# Overview of Hierarchical Modeling

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) SPM8 Nonsphericity Modelling

(SPM96,99,2,5,8) Data exploration Conclusions 2 Overview Mixed effects motivation Evaluating mixed effects methods Two methods Summary statistic approach (HF) SPM8 Nonsphericity Modelling (SPM96,99,2,5,8) 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 Fixed vs. Random Effects in fMRI 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 subjects estimated effect 2FFX Subj. 1 Subj. 2 Subj. 3 Subj. 4 Subj. 5 Subj. 6

0 2RFX Distribution of population effect 6 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 isnt wrong, just usually isnt 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) ~ 2n Good: Doesnt 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) SPM8 Nonsphericity Modelling (SPM96,99,2,5,8) 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) SPM8 Nonsphericity Modelling

(SPM96,99,2,5,8) Data exploration Conclusions 19 Overview Mixed effects motivation Evaluating mixed effects methods Two methods Summary statistic approach (HF) SPM8 Nonsphericity Modelling (SPM96,99,2,5,8) 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 cbi, contrast estimate Analyze {cbi}i 21 Holmes & Friston motivation... Fixed effects... estimated mean activation image ^

1 ^ ^ 2 ^ ^ 3 ^ ^

4 ^ ^ 5 p < 0.001 (uncorrected) ^ c.f. 2 / nw c.f. n subjects w error DF

p < 0.05 (corrected) ^ ^ 6 ^ SPM{t} ...powerful but wrong inference SPM{t}

22 level-one Holmes & Friston Random Effects level-two (within-subject) ^ 1 (between-subject) ^

^ 2 ^ ^ 3 (no voxels significant at p < 0.05 (corrected)) ^

^ 4 variance ^2 an estimate of the mixed-effects model variance 2 + 2 / w ^

^ c.f. 2/n = 2 /n + 2 / nw c.f. ^ 5 ^ ^ 6 ^

timecourses at [ 03, -78, 00 ] p < 0.001 (uncorrected) contrast images SPM{t} 23 Holmes & Friston Assumptions Distribution Normality Independent subjects Homogeneous Variance

Intrasubject variance homogeneous 2FFX 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 2FFX differs between subjects Reduced efficiency Here, optimal requires down-weighting the 3 highly variable subjects 0

26 Holmes & Friston Validity 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

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 Dramatic imbalance or heteroscedasticity

28 Overview Mixed effects motivation Evaluating mixed effects methods Two methods Summary statistic approach (HF) SPM8 Nonsphericity Modelling (SPM96,99,2,5,8) Data exploration Conclusions 29 SPM8 Nonsphericity Modelling 1 effect per subject

Uses Holmes & Friston approach >1 effect per subject Cant use HF; must use SPM8 Nonsphericity Modelling Variance basis function approach used... 30 SPM8 Notation: iid case Cor() = I y = X + N1 Np p1

N1 12 subjects, 4 conditions X Error covariance N Use F-test to find differences btw conditions Standard Assumptions Identical distn Independence Sphericity... but here not realistic!

N 31 Multiple Variance Components Cor() =k kQk y = X + N1 Np p1 N1 Error covariance 12 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 N N 32 Non-Sphericity Modeling Errors are not independent and not identical

Error Covariance Qks: 33 Non-Sphericity Modeling Errors are independent but not identical Eg. Two Sample T Two basis elements Qks: Error Covariance 34

SPM8 Nonsphericity Modelling Assumptions & Limitations Cor() =k kQk assumed to globally homogeneous ks 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) SPM8 Nonsphericity Modelling (SPM96,99,2,5,8) 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, 646430 volumes, 334mm vx. Experiment (Block Design only) Passive sentence listening 22 Factorial Design Sentence Effect: Same sentence repeated vs different Speaker Effect: Same speaker vs. different Analysis

Slice time correction, motion correction, sptl. norm. 555 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 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! 51 52 References for four RFX 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. 53 NI 21:1732-1747 (2004)

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