressors Model is specified by 1. Design matrix 2. Assumptions about ε Random effects & variance components x Subject1 Subject2 Subject3 Subject1 Subject2 Subject3 residuals Fixed effects Are you confident that a new observation from any of of subjects will be around their mean? Yes! using within-subjects variance infer for these subjects case study Random effects Are you confident that a new observation from a new subject will be around the mean of of first 3? o! using between-subjects variance infer for any subject population Fixed vs. Random effects Fixed Effects Intra-subject variation suggests all these subjects different from zero Random Effects Intersubject variation suggests population not very different from zero Distribution of each subject s effect Subj. 1 Subj. 2 Subj. 3 Subj. 4 Subj. 5 Subj. 6 0 σ 2 FF σ 2 RF

3 Fixed vs. Random Contents Fixed isn t wrong, just usually isn t t of interest as limited to case study. Fixed Effects Inference I I can see this effect in this cohort Random Effects Inference If I were to sample a new cohort from the population I would get the same result Variance components Hierarchical model RF and summary statistics Variance/covariance matrix «Tae home» message Hierarchical model y = = ( n 1) = (2) (2) (2) M Hierarchical model Multiple variance components at each level C At each level, distribution of parameters is given by level above. = λ Q ( i ) ( i ) ( i ) ε y y = = () 1 ( 1) ( 1) ( 1) ( 2) ( 2) ( 2) = 1 Example: two level model 2 ( 1) ( 1 ) ( 1) = ( 2) ( 2 ) ( 2) What we don t now: distribution of parameters and variance parameters. 3 First level Second level

4 Estimation Group analysis in in practice Hierarchical model y = ( n 1) = = (2) M (2) (2) Many 2-level models are just too big to compute. And even if, it taes a long time! Single-level model y (2) = ε + ε K ( n 1) K = + e ε + Is there a fast approximation? Contents Multi-subject analysis? Variance components Hierarchical model RF and summary statistics Variance/covariance matrix «Tae home» message α 1 α 2 α 3 α 4 α 5 estimated mean activation image α c.f. / nw c.f. p < (uncorrected) SPM{t} p < 0.05 (corrected) α 6 SPM{t}

5 Two-stage analysis of random effect level-one (within-subject) α 1 α 2 α 3 α 4 α 5 α 6 timecourses at [ 03, -78, 00 ] contrast images level-two (between-subject) variance σ 2 an estimate of the mixed-effects model variance σ 2 α + σ2 ε / w (no voxels significant at p < 0.05 (corrected)) α c.f. σ 2 /n = σ 2 α /n + σ2 ε / nw c.f. p < (uncorrected) SPM{t} vs. Two stage random effects group comparison level-one (within-subject) 12 subjects two-sample t-test contrast images level-two (between-subject) Summary Auditory Data Analyse subjects individually Build within-subject models Calculate contrast(s) of interest Use contrast images in a 2 level (Random Effect,, RF) analysis Build between-subject model Calculates SPMs of interest Draw conclusions for the population Summary statistics Hierarchical Model Friston Fristonet et al. al. (2004) (2004) Mixed Mixed effects effects and and fmri fmri studies, studies, euroimage euroimage

6 Contents Variance-Covariance matrix Height of Swedish men Weight of Swedish men Variance components Hierarchical model RF and summary statistics Variance/covariance matrix «Tae home» message μ=180cm, σ=14cm (σ 2 =200) μ=80g, σ=14g (σ 2 =200) Each completely characterised by μ (mean) and σ 2 (variance), i.e. we can calculate p(l μ,σ 2 ) for any l Source: J. Andersson Variance-Covariance matrix non-sphericity 4 Cov(ε ) = ow let us view height and weight as a 2-2 dimensional stochastic variable (p(l,w)). non-sphericity means that that the theerrorcovariancedoesn t error loo loolie liethis: 2 Cov( ε ) = σ I μ = Σ = p(l,w μ,σ) Source: J. Andersson 1 Cov(ε ) = Cov(ε ) = 1 1 2

7 Variance quiz Variance quiz Height Height Weight Weight # hours watching telly per day # hours watching telly per day Variance quiz Variance quiz Height Weight # hours watching telly per day Shoe size Height Weight # hours watching telly per day Shoe size

8 Example I Population differences Stimuli: Auditory Presentation (SOA = 4 secs) of of (i) (i) words and and (ii) (ii) words spoen bacwards 1 st st level: Controls Blinds Subjects: e.g. e.g. Boo and and Koob (i) (i) control subjects (ii) (ii) blind subjects 2 nd nd level: V T c = [ 1 1] Scanning: fmri, scans per per subject, bloc design U. U. oppeney et et al. al. Example II SPM2 otation: iid case Stimuli: Subjects: Scanning: Question: Auditory Presentation (SOA = 4 secs) of of words Motion jump Sound clic (i) (i) control subjects fmri, scans per per subject, bloc design What regions are are affected by by the the semantic content of of the the words? Visual pin Action turn U. U. oppeney et et al. al. y = 1 p p subjects, 4 conditions Use F-test F to find differences btw conditions Standard Assumptions Identical distribution Independence Sphericity... but here not realistic! Cor( ε ) = λi Error covariance

9 Multiple Variance Components y = 1 p p subjects, 4 conditions Measurements btw subjects uncorrelated Measurements w/in subjects correlated Errors can now have different variances and there can be correlations Allows for nonsphericity Cor(ε) = λ Q Error covariance 1 st st level: 2 nd nd level: Repeated measures Anova Motion? = Sound V? = Visual? = T c Action 1 1 = Contents Summary Design efficiency Experimental design Random effect analysis «Tae home» message Linear hierarchical models are general enough for typical multi-subject imaging data (PET, fmri, EEG/MEG). Summary statistics are robust approximation for group analysis. Also accomodates multiple contrasts per subject.

10 Regression example Regression example = β 1 + β 2 + = β + β β 1 = 1 β 2 = 100 Fit the GLM β 1 = 5 β 2 = 100 Fit the GLM voxel time series box-car reference functionmean value voxel time series box-car reference functionmean value otes! Coefficients (= parameters) are estimated using the Ordinary Least Squares (OLS) by minimizing the fluctuations, - variability variance of the noise the residuals Because the parameters depend on the scaling of the regressors included in the model, one should be careful in comparing manually entered regressors The residuals, their sum of squares s and the resulting tests (t,f), do not depend on the scaling of the regressors. Top Ten Things Sex and Brain Imaging Have in Common 10. It's not how big the region is, it's what you do with it. 9. Both involve heavy PETting or powerful magnetism. 8. It's important to select regions of interest. 7. Experts agree that timing is critical. 6. Both require correction for motion. 5. Experimentation is everything. 4. You often can't get access when you need it. 3. You always hope for multiple activations. 2. Both mae a lot of noise. 1. Both are better when the assumptions are met.