Download - Variance components
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