Mixed effects and Group Modeling for fMRI data

Mixed effects and Group Modelingfor fMRI data Thomas Nichols, Ph.D. Department of Statistics & Warwick Manufacturing Group University of Warwick Zurich SPM CourseFebruary 16, 2012

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

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

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)

Distribution of each subject’s estimated effect Fixed vs.RandomEffects in fMRI 2FFX Subj. 1 Subj. 2 • Fixed Effects • Intra-subject variation suggests all these subjects different from zero • Random Effects • Intersubject variation suggests population not very different from zero Subj. 3 Subj. 4 Subj. 5 Subj. 6 0 2RFX Distribution of population effect

Fixed Effects • Only variation (over sessions) is measurement error • True Response magnitude is fixed

Random/Mixed Effects • Two sources of variation • Measurement error • Response magnitude • Response magnitude is random • Each subject/session has random magnitude

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

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”

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

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) ~ 2n • Good: Doesn’t assume common variance • Bad: Not implemented in software Hard to interrogate statistic maps

Assessing RFX ModelsIssues 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?

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 cbi, contrast estimate • Analyze {cbi}i

^ 1 ^  ^ 2 ^  ^ 3 ^  ^ 4 ^  ^ 5 ^  ^ 6 ^  Holmes & Fristonmotivation... estimated mean activation image Fixed effects... p < 0.001 (uncorrected) — ^ •– c.f. 2 / nw SPM{t} – c.f. n – subjects w – error DF p < 0.05 (corrected) ...powerful but wrong inference SPM{t}

^ 1 ^  ^ 2 ^  ^ 3 ^  ^ 4 ^  ^ 5 ^  ^ 6 ^  Holmes & FristonRandom Effects level-one(within-subject) level-two(between-subject)  an estimate of the mixed-effects model variance 2+2/w  ^ variance 2 (no voxels significant at p < 0.05 (corrected))  — ^ •– c.f. 2/n = 2 /n + 2 / nw  – c.f.  p < 0.001 (uncorrected)  SPM{t} contrast images timecourses at [ 03, -78, 00 ]

Holmes & FristonAssumptions • Distribution • Normality • Independent subjects • Homogeneous Variance • Intrasubject variance homogeneous • 2FFX same for all subjects • Balanced designs

Holmes & FristonLimitations • 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

Holmes & FristonEfficiency • If assumptions true • Optimal, fully efficient • If 2FFX differs between subjects • Reduced efficiency • Here, optimal requires down-weighting the 3 highly variable subjects 0

Holmes & FristonValidity • If assumptions true • Exact P-values • If 2FFX differs btw subj. • Standard errors not OK • Est. of 2RFX may be biased • DF not OK • Here, 3 Ss dominate • DF < 5 = 6-1 0 2RFX

Holmes & FristonRobustness • In practice, Validity & Efficiency are excellent • For one sample case, HF almost impossible to break • 2-sample & correlation might give trouble • Dramatic imbalance or heteroscedasticity • 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.

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...

12 subjects,4 conditions Use F-test to find differences btw conditions Standard Assumptions Identical distn Independence “Sphericity”... but here not realistic! SPM8 Notation: iid case Cor(ε) = λ I y = X + e N 1 N  pp  1 N  1 X Error covariance N N

12 subjects, 4 conditions Measurements btw subjects uncorrelated Measurements w/in subjects correlated Multiple Variance Components Cor(ε) =ΣkλkQk y = X + e N 1 N  pp  1 N  1 Error covariance N N Errors can now have different variances and there can be correlations Allows for ‘nonsphericity’

Errors are not independent and not identical Non-Sphericity Modeling Error Covariance Qk’s:

Errors are independent but not identical Eg. Two Sample T Two basis elements Error Covariance Non-Sphericity Modeling Qk’s:

SPM8 Nonsphericity Modelling • Assumptions & Limitations • assumed to globallyhomogeneous • lk’s only estimated from voxels with large F • Most realistically, Cor(e) spatially heterogeneous • Intrasubject variance assumed homogeneous Cor(ε) =ΣkλkQk

SPM8 Nonsphericity Modelling • Efficiency & Power • If assumptions true, fully efficient • Validity & Robustness • P-values could be wrong (over or under) if local Cor(e) very different from globally assumed • Stronger assumptions than Holmes & Friston

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 Deviationalso useful • Variancelowest inwhite matter • Highest around ventricles

Look at the Data! • Then the functionals • Set same intensity window for all [-10 10] • Last 6 subjects good • Some variability in occipital cortex

Feel the Void! • Compare functional with anatomical to assess extent of signal voids

Conclusions • Random Effects crucial for pop. inference • When question reduces to one contrast • HF summary statistic approach • When question requires multiple contrasts • Repeated measures modelling • Look at the data!

References for fourRFX Approaches in fMRI • Holmes & Friston (HF) • Summary Statistic approach (contrasts only) • Holmes & Friston (HBM 1998). Generalisability, Random Effects & Population Inference. NI, 7(4 (2/3)):S754, 1999. • Holmes et al. (SnPM) • Permutation inference on summary statistics • Nichols & Holmes (2001). Nonparametric Permutation Tests for Functional Neuroimaging: A Primer with Examples. HBM, 15;1-25. • Holmes, Blair, Watson & Ford (1996). Nonparametric Analysis of Statistic Images from Functional Mapping Experiments. JCBFM, 16:7-22. • Friston et al. (SPM8 Nonsphericity Modelling) • Empirical Bayesian approach • Friston et al. Classical and Bayesian inference in neuroimaging: theory. NI 16(2):465-483, 2002 • Friston et al. Classical and Bayesian inference in neuroimaging: variance component estimation in fMRI. NI: 16(2):484-512, 2002. • Beckmann et al. & Woolrich et al. (FSL3) • Summary Statistics (contrast estimates and variance) • Beckmann, Jenkinson & Smith. General Multilevel linear modeling for group analysis in fMRI. NI 20(2):1052-1063 (2003) • Woolrich, Behrens et al. Multilevel linear modeling for fMRI group analysis using Bayesian inference. NI 21:1732-1747 (2004)

Mixed effects and Group Modeling for fMRI data