Group Modeling
for fMRI
Thomas Nichols, Ph.D.
Assistant Professor
Department of Biostatistics
http://www.sph.umich.edu/~nichols
IPAM MBI
July 21, 2004
1
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
– Summary statistic approach (HF)
– SPM2
– FSL3
• Conclusions
2
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
–
–
–
–
Summary statistic approach (HF)
SnPM
SPM2
FSL3
• 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
Random Effects
Illustration
3 Ss, 5 replicated RT’s
x
• Standard linear model
Y  X  
assumes only one source
of iid random variation
• Consider this RT data
• Here, two sources
– Within subject var.
– Between subject var.
– Causes dependence in 
Subject1
Subject2
Subject3
Residuals
Subject1
Subject2
Subject3
5
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 subjects/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
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
–
–
–
–
Summary statistic approach (HF)
SnPM
SPM2
FSL3
• Conclusions
11
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?
12
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)
13
Issues: Soft Assumptions
Regularization
• 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
14
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 2RFX
• Usually DF and n yoked, e.g. DF = n-p
15
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 = a
– Valid control (possibly conservative)
• FPR  a
• Problems when
– Standard Errors inaccurate
– Degrees of freedom inaccurate
16
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
–
–
–
–
Summary statistic approach (HF)
SnPM
SPM2
FSL3
• Conclusions
17
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):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)
18
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
–
–
–
–
Summary statistic approach (HF)
SnPM
SPM2
FSL3
• Conclusions
19
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
20
Case Studies: HF
Assumptions
• Distribution
– Normality
– Independent subjects
• Homogeneous Variance
– Intrasubject variance homogeneous
• 2FFX same for all subjects
– Balanced designs
21
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...
From HBM
Poster
WE 253
22
Case Studies: HF
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
23
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
0
2RFX
24
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
25
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
–
–
–
–
Summary statistic approach (HF)
SnPM
SPM2
FSL3
• Conclusions
26
Case Studies: SnPM
• No Gaussian assumption
• Instead, uses data to find empirical
distribution
5%
5%
Parametric Null Distribution
Nonparametric Null Distribution
27
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...
A
B
A
B
A
B
103.00
90.48
99.93
87.83
99.76
96.06
• Null hypothesis Ho
– No experimental effect, A & B labels arbitrary
• Statistic
– Mean difference
28
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
29
Permutation Test
Toy Example
• Under Ho
– Consider all equivalent relabelings
– Compute all possible statistic values
AAABBB 4.82
ABABAB 9.45
BAAABB -1.48
BABBAA -6.86
AABABB -3.25
ABABBA 6.97
BAABAB 1.10
BBAAAB 3.15
AABBAB -0.67
ABBAAB 1.38
BAABBA -1.38
BBAABA 0.67
AABBBA -3.15
ABBABA -1.10
BABAAB -6.97
BBABAA 3.25
ABAABB 6.86
ABBBAA 1.48
BABABA -9.45
BBBAAA -4.82
30
Permutation Test
Toy Example
• Under Ho
– Consider all equivalent relabelings
– Compute all possible statistic values
– Find 95%ile of permutation distribution
AAABBB 4.82
ABABAB 9.45
BAAABB -1.48
BABBAA -6.86
AABABB -3.25
ABABBA 6.97
BAABAB 1.10
BBAAAB 3.15
AABBAB -0.67
ABBAAB 1.38
BAABBA -1.38
BBAABA 0.67
AABBBA -3.15
ABBABA -1.10
BABAAB -6.97
BBABAA 3.25
ABAABB 6.86
ABBBAA 1.48
BABABA -9.45
BBBAAA -4.82
31
Permutation Test
Toy Example
• Under Ho
– Consider all equivalent relabelings
– Compute all possible statistic values
– Find 95%ile of permutation distribution
AAABBB 4.82
ABABAB 9.45
BAAABB -1.48
BABBAA -6.86
AABABB -3.25
ABABBA 6.97
BAABAB 1.10
BBAAAB 3.15
AABBAB -0.67
ABBAAB 1.38
BAABBA -1.38
BBAABA 0.67
AABBBA -3.15
ABBABA -1.10
BABAAB -6.97
BBABAA 3.25
ABAABB 6.86
ABBBAA 1.48
BABABA -9.45
BBBAAA -4.82
32
Permutation Test
Toy Example
• Under Ho
– Consider all equivalent relabelings
– Compute all possible statistic values
– Find 95%ile of permutation distribution
-8
-4
0
4
8
33
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
– Subjects independent
34
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!
