Transcript Group analyses Will Penny SPM Short Course, Zurich 2008 WellcomeTrust Centre for Neuroimaging University College London.

Group analyses
Will Penny
SPM Short Course, Zurich 2008
WellcomeTrust Centre for
Neuroimaging
University College London
Overview
Why hierarchical models?
The model
Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples
Overview
Why hierarchical models?
The model
Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples
Data
fMRI, single subject
fMRI, multi-subject
EEG/MEG, single subject
ERP/ERF, multi-subject
Hierarchical model for all
imaging data!
Reminder: voxel by voxel
model
specification
Time
parameter
estimation
hypothesis
statistic
Intensity
single voxel
time series
SPM
Overview
Why hierarchical models?
The model
Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples
General Linear Model
p
1
y  X  
1
1

y
N
=
X
N
N: number of scans
p: number of regressors
p
+

N
Model is specified by
1. Design matrix X
2. Assumptions about 
Linear hierarchical model
Hierarchical model


(1 )

(2)
y  X
(1 )
 X
(2)
(1 )


(1 )
(2)
C


( n 1 )
 X
(n)

(n)

Multiple variance
components at each level
(i)

  Q
(n)
At each level, distribution of
parameters is given by level above.
What we don’t know: distribution of
parameters and variance parameters.
(i)
k
k
(i)
k
Example: Two level model
1 1
yX 

1
X

2  2 

(1)
y
=


X1
1
2 
1
 2 
+  
1
(1)
X2
 1 = X 2 
+  2 
(1)
X3
Second level
First level
Algorithmic Equivalence
Hierarchical
model


(1 )

(2)

(2)
(n)

(n)
y  X
(1 )
 X
(2)
(1 )

(1 )


( n 1 )
 X
(n)

Parametric
Empirical
Bayes (PEB)
EM = PEB = ReML
Single-level
model
y 

X
(1 )
 X 
... 
(1 )
( n 1 )
(2)

 
(1 )
(n) (n)
X X 
(1 )
X
(n)
Restricted
Maximum
Likelihood
(ReML)
Overview
Why hierarchical models?
The model
Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples
Estimation
y  X
N 1
Np
  
p 1
EM-algorithm
N 1
1
C |y  ( X C  X )
T
 |y  C |y X C  y
T
maximise
L  ln p(y| λ )
g
J
d
M-step
d L
d
2
1
 J g
k
Assume, at voxel j:
E-step
dL
2
C   k Qk
1
1
 jk   j k
Friston et al. 2002,
Neuroimage
And in practice?
Most 2-level models are just too big to
compute.
And even if, it takes a long time!
Moreover, sometimes we‘re only
interested in one specific effect and
don‘t want to model all the data.
Is there a fast approximation?
Summary Statistics approach
First level
Data
Design Matrix
ˆ1
ˆ12
Second level
Contrast Images
t
T
c ˆ
T
Vaˆr ( c ˆ )
SPM(t)
ˆ 2
ˆ 22
ˆ11
ˆ11
2
ˆ12
ˆ12
2
One-sample
t-test @ 2nd level
Validity of 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
All other cases: Summary stats approach seems to
be robust against typical violations.
Mixed-effects
y
X  [X
(0)
(1 )
X
data
X  [X
]
Q  {Q
V  I
Summary
statistics
(1 )
1
X
(1 )
, , X
X
(1 )
(2)
Q
]
(2)
1
X
(1 ) T
Step 1
ˆ
(1 )
 (X V
T
1
X)
1
T
X V
1
y
  REML { yy
Y  ˆ
V 
T
n , X , Q}
(1 )
X  X
(2)

(1 )
i
X
(1 ) 
(1 )
Qi X
(1 )  T

i
EM
approach
(0)

(2)
j
(2)
Qj
j
Step 2
ˆ
(2)
ˆ
 (X V
T
(2)
1
X)
1
T
X V
1
y
Friston et al. (2004)
Mixed effects and fMRI
studies, Neuroimage
, }
Overview
Why hierarchical models?
The model
Estimation
Expectation-Maximization
Summary Statistics approach
Variance components
Examples
Sphericity
C  Cov( )  E ( )
T
y  X  
Scans
‚sphericity‘ means:
Cov( )   I
2
Var ( i )  
 1
2
Scans
i.e.
2
2nd level: Non-sphericity
Error
covariance
Errors are independent
but not identical
Errors are not independent
and not identical
Example I
Stimuli:
Auditory Presentation (SOA = 4 secs) of
(i) words and (ii) words spoken backwards
e.g.
“Book”
and
“Koob”
Subjects:
Scanning:
(i) 12 control subjects
(ii) 11 blind subjects
fMRI, 250 scans per
subject, block design
U. Noppeney et al.
Population differences
1st level:
Controls
Blinds
2nd level:
V
c  [1  1]
T
X
Example II
Stimuli:
Auditory Presentation (SOA = 4 secs) of words
Motion Sound
Visual
Action
“jump” “click” “pink”
“turn”
Subjects:
(i) 12 control subjects
Scanning:
fMRI, 250 scans per
subject, block design
Question:
What regions are affected
by the semantic content of
the words?
U. Noppeney et al.
Repeated measures Anova
1st level:
1.Motion
2.Sound
?
=
3.Visual
?
?
=
=
X
2nd level:
4.Action
Repeated measures Anova
1st level:
Motion
Sound
?
Visual
?
?
=
Action
=
=
X
2nd level:
1

T
c  0
0

V
X
1
0
1
1
0
1
0 

0 
 1
Some practical points
RFX: If not using multi-dimensional contrasts at 2nd level (Ftests), use a series of 1-sample t-tests at the 2nd level.
Use mixed-effects model only, if seriously in doubt about
validity of summary statistics approach.
Conclusion
Linear hierarchical models are general enough for typical
multi-subject imaging data (PET, fMRI, EEG/MEG).
Summary statistics are robust approximation to mixed-effects
analysis.
To minimize number of variance components to be estimated
at 2nd level, compute relevant contrasts at 1st level and use
simple test at 2nd level.