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 17, 2011
1
Outline
• Mixed effects motivation
• Evaluating mixed effects methods
• Two methods
– Summary statistic approach (HF)
– SPM8 Nonsphericity Modelling
(SPM96,99,2,5,8)
• Data exploration
• Conclusions
2
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Two methods
– Summary statistic approach (HF)
– SPM8 Nonsphericity Modelling
(SPM96,99,2)
• Data exploration
• Conclusions
3
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
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
2FFX
Subj. 1
Subj. 2
Subj. 3
Subj. 4
Subj. 5
Subj. 6
0
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)
– SPM8 Nonsphericity Modelling
(SPM96,99,2)
• 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)
– SPM8 Nonsphericity Modelling
(SPM96,99,2,5,8)
• Data exploration
• Conclusions
19
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Two methods
– Summary statistic approach (HF)
– SPM8 Nonsphericity Modelling
(SPM96,99,2,5,8)
• Data exploration
• Conclusions
20
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
Holmes & Friston
motivation...
Fixed effects...
estimated mean
activation image
^

1
^ 
^

2
^


^

3
^


^

4
^


^

5
—
^ – c.f. 2 / nw

•

^


SPM{t}
– c.f.
n – subjects
w – error DF
p < 0.05 (corrected)
^


^

6
p < 0.001 (uncorrected)
...powerful but
wrong inference
SPM{t}
22
Holmes & Friston
Random Effects
level-one
level-two
(within-subject)
1
^
(between-subject)

^


^

2

^


^

3
(no voxels significant at p < 0.05 (corrected))

^


^

4
—

^


5
variance^2
an estimate of the
mixed-effects
model variance
2 + 2 / w
^ – c.f. 2/n = 2 /n + 2 / nw

•


– c.f.
^

^


^

6
^


timecourses at [ 03, -78, 00 ]
p < 0.001 (uncorrected)

contrast images
SPM{t}
23
Holmes & Friston
Assumptions
• Distribution
– Normality
– Independent subjects
• Homogeneous Variance
– Intrasubject variance homogeneous
• 2FFX same for all subjects
– Balanced designs
24
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
Holmes & Friston
Efficiency
• If assumptions true
– Optimal, fully efficient
• If 2FFX differs between
subjects
– Reduced efficiency
– Here, optimal ˆ requires
down-weighting the 3
highly variable subjects
0
ˆ
26
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
0
2RFX
• Here, 3 Ss dominate
• DF < 5 = 6-1
27
Holmes & Friston
Robustness
• In practice, Validity & Efficiency are excellent
– For one sample case, HF almost impossible to break
False Positive Rate
Power Relative to Optimal
(outlier severity)
(outlier severity)
Mumford & Nichols. Simple group fMRI modeling and inference. Neuroimage, 47(4):1469--1475, 2009.
• 2-sample & correlation might give trouble
– Dramatic imbalance or heteroscedasticity
28
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Two methods
– Summary statistic approach (HF)
– SPM8 Nonsphericity Modelling
(SPM96,99,2,5,8)
• Data exploration
• Conclusions
29
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...
30
SPM8 Notation: iid case
y=X  +
N1
Np
p1
Cor(ε) = λ I
N1
• 12 subjects,
4 conditions
X
Error covariance
N
– Use F-test to find
differences btw conditions
• Standard Assumptions
– Identical distn
– Independence
– “Sphericity”... but here
not realistic!
N
31
Multiple Variance Components
y=X  +
N1
Np
p1
Cor(ε) =Σk λkQk
N1
Error covariance
• 12 subjects, 4 conditions
• Measurements btw
subjects uncorrelated
• Measurements w/in
subjects correlated
Errors can now have
different variances and
there can be correlations
Allows for ‘nonsphericity’
N
N
32
Non-Sphericity Modeling
• Errors are
independent but not
identical
– Eg. Two Sample T
Two basis elements
Qk’s:
Error Covariance
33
Non-Sphericity Modeling
• Errors are not
independent and not
identical
Error Covariance
Qk’s:
34
SPM8 Nonsphericity
Modelling
• Assumptions & Limitations
– Cor(ε) =Σk λkQk
assumed to globally
homogeneous
– lk’s only estimated from voxels with large F
– Most realistically, Cor() spatially heterogeneous
– Intrasubject variance assumed homogeneous
35
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
36
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Two methods
– Summary statistic approach (HF)
– SPM8 Nonsphericity Modelling
(SPM96,99,2,5,8)
• Data exploration
• Conclusions
44
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
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
52
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):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)
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