Mixed effects and Group Modeling for fMRI data Thomas Nichols, - PowerPoint PPT Presentation

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