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Group Modeling for fMRI

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Presentation on theme: "Group Modeling for fMRI"— Presentation transcript:

1 Group Modeling for fMRI
Thomas Nichols, Ph.D. Assistant Professor Department of Biostatistics IPAM MBI July 21, 2004

2 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SPM2 FSL3 Conclusions

3 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SnPM SPM2 FSL3 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
3 Ss, 5 replicated RT’s 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  Residuals

6 Fixed vs. Random Effects in fMRI
Distribution of each subject’s estimated effect 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 2RFX Distribution of population effect

7 Fixed Effects Only variation (over subjects/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 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SnPM SPM2 FSL3 Conclusions

12 Assessing RFX Models Issues to Consider
Assumptions & Limitations What must I assume? 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?

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

14 Issues: Soft Assumptions Regularization
Weakened homogeneity assumption Usually variance/autocorrelation regularized over space Examples fmristat - local pooling (smoothing) of (2RFX)/(2FFX) SnPM - local pooling (smoothing) of 2RFX FSL3 - Bayesian (noninformative) prior on 2RFX

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

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

17 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SnPM SPM2 FSL3 Conclusions

18 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. (SPM2) Empirical Bayesian approach Friston et al. Classical and Bayesian inference in neuroimagining: theory. NI 16(2): , 2002 Friston et al. Classical and Bayesian inference in neuroimaging: variance component estimation in fMRI. NI: 16(2): , 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): (2003) Woolrich, Behrens et al. Multilevel linear modeling for fMRI group analysis using Bayesian inference. NI 21: (2004)

19 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SnPM SPM2 FSL3 Conclusions

20 Case Studies: 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 Case Studies: HF Assumptions
Distribution Normality Independent subjects Homogeneous Variance Intrasubject variance homogeneous 2FFX same for all subjects Balanced designs

22 Case Studies: HF Limitations
From HBM Poster WE 253 Case Studies: HF 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 conditions Though nonsphericity itself may be of interest...

23 Case Studies: HF Efficiency
If assumptions true Optimal, fully efficient If 2FFX differs between subjects Reduced efficiency Here, optimal requires down-weighting the 3 highly variable subjects

24 Case Studies: HF Validity
If assumptions true Exact P-values If 2FFX differs btw subj. Standard errors OK Est. of 2RFX unbiased DF not OK Here, 3 Ss dominate DF < 5 = 6-1 2RFX

25 Case Studies: HF Robustness
Heterogeneity of 2FFX across subjects... How bad is bad? Dramatic imbalance (rough rules of thumb only!) Some subjects missing 1/2 or more sessions Measured covariates of interest having dramatically different efficiency E.g. Split event related predictor by correct/incorrect One subj 5% trials correct, other subj 80% trials correct Dramatic heteroscedasticity A “bad” subject, e.g. head movement, spike artifacts

26 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SnPM SPM2 FSL3 Conclusions

27 Case Studies: SnPM No Gaussian assumption
Instead, uses data to find empirical distribution 5% Parametric Null Distribution 5% Nonparametric Null Distribution

28 Permutation Test Toy Example
Data from V1 voxel in visual stim. experiment A: Active, flashing checkerboard B: Baseline, fixation 6 blocks, ABABAB Just consider block averages... Null hypothesis Ho No experimental effect, A & B labels arbitrary Statistic Mean difference A B 103.00 90.48 99.93 87.83 99.76 96.06

29 Permutation Test Toy Example
Under Ho Consider all equivalent relabelings AAABBB ABABAB BAAABB BABBAA AABABB ABABBA BAABAB BBAAAB AABBAB ABBAAB BAABBA BBAABA AABBBA ABBABA BABAAB BBABAA ABAABB ABBBAA BABABA BBBAAA

30 Permutation Test Toy Example
Under Ho Consider all equivalent relabelings Compute all possible statistic values AAABBB ABABAB BAAABB BABBAA AABABB ABABBA BAABAB BBAAAB AABBAB ABBAAB BAABBA BBAABA AABBBA ABBABA BABAAB BBABAA ABAABB ABBBAA BABABA BBBAAA

