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1
Mixed effects and Group Modeling
for fMRI data
Thomas Nichols, Ph.D.Department of Statistics
Warwick Manufacturing GroupUniversity of Warwick
Zurich SPM CourseFebruary 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)
6
Subj. 1
Subj. 2
Subj. 3
Subj. 4
Subj. 5
Subj. 6
0
Fixed vs.RandomEffects 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
2FFX
2RFX
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) ~ 2n
– 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 ModelsIssues 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?
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 cbi, contrast estimate
– Analyze {cbi}i
22
Holmes & Fristonmotivation...
p < 0.001 (uncorrected)
p < 0.05 (corrected)
SPM{t}
SPM{t}
1^
2^
3^
4^
5^
6^
^
• – c.f. 2 / nw
—
^
^
^
^
^
– c.f.
estimated mean activation imageFixed effects...
...powerful but wrong inference
n – subjects
w – error DF
23
^
Holmes & FristonRandom Effects
1^
2^
3^
4^
5^
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 < 0.001 (uncorrected)
SPM{t}
(no voxels significant at p < 0.05 (corrected))
24
Holmes & FristonAssumptions
• Distribution– Normality– Independent subjects
• Homogeneous Variance– Intrasubject variance homogeneous
2FFX same for all subjects
– Balanced designs
25
Holmes & FristonLimitations• 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 & FristonEfficiency
• If assumptions true– Optimal, fully efficient
• If 2FFX differs between
subjects– Reduced efficiency– Here, optimal requires
down-weighting the 3 highly variable subjects
0
27
Holmes & FristonValidity• If assumptions true
– Exact P-values
• If 2FFX differs btw subj.
– Standard errors not OK• Est. of 2
RFX may be biased
– DF not OK• Here, 3 Ss dominate
• DF < 5 = 6-1
0
2RFX
• 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 & FristonRobustness
(outlier severity)
Mumford & Nichols. Simple group fMRI modeling and inference. Neuroimage, 47(4):1469--1475, 2009.
False Positive Rate Power 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) =Σk λkQk
33
Non-Sphericity Modeling
• Errors are independent but not identical– Eg. Two Sample T
Two basis elements
Error Covariance
Qk’s:
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) =Σk λkQk
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
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, 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 Deviationalso useful– Variance
lowest inwhite 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
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!
53
References for fourRFX 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.
NI 21:1732-1747 (2004)