1 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 18, 2010

2 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

3 Overview Mixed effects motivation Evaluating mixed effects methods Two methods –Summary statistic approach (HF) (SPM96,99,2) –SPM8 Nonsphericity Modelling Data exploration Conclusions

4 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)

5 Random Effects Illustration Standard linear model assumes only one source of iid random variation Consider this RT data Here, two sources –Within subject var. –Between subject var. –Causes dependence in  3 Ss, 5 replicated RT’s Residuals

6 Subj. 1 Subj. 2 Subj. 3 Subj. 4 Subj. 5 Subj. 6 0 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 subject’s estimated effect Distribution of population effect  2 FFX  2 RFX

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

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

9 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

10 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”

11 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

12 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

13 Overview Mixed effects motivation Evaluating mixed effects methods Two methods –Summary statistic approach (HF) (SPM96,99,2) –SPM8 Nonsphericity Modelling Data exploration Conclusions

14 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?

15 Issues: Assumptions Distributional Assumptions –Gaussian? Nonparametric? Homogeneous Variance –Over subjects? –Over conditions? Independence –Across subjects? –Across conditions/repeated measures –Note: Nonsphericity = (Heterogeneous Var) or (Dependence)

16 Issues: Soft Assumptions Regularization Regularization –Weakened homogeneity assumption –Usually variance/autocorrelation regularized over space Examples –fmristat - local pooling (smoothing) of (  2 RFX )/(  2 FFX ) –SnPM - local pooling (smoothing) of  2 RFX –FSL - Bayesian (noninformative) prior on  2 RFX –SPM – global pooling (averaging) of  MFX i,j

17 Issues: Efficiency & Power Efficiency: 1/(Estmator Variance) –Goes up with n Power: Chance of detecting effect –Goes up with n –Also goes up with degrees of freedom (DF) DF accounts for uncertainty in estimate of  2 RFX Usually DF and n yoked, e.g. DF = n-p

18 Issues: Validity Are P-values accurate? –I reject my null when P < 0.05 Is my risk of false positives controlled at 5%? –“Exact” control FPR =  –Valid control (possibly conservative) FPR   Problems when –Standard Errors inaccurate –Degrees of freedom inaccurate

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 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

21 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

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

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

24 Holmes & Friston Assumptions Distribution –Normality –Independent subjects Homogeneous Variance –Intrasubject variance homogeneous  2 FFX same for all subjects –Balanced designs

25 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

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

27 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 –DF not OK Here, 3 Ss dominate DF < 5 =  2 RFX

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 28 Holmes & Friston Robustness (outlier severity) Mumford & Nichols. Simple group fMRI modeling and inference. Neuroimage, 47(4): , False Positive RatePower Relative to Optimal (outlier severity)

29 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

30 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...

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

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

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

34 Non-Sphericity Modeling Errors are not independent and not identical Q k ’s: Error Covariance

35 SPM8 Nonsphericity Modelling Assumptions & Limitations – assumed to globally homogeneous – k ’s only estimated from voxels with large F –Most realistically, Cor(  ) spatially heterogeneous –Intrasubject variance assumed homogeneous Cor(ε) =Σ k λ k Q k

36 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

37 Overview Mixed effects motivation Evaluating mixed effects methods Two Three methods –Summary statistic approach (HF) (SPM96,99,2,5,8) –SPM8 Nonsphericity Modelling –FSL Data exploration Conclusions

38 FSL3: Full Mixed Effects Model First-level, combines sessions Second-level, combines subjects Third-level, combines/compares groups

39 FSL3: Summary Statistics

Summary Stats Equivalence Crucially, summary stats here are not just estimated effects. Summary Stats needed for equivalence: Beckman et al., 2003

41 Case Study: FSL3’s FLAME Uses summary-stats model equivalent to full Mixed Effects model Doesn’t assume intrasubject variance is homogeneous –Designs can be unbalanced –Subjects measurement error can vary

42 Case Study: FSL3’s FLAME Bayesian Estimation –Priors, priors, priors –Uses reference prior Final inference on posterior of  –  | y has Multivariate T dist n (MVT) but with unknown dof

Approximating MVTs Samples MVT dof? SLOW Estimate Model FAST BIDET MCMC Gaussian BIDET = Bayesian Inference with Distribution Estimation using T

44 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

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

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

50 Overview Mixed effects motivation Evaluating mixed effects methods Two methods –Summary statistic approach (HF) (SPM96,99,2,5,8 ) –SPM8 Nonsphericity Modelling –FSL3 Conclusions

51 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!

52

53 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, 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): , 2002 –Friston et al. Classical and Bayesian inference in neuroimaging: variance component estimation in fMRI. NI: 16(2): , 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): (2003) –Woolrich, Behrens et al. Multilevel linear modeling for fMRI group analysis using Bayesian inference. NI 21: (2004)