Mixed effects and Group Modeling for fmri data

Transcription

1 Mixed effects and Group Modeling for fmri data Thomas Nichols, Ph.D. Department of Statistics Warwick Manufacturing Group University of Warwick Warwick fmri Reading Group May 19,

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 2

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

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) 4

5 Random Effects 3 Ss, 5 replicated RT s 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 ε Residuals 5

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

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

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

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 9

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 10

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 11

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) ~ χ 2 n Good: Doesn t assume common variance Bad: Not implemented in software Hard to interrogate statistic maps 12

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

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? 14

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) 15

16 Issues: Soft Assumptions Regularization Regularization Weakened homogeneity assumption Usually variance/autocorrelation regularized over space Examples fmristat - local pooling (smoothing) of (σ 2 RFX )/(σ2 FFX ) SnPM - local pooling (smoothing) of σ 2 RFX FSL - Bayesian (noninformative) prior on σ 2 RFX SPM global pooling (averaging) of σ MFX i,j 16

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 σ 2 RFX Usually DF and n yoked, e.g. DF = n-p 17

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 = α Valid control (possibly conservative) FPR α Problems when Standard Errors inaccurate Degrees of freedom inaccurate 18

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 19

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 20

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 21

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

23 Holmes & Friston level-one (within-subject) ^ α 1 ^ σ 2 ε ^ α 2 ^ σ 2 ε ^ α 3 ^ σ 2 ε ^ α 4 ^ σ 2 ε ^ α 5 ^ σ 2 ε Random Effects level-two (between-subject) variance ^ σ 2 α ^ c.f. σ 2 /n = σ 2 α /n + σ2 ε / nw c.f. an estimate of the mixed-effects model variance σ 2 α + σ2 ε / w (no voxels significant at p < 0.05 (corrected)) p < (uncorrected) ^ α 6 ^ σ 2 ε timecourses at [ 03, -78, 00 ] contrast images SPM{t} 23

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

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 25

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

27 Holmes & Friston Validity If assumptions true Exact P-values If σ 2 FFX differs btw subj. Standard errors not OK Est. of σ 2 RFX biased DF not OK may be Here, 3 Ss dominate DF < 5 = σ 2 RFX 27

28 Holmes & Friston In practice, Validity & Efficiency are excellent For one sample case, HF almost impossible to break False Positive Rate Robustness Power Relative to Optimal (outlier severity) (outlier severity) Mumford & Nichols. Simple group fmri modeling and inference. Neuroimage, 47(4): , sample & correlation might give trouble Dramatic imbalance or heteroscedasticity 28

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 29

30 SPM8 Nonsphericity 1 effect per subject Modelling Uses Holmes & Friston approach >1 effect per subject Can t use HF; must use SPM8 Nonsphericity Modelling Variance basis function approach used... 30

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

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

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

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

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

36 SPM8 Nonsphericity Efficiency & Power Modelling If assumptions true, fully efficient Validity & Robustness P-values could be wrong (over or under) if local Cor(ε) very different from globally assumed Stronger assumptions than Holmes & Friston 36

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 37

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

39 FSL3: Summary Statistics 39

40 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 41

42 Case Study: FSL3 s FLAME Bayesian Estimation Priors, priors, priors Uses reference prior Final inference on posterior of β β y has Multivariate T dist n (MVT) but with unknown dof 42

43 Approximating MVTs FAST Gaussian Estimate dof? MVT Model MCMC BIDET Samples SLOW 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 44

45 Data: FIAC Data Acquisition 3 TE Bruker Magnet For each subject: 2 (block design) sessions, 195 EPI images each TR=2.5s, TE=35ms, 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 mm FWHM Gaussian smoothing Box-car convolved w/ canonical HRF Drift fit with DCT, 1/128Hz

46 Look at the Data! With small n, really can do it! Start with anatomical Alignment OK? Yup Any horrible anatomical anomalies? Nope

47 Look at the Data! Mean & Standard Deviation also useful Variance lowest in white matter Highest around ventricles

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

49 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 50

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! 51

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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) 53