31 Permutation Test Toy Example
Under Ho Consider all equivalent relabelings Compute all possible statistic values Find 95%ile of permutation distribution AAABBB ABABAB BAAABB BABBAA AABABB ABABBA BAABAB BBAAAB AABBAB ABBAAB BAABBA BBAABA AABBBA ABBABA BABAAB BBABAA ABAABB ABBBAA BABABA BBBAAA

32 Permutation Test Toy Example
Under Ho Consider all equivalent relabelings Compute all possible statistic values Find 95%ile of permutation distribution AAABBB ABABAB BAAABB BABBAA AABABB ABABBA BAABAB BBAAAB AABBAB ABBAAB BAABBA BBAABA AABBBA ABBABA BABAAB BBABAA ABAABB ABBBAA BABABA BBBAAA

33 Permutation Test Toy Example
Under Ho Consider all equivalent relabelings Compute all possible statistic values Find 95%ile of permutation distribution -8 -4 4 8

34 Case Studies: SnPM Assumptions
1-Sample t on difference data Under null, differences distn symmetric about 0 OK if 2FFX differs btw subjects! Subjects independent 2-Sample t Under null, distn of all data the same Implies 2FFX must be the same across subjects

35 Case Studies: SnPM Efficiency
Just as with HF... Lower efficiency if heterogeneous variance Efficiency increases with n Efficiency increases with DF How to increase DF w/out increasing n? Variance smoothing!

36 Case Studies: SnPM Smoothed Variance t
For low df, variance estimates poor Variance image “spikey” Consider smoothed variance t statistic mean difference t-statistic variance

37 Permutation Test Smoothed Variance t
Smoothed variance estimate better Variance “regularized” df effectively increased, but no RFT result mean difference Smoothed Variance t-statistic smoothed variance

38 Small Group fMRI Example
Smoothed Variance t Statistic, Nonparametric Threshold 378 sig. vox. Permutation Distn Maximum t11 12 subject study Item recognition task 1-sample t test on diff. imgs FWE threshold t11 Statistic, RF & Bonf. Threshold uRF = 9.87 uBonf = sig. vox. t11 Statistic, Nonparametric Threshold uPerm = sig. vox. Working memory data: Marshuetz et al (2000)

39 Case Studies: SnPM Validity
If assumptions true P-values’s exact Note on “Scope of inference” SnPM can make inference on population Simply need to assume random sampling of population Just as with parametric methods!

40 Case Studies: SnPM Robustness
If data not exchangeable under null Can be invalid, P-values too liberal More typically, valid but conservative

41 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SnPM SPM2 FSL3 Conclusions

42 Case Study: SPM2 1 effect per subject >1 effect per subject
Uses Holmes & Friston approach >1 effect per subject Variance basis function approach used...

43 SPM2 Notation: iid case X y = X  + e 12 subjects, 4 conditions
N  N  p p  N  1 X Error covariance 12 subjects, 4 conditions Use F-test to find differences btw conditions Standard Assumptions Identical distn Independence “Sphericity”... but here not realistic! N N

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

45 Non-Sphericity Modeling
Errors are independent but not identical Errors are not independent and not identical Error Covariance

46 Case Study: SPM2 Assumptions & Limitations
assumed to globally homogeneous lk’s only estimated from voxels with large F Most realistically, Cor(e) spatially heterogeneous Intrasubject variance assumed homogeneous

47 Case Study: SPM2 Efficiency & Power Validity & Robustness
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

48 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SnPM SPM2 FSL3 Conclusions

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

50 FSL3: Summary Statistics

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

52 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

53 Case Study: FSL3’s FLAME
Bayesian Estimation Priors, priors, priors Use reference prior Final inference on posterior of b b | y has Multivariate T distn (MVT) but with unknown dof

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

55 Overview Mixed effects motivation Evaluating mixed effects methods
Case Studies Summary statistic approach (HF) SnPM SPM2 FSL3 Conclusions

56 Conclusions Random Effects crucial for pop. inference
Don’t fall in love with your model! Evaluate... Assumptions Efficiency/Power Validity & Robustness

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