Mixed effects and Group Modeling for fMRI data Thomas Nichols, Ph.D. Department of Statistics & Warwick Manufacturing Group University of Warwick Zurich SPM Course February 16, 2012 1
Outline • Mixed effects motivation • Evaluating mixed effects methods • Two methods – Summary statistic approach (HF) (SPM96,99,2,5,8) – SPM8 Nonsphericity Modelling • Data exploration • Conclusions 2
Overview • Mixed effects motivation • Evaluating mixed effects methods • Two methods – Summary statistic approach (HF) (SPM96,99,2,5,8) – SPM8 Nonsphericity Modelling • Data exploration • Conclusions 3
Lexicon Hierarchical Models • Mixed Effects Models • Random Effects (RFX) Models • Components of Variance ... all the same ... all alluding to multiple sources of variation (in contrast to fixed effects) 4
Distribution of Fixed vs. each subject ’ s Random estimated effect 2 FFX Effects in fMRI Subj. 1 • Fixed Effects Subj. 2 – Intra-subject Subj. 3 variation suggests Subj. 4 all these subjects Subj. 5 different from zero • Random Effects Subj. 6 0 – Intersubject 2 variation suggests RFX population not very different from Distribution of zero 6 population effect
Fixed Effects • Only variation (over sessions) is measurement error • True Response magnitude is fixed 7
Random/Mixed Effects • Two sources of variation – Measurement error – Response magnitude • Response magnitude is random – Each subject/session has random magnitude – 8
Random/Mixed Effects • Two sources of variation – Measurement error – Response magnitude • Response magnitude is random – Each subject/session has random magnitude – But note, population mean magnitude is fixed 9
Fixed vs. Random • Fixed isn ’ t “ wrong, ” just usually isn ’ t of interest • Fixed Effects Inference – “ 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 ” 10
Two Different Fixed Effects Approaches • Grand GLM approach – Model all subjects at once – Good: Mondo DF – Good: Can simplify modeling – Bad: Assumes common variance over subjects at each voxel – Bad: Huge amount of data 11
Two Different Fixed Effects Approaches • Meta Analysis approach – Model each subject individually – Combine set of T statistics • mean(T) n ~ N(0,1) • sum(-logP) ~ 2 n – Good: Doesn ’ t assume common variance – Bad: Not implemented in software Hard to interrogate statistic maps 12
Overview • Mixed effects motivation • Evaluating mixed effects methods • Two methods – Summary statistic approach (HF) (SPM96,99,2,5,8) – SPM8 Nonsphericity Modelling • Data exploration • Conclusions 13
Assessing RFX Models Issues to Consider • Assumptions & Limitations – What must I assume? • Independence? • “ Nonsphericity ” ? (aka independence + homogeneous var.) – When can I use it • Efficiency & Power – How sensitive is it? • Validity & Robustness – Can I trust the P-values? – Are the standard errors correct? – If assumptions off, things still OK? 14
Overview • Mixed effects motivation • Evaluating mixed effects methods • Two methods – Summary statistic approach (HF) (SPM96,99,2,5,8) – SPM8 Nonsphericity Modelling • Data exploration • Conclusions 19
Overview • Mixed effects motivation • Evaluating mixed effects methods • Two methods – Summary statistic approach (HF) (SPM96,99,2,5,8) – SPM8 Nonsphericity Modelling • Data exploration • Conclusions 20
Holmes & Friston • Unweighted summary statistic approach • 1- or 2-sample t test on contrast images – Intrasubject variance images not used (c.f. FSL) • Proceedure – Fit GLM for each subject i – Compute cb i , contrast estimate – Analyze { cb i } i 21
Holmes & Friston motivation... estimated mean Fixed effects... activation image 1 ^ ^ 2 ^ ^ p < 0.001 (uncorrected) — • – c.f. 2 ^ / nw 3 ^ SPM{ t } – c.f. ^ 4 ^ n – subjects ^ w – error DF 5 ^ p < 0.05 (corrected) ^ ...powerful but SPM{ t } 6 ^ wrong inference 22 ^
Holmes & Friston Random Effects level-one level-two (within-subject) (between-subject) ^ 1 an estimate of the ^ mixed-effects model variance 2 ^ 2 + 2 variance 2 ^ / w ^ (no voxels significant at p < 0.05 (corrected) ) 3 ^ ^ — • – c.f. 2 / n = 2 ^ / n + 2 / nw 4 ^ ^ – c.f. 5 ^ ^ p < 0.001 (uncorrected) 6 ^ SPM{ t } ^ 23 timecourses at [ 03, -78, 00 ] contrast images
