Group analyses of fMRI data
Klaas Enno Stephan
Translational Neuromodeling Unit (TNU)
Institute for Biomedical Engineering, University of Zurich & ETH Zurich
Laboratory for Social & Neural Systems Research (SNS), University of Zurich
Wellcome Trust Centre for Neuroimaging, University College London
With many thanks for slides & images to:
FIL Methods group,
particularly Will Penny & Tom Nichols
Methods & models for fMRI data analysis
November 2012
Overview of SPM
Image time-series
Realignment
Kernel
Design matrix
Smoothing
General linear model
Statistical parametric map (SPM)
Statistical
inference
Normalisation
Gaussian
field theory
p <0.05
Template
Parameter estimates
Reminder: voxel-wise time series analysis!
model
specification
Time
parameter
estimation
hypothesis
statistic
BOLD signal
single voxel
time series
SPM
The model: voxel-wise GLM
p
1
1
1

p
y
N
=
N
X
y  X  e
e ~ N (0,  I )
2
+ e
N
Model is specified by
1. Design matrix X
2. Assumptions about e
N: number of scans
p: number of regressors
The design matrix embodies all available knowledge about
experimentally controlled factors and potential confounds.
GLM assumes Gaussian “spherical” (i.i.d.) errors
sphericity = iid:
error covariance is
scalar multiple of
identity matrix:
Cov(e) = 2I
Examples for non-sphericity:
 4 0
Cov(e)  

 0 1
non-identity
1 0
Cov(e)  

0
1


2 1
Cov(e)  

1
2


non-independence
Multiple covariance components at 1st level
V  Cov( e)
e ~ N (0,  V )
2
enhanced noise model
V
= 1
V   iQi
error covariance components Q
and hyperparameters 
Q1
+ 2
Q2
Estimation of hyperparameters  with ReML (restricted maximum
likelihood).
t-statistic based on ML estimates
Wy  WX  We
̂  (WX ) Wy

c=10000000000
c ˆ
t
T ˆ
ˆ
st d ( c  )
T
W V
stˆd (cT ˆ ) 
ˆ c (WX ) (WX ) c
1 / 2
 2V  Cov (e)

2 T
ˆ
2



T
Wy  WXˆ

2
tr ( R)
R  I  WX (WX ) 
X
V 
 Q
i
i
For brevity:
ReMLestimates
(WX )   ( X TWX ) 1 X T
Fixed vs.
random effects
analysis
• Fixed Effects
– Intra-subject variation
suggests most subjects
different from zero
• Random Effects
– Inter-subject variation
suggests population is
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
8
Fixed Effects
• Assumption: variation (over subjects) is only due to
measurement error
• parameters are fixed properties of the population (i.e., they are
the same in each subject)
Random/Mixed Effects
• two sources of variation (over subjects)
– measurement error
– Response magnitude: parameters are probabilistically
distributed in the population
• effect (response magnitude) in each subject is randomly
distributed
Random/Mixed Effects
• two sources of variation (over subjects)
– measurement error
– Response magnitude: parameters are probabilistically
distributed in the population
• effect (response magnitude) in each subject is randomly
distributed
• variation around population mean
Group level inference: fixed effects (FFX)
• assumes that parameters are “fixed properties of the
population”
• all variability is only intra-subject variability, e.g. due to
measurement errors
• Laird & Ware (1982): the probability distribution of the data
has the same form for each individual and the same
parameters
• In SPM: simply concatenate the data and the design
matrices
 lots of power (proportional to number of scans),
but results are only valid for the group studied and
cannot be generalized to the population
Group level inference: random effects (RFX)
• assumes that model parameters are probabilistically
distributed in the population
• variance is due to inter-subject variability
• Laird & Ware (1982): the probability distribution of the data
has the same form for each individual, but the parameters
vary across individuals
• hierarchical model
 much less power (proportional to number of
subjects), but results can be generalized to the
population
FFX vs. RFX
• FFX is not "wrong", it makes different assumptions and
addresses a different question than RFX
• For some questions, FFX may be appropriate (e.g., low-level
physiological processes).
• For other questions, RFX is much more plausible (e.g., cognitive
tasks, disease processes in heterogeneous populations).
Hierachical models
fMRI, single subject
fMRI, multi-subject
EEG/MEG, single subject
ERP/ERF, multi-subject
Hierarchical models for all imaging
data!
Linear hierarchical model
Hierarchical model
Multiple variance components
at each level
y  X (1) (1)   (1)
 (1)  X ( 2) ( 2)   ( 2)

C   Q
(i)
(i)

k
k
 ( n 1)  X ( n ) ( n )   ( n )
At each level, distribution of parameters
is given by level above.
What we don’t know: distribution of parameters
and variance parameters (hyperparameters).
(i)
k
Example: Two-level model
1 1
yX 