35
Case Studies: SnPM
Smoothed Variance t
• For low df, variance estimates poor
– Variance image “spikey”
• Consider smoothed variance t statistic
mean difference
variance
t-statistic
36
Permutation Test
Smoothed Variance t
• Smoothed variance estimate better
– Variance “regularized”
– df effectively increased, but no RFT result
mean difference
smoothed
variance
Smoothed
Variance
t-statistic
37
Small Group
fMRI Example
• 12 subject study
– Item recognition task
– 1-sample t test on diff. imgs
– FWE threshold
uRF = 9.87
uBonf = 9.80
5 sig. vox.
t11 Statistic, RF & Bonf. Threshold
378 sig. vox.
Smoothed Variance
t Statistic,
Permutation Distn Maximum t11
Nonparametric Threshold
uPerm = 7.67
58 sig. vox.
t11 Statistic, Nonparametric Threshold
38
Working memory data: Marshuetz et al (2000)
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!
39
Case Studies: SnPM
Robustness
• If data not exchangeable under null
– Can be invalid, P-values too liberal
– More typically, valid but conservative
40
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
–
–
–
–
Summary statistic approach (HF)
SnPM
SPM2
FSL3
• Conclusions
41
Case Study: SPM2
• 1 effect per subject
– Uses Holmes & Friston approach
• >1 effect per subject
– Variance basis function approach used...
42
SPM2 Notation: iid case
Cor( )  I
y=X  +
N1
Np
p1
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
43
Multiple Variance Components
Cor ( )    k Qk
y=X  +
N1
Np
p1
N1
k
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
44
Non-Sphericity Modeling
Error Covariance
– Errors are
independent
but not identical
– Errors are not
independent
and not identical
45
Case Study: SPM2
• Assumptions & Limitations
– Cor ( )    k Qk assumed to globally
k
homogeneous
– k’s only estimated from voxels with large F
– Most realistically, Cor() spatially heterogeneous
– Intrasubject variance assumed homogeneous
46
Case Study: SPM2
• 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
47
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
–
–
–
–
Summary statistic approach (HF)
SnPM
SPM2
FSL3
• Conclusions
48
FSL3: Full Mixed Effects Model
Y  X K  K   K First-level, combines sessions
 K  X G  G   G Second-level, combines subjects
G  X H  H   H Third-level, combines/compares groups
49
FSL3: Summary Statistics
50
Summary Stats Equivalence
ct  H  0
ct  H  0
Crucially, summary stats here are not just
estimated effects.
Summary Stats needed for equivalence:
ˆ
cov( ˆ )
dof
Beckman et al.,
2003
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
52
Case Study: FSL3’s FLAME
• Bayesian Estimation
– Priors, priors, priors
– Use reference prior
• Final inference on posterior of 
–  | y has Multivariate T distn (MVT)
but with unknown dof
53
Approximating MVTs
Gaussian
FAST
Estimate

p(  | Y , ˆ )
p(  | Y ) MVT
dof?
Model
MCMC
BIDET
p(  | Y )
Samples
SLOW
BIDET =
Bayesian Inference with
Distribution Estimation using T
Overview
• Mixed effects motivation
• Evaluating mixed effects methods
• Case Studies
–
–
–
–
Summary statistic approach (HF)
SnPM
SPM2
FSL3
• Conclusions
55
Conclusions
• Random Effects crucial for pop. inference
• Don’t fall in love with your model!
• Evaluate...
– Assumptions
– Efficiency/Power
– Validity & Robustness
56