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. 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

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 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. 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: Doesn’t 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) (SPM96,99,2)
SPM8 Nonsphericity Modelling
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) (SPM96,99,2,5,8)
SPM8 Nonsphericity Modelling
Data exploration
Conclusions

15. 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

16. 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

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

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

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

20. Holmes & Friston 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

21. 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

22. 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
2RFX

23. Holmes & Friston Robustness
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
(outlier severity)
Mumford & Nichols. Simple group fMRI modeling and inference. Neuroimage, 47(4): , 2009.
False Positive Rate
Power Relative to Optimal

24. 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

25. 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...

26. SPM8 Notation: iid case X y = X  + e Cor(ε) = λ I
N  N  p p  N  1
Cor(ε) = λ I 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

27. Multiple Variance Components
y = X  + e
N  N  p p  N  1
Cor(ε) =Σk λkQk
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’

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

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

30. SPM8 Nonsphericity Modelling
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
Cor(ε) =Σk λkQk

31. 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

32. 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

33. Data: FIAC Data Acquisition Experiment (Block Design only) Analysis
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

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

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

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

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

38. 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!

40. 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): , 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)