1
 1  X 2  2    2 
 1
X 1(1)
y =
 2 
+  
1
X 2(1)
 1 = X 2 
+  2 
X 3(1)
Second level
First level
Two-level model
y  X (1) (1)   (1)
 (1)  X (2) (2)   (2)
y  X (1)  X (2) (2)   (2)    (1)
 X (1) X (2) (2)  X (1) (2)   (1)
fixed effects
Friston et al. 2002, NeuroImage
random effects
Mixed effects analysis
Non-hierarchical model
y  X (1) X (2) (2)  X (1) (2)   (1)
ˆ(1)  X (1) y
 X (2) (2)   (2)  X (1) (1)
Estimating 2nd level effects
 X (2) (2)   (2)
Variance components at 2nd
level
Cov 
(2)
C
(2)
X
(1)
(1)
C X
(1)T
within-level
between-level
non-sphericity non-sphericity
Within-level non-sphericity at
both levels: multiple
covariance components
C
(i )
   k Qk(i )
(i )
k
Friston et al. 2005, NeuroImage
Algorithmic equivalence
y  X (1) (1)   (1)
Hierarchical
model

(1)
X 
( 2)
( 2)

( 2)

Parametric
Empirical
Bayes (PEB)
 ( n 1)  X ( n ) ( n )   ( n )
EM = PEB = ReML
Single-level
model
y   (1)  X (1) ( 2 ) 
... 
X (1)  X ( n 1) ( n ) 
X (1)  X ( n ) ( n )
Restricted
Maximum
Likelihood
(ReML)
Estimation by Expectation Maximisation (EM)
y  X   
N 1
N  p p1
N 1
EM-algorithm
C | y 1  XT C 1X  C 1
η | y  C | y  X C y  C η 
C   k Qk
T
1
E-step
k
maximise L  ln p( y | λ)
•
E-step: finds the (conditional) expectation
of the parameters, holding the
hyperparameters fixed
•
M-step: updates the maximum likelihood
estimate of the hyperparameters, keeping
the parameters fixed
dL
g
d
d 2L
J 2
d
    J 1 g
M-step
Gauss-Newton
gradient ascent
Friston et al. 2002, NeuroImage
Practical problems
• Full MFX inference using REML or EM for a wholebrain 2-level model has enormous computational costs
• for many subjects and scans, covariance matrices become
extremely large
• nonlinear optimisation problem for each voxel
• Moreover, sometimes we are only interested in one
specific effect and do not want to model all the data.
• Is there a fast approximation?
Summary statistics approach: Holmes & Friston 1998
First level
Data
Design Matrix
̂1
̂ 12
Second level
Contrast Images
t
cT ˆ
Vaˆr (cT ˆ )
SPM(t)
̂ 2
ˆ 22
̂11
̂ 112
̂12
̂ 122
One-sample
t-test @ 2nd level
Validity of the summary statistics approach
The summary stats approach is exact if for each
session/subject:
Within-session covariance the same
First-level design the same
One contrast per session
But:
Summary stats approach is fairly robust
against violations of these conditions.
Mixed effects analysis: spm_mfx
y  data
X  [ X (0)
V I
Summary
statistics
non-hierarchical model
X  [ X ( 0)
X (1) ]
X (1) X ( 2) ]
Q  {Q1(1) ,, X (1) Q1( 2) X (1)T ,}
Step 1
ˆ (1)  ( X TV 1 X ) 1 X TV 1 y
  REML{ yyT n , X , Q}
Y  ˆ (1)
X  X ( 2)
V   (i1) X (1) Qi(1) X (1) T   (j2 )Q (j 2 )
i
EM
approach
Friston et al. 2005, NeuroImage
j
1st level
non-sphericity
2nd level
non-sphericity
Step 2
ˆ ( 2)  ( X TV 1 X ) 1 X TV 1 y
ˆ(2)
pooling over
voxels
2nd level non-sphericity modeling in SPM8
• 1 effect per subject
→ use summary statistics approach
• >1 effect per subject
→ model sphericity at 2nd level using variance basis
functions
Reminder: sphericity
C  Cov( )  E ( )
T
y  X  
„sphericity“ means:
Scans
Cov( )   I
2
i.e. Var ( i )  
1 0
Cov( )  

0
1


Scans
2
2nd level: non-sphericity
Error
covariance
Errors are independent
but not identical:
e.g. different groups (patients,
controls)
Errors are not independent
and not identical:
e.g. repeated measures for each
subject (multiple basis functions,
multiple conditions etc.)
Example of 2nd level non-sphericity
Error Covariance
Qk:
...
...
Example of 2nd level non-sphericity
y=X  +e
N1
Np
p1
Cor(ε) =Σk λkQk
N1
error covariance
• 12 subjects, 4 conditions
N
• Measurements between
subjects uncorrelated
• Measurements within
subjects correlated
• Errors can have different
variances across subjects
N
30
2nd level non-sphericity modeling in SPM8:
assumptions and limitations
• Cor() assumed to be globally homogeneous
• k’s only estimated from voxels with large F
• intrasubject variance assumed homogeneous
Practical conclusions
• Linear hierarchical models are used for group analyses of multisubject imaging data.
• The main challenge is to model non-sphericity (i.e. non-identity
and non-independence of errors) within and between levels of
the hierarchy.
• This is done by estimating hyperparameters using EM or ReML
(which are equivalent for linear models).
• The summary statistics approach is robust approximation to a
full mixed-effects analysis.
– Use mixed-effects model only, if seriously in doubt about validity of
summary statistics approach.
Recommended reading
Linear hierarchical models
Mixed effect models
Thank you