Transcript Variance components Stefan Kiebel Wellcome Dept. of Imaging Neuroscience Institute of Neurology, UCL, London.

Variance components
Stefan Kiebel
Wellcome Dept. of Imaging Neuroscience
Institute of Neurology, UCL, London
Modelling in SPM
functional data
design matrix
hypotheses
smoothed
normalised
data
pre-processing
templates
general
linear
model
variance components
parameter
estimation
SPMs
adjusted
P-values
y  X  
general linear model
p
1
1
1

y
N
=
X
N
N: number of observations
p: number of regressors
p
+

error 
normally
distributed
N
model specified by
1. design matrix X
2. assumptions about 
Summary
Sphericity/non-sphericity
Restricted Maximum Likelihood (ReML)
Estimation in SPM2
Summary
Sphericity/non-sphericity
Restricted Maximum Likelihood (ReML)
Estimation in SPM2
‚sphericity‘
C  Cov( )  E ( )
T
y  X  
Scans
‚sphericity‘ means:
Cov( )   I
2
i.e. Var ( )  
i
2
Scans
 2 1
‚non-sphericity‘
 4 0
Cov( )  

0 1 
non-sphericity means that
the error covariance doesn‘t
look like this*:
Cov( )   I
2
1 0
Cov( )  

0
1


*: or can be brought through a
linear transform to this form
2 1 
Cov( )  

1 2 
Example: serial correlations
 t  a t 1  t with t ~ N (0,  2 )
autoregressive process of
order 1 (AR(1))
N
Cov( )
autocovariancefunction
N
Summary
Sphericity/non-sphericity
Restricted Maximum Likelihood (ReML)
Estimation in SPM2
Restricted Maximum Likelihood
y  X  
Cov( ) ?
observed
Q1
T
y
y
 j j
voxel j
ReML
estimated
Q2
ˆ1Q1  ˆ2Q2
t-statistic (OLS estimator)
c ˆ
t
T ˆ
ˆ
St d (c  )
y  X  
T
̂  X  y
 2V  Cov( )
2 T

T
ˆ
Stˆd (c  )  ˆ c X VX c
T
c = +1 0 0 0 0 0 0 0 0 0 0
X
ˆ 2



y  Xˆ

2
V
tr ( RV )
R  I  XX 
approximate degrees of
freedom following
Satterthwaite
ReMLestimate
Variance components
y  X  
V  Cov( )  1Q1  2Q2    K QK
The variance parameters  are
estimated by ReML.
Variance components Q
model the error
Q1 
Q1 
Q2 
1   and Q1  I
2
model for sphericity
model for inhomogeneous
variances (2 groups)
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
cT  [1  1]
X
Summary
Sphericity/non-sphericity
Restricted Maximum Likelihood (ReML)
Estimation in SPM2
Estimating variances
y  X   
N 1
N p
p1
EM-algorithm
N 1
C | y  ( X T C1 X ) 1
  | y  C | y X C y
T
maximise
1

E-step
L  ln p(y|λ)
dL
d
d 2L
J 2
d
    J 1 g
g
C   k Qk
k
Assume, at voxel j:
M-step
 jk   j k
K. Friston et al. 2002,
Neuroimage
voxelwise
model
specification
Time
parameter
estimation
hypothesis
statistic
Intensity
Time series in
one voxel
SPM
Spatial ‚Pooling‘
•
•
Assumptions in SPM2:
global correlation matrix V
local variance
estimated
observed
ReML
T
y
y
 j j
Cˆ  ˆ1Q1  ˆ2Q2
voxel j
local in voxel j:
ˆ 
2
j
T
j j
Cˆ j  ˆ 2j V
r r
tr ( R)
rj  RV 1/ 2 y j , R  I  V 1/ 2 X (V 1/ 2 X ) 
Cˆ  n
V
,
ˆ
trace(C )
global
where V is N  N  Matrix
Estimation in SPM2
y j  X j   j
Cˆ  Coˆv( )  ReML(
T
y
y
 j j , X , Q)
voxel j
ReML (pooled estimate)
̂ j ,OLS  X y j

Ordinary least-squares
•optional in SPM2
•one pass through data
•statistic using (approximated)
effective degrees of freedom
ˆ j ,ML  ( X TV 1 X ) 1 X TV 1 y j
‚quasi‘-Maximum Likelihood
•2 passes (first pass for selection of
voxels)
•more precise estimate of V
t-statistic (ML-estimate)
y  X  
W  V 1/ 2
c ˆ
2
t

V  Cov( )
T ˆ
Stˆd (c  )
T ˆ
2 T

T
Stˆd (c  )  ˆ c (WX ) (WX ) c
T
̂  (WX )  Wy
c = +1 0 0 0 0 0 0 0 0 0 0
X
V
ˆ 
2

 Wy  WXˆ

2
tr( R)
R  I  WX (WX ) 
ReMLestimate
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:
Motion
Sound
?
Visual
?
?
=
=
2nd level:
Action
=
X
 1 1 0 0 


cT   0 1  1 0 
 0 0 1  1


V
X