Holmes & Friston Assumptions • Distribution – Normality – Independent subjects • Homogeneous Variance – Intrasubject variance homogeneous • 2 FFX same for all subjects – Balanced designs 24
Holmes & Friston Limitations • Limitations – Only single image per subject – If 2 or more conditions, Must run separate model for each contrast • Limitation a strength! – No sphericity assumption made on different conditions when each is fit with separate model 25
Holmes & Friston Efficiency • If assumptions true – Optimal, fully efficient • If 2 FFX differs between subjects – Reduced efficiency ˆ 0 – Here, optimal requires down-weighting the 3 highly variable subjects ˆ 26
Holmes & Friston Validity • If assumptions true – Exact P-values • If 2 FFX differs btw subj. – Standard errors not OK • Est. of 2 RFX may be biased 0 – DF not OK 2 RFX • Here, 3 Ss dominate • DF < 5 = 6-1 27
Holmes & Friston Robustness • In practice, Validity & Efficiency are excellent – For one sample case, HF almost impossible to break False Positive Rate Power Relative to Optimal (outlier severity) (outlier severity) Mumford & Nichols. Simple group fMRI modeling and inference. Neuroimage , 47(4):1469--1475, 2009. • 2-sample & correlation might give trouble 28 – Dramatic imbalance or heteroscedasticity
Overview • Mixed effects motivation • Evaluating mixed effects methods • Two methods – Summary statistic approach (HF) (SPM96,99,2,5,8) – SPM8 Nonsphericity Modelling • Data exploration • Conclusions 29
SPM8 Nonsphericity Modelling • 1 effect per subject – Uses Holmes & Friston approach • >1 effect per subject – Can ’ t use HF; must use SPM8 Nonsphericity Modelling – Variance basis function approach used... 30
SPM8 Notation: iid case y = X + Cor( ε ) = λ I N 1 N p p 1 N 1 X • 12 subjects, Error covariance 4 conditions N – Use F-test to find differences btw conditions • Standard Assumptions N – Identical distn – Independence – “ Sphericity ” ... but here 31 not realistic!
Multiple Variance Components y = X + Cor( ε ) =Σ k λ k Q k N 1 N p p 1 N 1 Error covariance • 12 subjects, 4 conditions N • Measurements btw subjects uncorrelated • Measurements w/in subjects correlated N Errors can now have different variances and there can be correlations Allows for ‘ nonsphericity ’ 32
Non-Sphericity Modeling • Error Covariance Errors are not independent and not identical Q k ’ s: 33
Non-Sphericity Modeling Q k ’ s: • Errors are independent but not identical Error Covariance – Eg. Two Sample T Two basis elements 34
SPM8 Nonsphericity Modelling • Assumptions & Limitations Cor( ε ) =Σ k λ k Q k – assumed to globally homogeneous – l k ’ s only estimated from voxels with large F – Most realistically, Cor( ) spatially heterogeneous – Intrasubject variance assumed homogeneous 35
SPM8 Nonsphericity Modelling • Efficiency & Power – If assumptions true, fully efficient • Validity & Robustness – P-values could be wrong (over or under) if local Cor( ) very different from globally assumed – Stronger assumptions than Holmes & Friston 36
Overview • Mixed effects motivation • Evaluating mixed effects methods • Two methods – Summary statistic approach (HF) (SPM96,99,2,5,8) – SPM8 Nonsphericity Modelling • Data exploration • Conclusions 44
Data: FIAC Data • Acquisition – 3 TE Bruker Magnet – For each subject: 2 (block design) sessions, 195 EPI images each – TR=2.5s, TE=35ms, 64 64 30 volumes, 3 3 4mm vx. • Experiment (Block Design only) – Passive sentence listening – 2 2 Factorial Design • Sentence Effect: Same sentence repeated vs different • Speaker Effect: Same speaker vs. different • Analysis – Slice time correction, motion correction, sptl. norm. – 5 5 5 mm FWHM Gaussian smoothing – Box-car convolved w/ canonical HRF – Drift fit with DCT, 1/128Hz
Look at the Data! • With small n , really can do it! • Start with anatomical – Alignment OK? • Yup – Any horrible anatomical anomalies? • Nope
Look at the Data! • Mean & Standard Deviation also useful – Variance lowest in white matter – Highest around ventricles
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