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Variance components. Stefan Kiebe l. Wellcome Dept. of Imaging Neuroscience Institute of Neurology, UCL, London. Modelling in SPM. functional data. design matrix. hypotheses. smoothed normalised data. parameter estimation. general linear model. pre-processing. SPMs. adjusted - PowerPoint PPT Presentation
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Variance componentsVariance components
Wellcome Dept. of Imaging NeuroscienceInstitute of Neurology, UCL, London
Stefan KiebeStefan Kiebell
Modelling in SPM
pre-processinggenerallinearmodel
SPMs
functional data
templates
smoothednormalised
data
design matrix
variance components
hypotheses
adjustedP-values
parameterestimation
general linear model Xy
=
+X
N
1
N N
1 1p
p
model specified by1. design matrix X2. assumptions about
N: number of observations p: number of regressors
error normally
distributedy
Summary
Sphericity/non-sphericity
Restricted Maximum Likelihood (ReML)
Estimation in SPM2
Summary
Sphericity/non-sphericity
Restricted Maximum Likelihood (ReML)
Estimation in SPM2
Sphericity/non-sphericity
‚sphericity‘
‚sphericity‘ means:
ICov 2)(
Xy )()( TECovC
Scans
Scan
si.e.
2)( iVar12
‚non-sphericity‘non-sphericity means that
the error covariance doesn‘t look like this*:
*: or can be brought through a linear transform to this form
ICov 2)(
1001
)(Cov
1004
)(Cov
2112
)(Cov
Example: serial correlations
withttt a 1 ),0(~ 2 Nt
autoregressive process of order 1 (AR(1))
)(Covautocovariance-
function
N
N
Restricted Maximum Likelihood (ReML)
Summary
Sphericity/non-sphericity
Estimation in SPM2
Restricted Maximum Likelihood
Xy ?)(Cov observed
ReMLestimated
2211ˆˆ QQ
j
Tjj yy
voxel
1Q
2Q
t-statistic (OLS estimator)
Xy
c = +1 0 0 0 0 0 0 0 0 0 0
)ˆ(ˆ
ˆ
T
T
cdtSct
cVXXccdtSTTT 2ˆ)ˆ(ˆ
)(
ˆˆ
2
2
RVtrXy
approximate degrees of freedom following
SatterthwaiteReML-estimate
yX ̂
)(2 CovV
XXIR
VX
Variance components
Variance components Q model the error
KKQQQCovV 2211)(
Xy
model for sphericity
IQ 12
1 and model for inhomogeneous
variances (2 groups)
1Q1Q 2Q
The variance parameters are estimated by ReML.
Example I
Stimuli: Auditory Presentation (SOA = 4 secs) of(i) words and (ii) words spoken backwards
Subjects:
e.g. “Book”
and “Koob”
fMRI, 250 scans per subject, block design
Scanning:U. Noppeney et al.
(i) 12 control subjects(ii) 11 blind subjects
Population differences1st level:
2nd level:
Controls Blinds
X
]11[ TcV
Estimation in SPM2
Summary
Sphericity/non-sphericity
Restricted Maximum Likelihood (ReML)
Estimating variances
111
NppNN
Xy EM-algorithm
yCXC
XCXCT
yy
Ty
1||
11| )(
gJd
LdJ
ddLg
1
2
2
E-step
M-step
K. Friston et al. 2002, Neuroimage
kk
kQC
Assume, at voxel j: kjjk
)lnL maximise p(y|λ
Time
Intensity
Tim
e
Time series inone voxel
voxelwise
model specification
parameterestimationhypothesis
statistic
SPM
Spatial ‚Pooling‘Assumptions in SPM2:
• global correlation matrix V • local variance
observed
ReML
estimated
2211ˆˆˆ QQC
jvoxel
Tjj yy
Matrix is where
, )ˆ(
ˆ
NNVCtracenCV
global
)( ,
)(ˆ
2/12/121
2
XVXVIRyRVr
Rtrrr
j/
j
jTj
j
local in voxel j: VC jj2ˆˆ
Estimation in SPM2
jjj Xy
jOLSj yX ,̂
),,ReML()(ˆˆ
QXyyvoCCjvoxel
Tjj
jTT
MLj yVXXVX 111, )(ˆ
‚quasi‘-Maximum LikelihoodOrdinary least-squares
ReML (pooled estimate)
•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
t-statistic (ML-estimate) Xy
c = +1 0 0 0 0 0 0 0 0 0 0
)ˆ(ˆ
ˆ
T
T
cdtSct
cWXWXccdtSTTT )()(ˆ)ˆ(ˆ 2
)(
ˆˆ
2
2
RtrWXWy
ReML-estimate
WyWX )(̂)(2
2/1
CovV
VW
)(WXWXIR
VX
Example II
Stimuli: Auditory Presentation (SOA = 4 secs) of words
Subjects:
fMRI, 250 scans persubject, block design
Scanning:
U. Noppeney et al.
(i) 12 control subjects
Motion Sound Visual Action“jump” “click” “pink” “turn”
Question:What regions are affectedby the semantic content ofthe words?
Repeated measures Anova1st level:
2nd level:
Visual Action
X
110001100011
Tc
?=
?=
?=
Motion Sound
V
X