Variance components
description
Transcript of Variance components
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Variance componentsVariance componentsVariance componentsVariance components
Wellcome Dept. of Imaging Neuroscience
Institute of Neurology, UCL, London
Stefan KiebeStefan Kiebell
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Modelling in SPM
pre-processingpre-processinggenerallinearmodel
generallinearmodel
SPMsSPMs
functional datafunctional data
templatestemplates
smoothednormalised
data
smoothednormalised
data
design matrixdesign matrix
variance componentsvariance components
hypotheseshypotheses
adjustedP-values
adjustedP-values
parameterestimation
parameterestimation
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general linear model Xy
=
+X
N
1
N N
1 1p
p
model specified by1. design matrix X2. assumptions about
model specified by1. design matrix X2. assumptions about
N: number of observations p: number of regressors
N: number of observations p: number of regressors
error normally
distributed
error normally
distributed
y
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Summary
Sphericity/non-sphericitySphericity/non-sphericity
Restricted Maximum Likelihood (ReML)Restricted Maximum Likelihood (ReML)
Estimation in SPM2Estimation in SPM2
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Summary
Sphericity/non-sphericitySphericity/non-sphericity
Restricted Maximum Likelihood (ReML)Restricted Maximum Likelihood (ReML)
Estimation in SPM2Estimation in SPM2
Sphericity/non-sphericitySphericity/non-sphericity
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‚sphericity‘
‚sphericity‘ means:‚sphericity‘ means:
ICov 2)(
Xy )()( TECovC
Scans
Sca
nsi.e.
2)( iVar12
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‚non-sphericity‘
non-sphericity means that the error covariance doesn‘t
look like this*:
non-sphericity means that the error covariance doesn‘t
look like this*:
*: or can be brought through a linear transform to this form
*: or can be brought through a linear transform to this form
ICov 2)(
10
01)(Cov
10
04)(Cov
21
12)(Cov
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Example: serial correlations
withwithttt a 1 ),0(~ 2 Nt
autoregressive process of order 1 (AR(1))
)(Covautocovariance-
function
N
N
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Restricted Maximum Likelihood (ReML)Restricted Maximum Likelihood (ReML)
Summary
Sphericity/non-sphericitySphericity/non-sphericity
Estimation in SPM2Estimation in SPM2
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Restricted Maximum Likelihood
Xy ?)(Cov observed
ReMLReMLestimated
2211ˆˆ QQ
j
Tjj yy
voxel
1Q
2Q
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t-statistic (OLS estimator)
Xy
c = +1 0 0 0 0 0 0 0 0 0 0c = +1 0 0 0 0 0 0 0 0 0 0
)ˆ(ˆ
ˆ
T
T
cdtS
ct
cVXXccdtSTTT 2ˆ)ˆ(ˆ
)(
ˆˆ
2
2
RVtr
Xy
approximate degrees of freedom following
Satterthwaite
approximate degrees of freedom following
SatterthwaiteReML-estimate
ReML-estimate
yX ̂
)(2 CovV
XXIR
VX
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Variance components
Variance components Q model the error
Variance components Q model the error
KKQQQCovV 2211)(
Xy
model for sphericitymodel for sphericity
IQ 12
1 and model for inhomogeneous
variances (2 groups)
model for inhomogeneousvariances (2 groups)
1Q1Q 2Q
The variance parameters are estimated by ReML.
The variance parameters are estimated by ReML.
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Example I
Stimuli:Stimuli: Auditory Presentation (SOA = 4 secs) of(i) words and (ii) words spoken backwards
Auditory Presentation (SOA = 4 secs) of(i) words and (ii) words spoken backwards
Subjects:Subjects:
e.g. “Book”
and “Koob”
e.g. “Book”
and “Koob”
fMRI, 250 scans per subject, block design
fMRI, 250 scans per subject, block design
Scanning:Scanning:
U. Noppeney et al.U. Noppeney et al.
(i) 12 control subjects(ii) 11 blind subjects
(i) 12 control subjects(ii) 11 blind subjects
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Population differences
1st level:1st level:
2nd level:2nd level:
ControlsControls BlindsBlinds
X
]11[ TcV
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Estimation in SPM2Estimation in SPM2
Summary
Sphericity/non-sphericitySphericity/non-sphericity
Restricted Maximum Likelihood (ReML)Restricted Maximum Likelihood (ReML)
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Estimating variances
111
NppNN
Xy EM-algorithmEM-algorithm
yCXC
XCXCT
yy
Ty
1||
11| )(
gJ
d
LdJ
d
dLg
1
2
2
E-stepE-step
M-stepM-step
K. Friston et al. 2002, Neuroimage
K. Friston et al. 2002, Neuroimage
kk
kQC
Assume, at voxel j:
Assume, at voxel j: kjjk
)lnL maximise p(y|λ
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Time
Intensity
Tim
e
Time series inone voxel
Time series inone voxel
voxelwise
model specification
model specification
parameterestimation
parameterestimation
hypothesishypothesis
statisticstatistic
SPMSPM
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Spatial ‚Pooling‘
Assumptions in SPM2:• global correlation matrix V • local variance
Assumptions in SPM2:• global correlation matrix V • local variance
observed
ReMLReML
estimated
2211ˆˆˆ QQC
jvoxel
Tjj yy
Matrix is where
, )ˆ(
ˆ
NNV
Ctrace
nCV
global
)( ,
)(ˆ
2/12/121
2
XVXVIRyRVr
Rtr
rr
j/
j
jTj
j
local in voxel j: VC jj2ˆˆ
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Estimation in SPM2
jjj Xy
jOLSj yX ,̂
),,ReML()(ˆˆ
QXyyvoCCjvoxel
Tjj
jTT
MLj yVXXVX 111, )(ˆ
‚quasi‘-Maximum Likelihood‚quasi‘-Maximum LikelihoodOrdinary least-squaresOrdinary least-squares
ReML (pooled estimate)ReML (pooled estimate)
•optional in SPM2•one pass through data•statistic using (approximated) effective degrees of freedom
•optional in SPM2•one pass through data•statistic using (approximated) effective degrees of freedom
•2 passes (first pass for selection of voxels)
•more precise estimate of V
•2 passes (first pass for selection of voxels)
•more precise estimate of V
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t-statistic (ML-estimate)
Xy
c = +1 0 0 0 0 0 0 0 0 0 0c = +1 0 0 0 0 0 0 0 0 0 0
)ˆ(ˆ
ˆ
T
T
cdtS
ct
cWXWXccdtSTTT )()(ˆ)ˆ(ˆ 2
)(
ˆˆ
2
2
Rtr
WXWy
ReML-estimate
ReML-estimate
WyWX )(̂)(2
2/1
CovV
VW
)(WXWXIR
VX
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Example II
Stimuli:Stimuli: Auditory Presentation (SOA = 4 secs) of words Auditory Presentation (SOA = 4 secs) of words
Subjects:Subjects:
fMRI, 250 scans persubject, block design
fMRI, 250 scans persubject, block designScanning:Scanning:
U. Noppeney et al.U. Noppeney et al.
(i) 12 control subjects(i) 12 control subjects
Motion Sound Visual Action
“jump” “click” “pink” “turn”
Question:Question:What regions are affectedby the semantic content ofthe words?
What regions are affectedby the semantic content ofthe words?
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Repeated measures Anova
1st level:1st level:
2nd level:2nd level:
VisualVisual ActionAction
X
1100
0110
0011Tc
?=
?=
?=
MotionMotion SoundSound
V
X