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



Estimation in SPM2


‚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


Summary


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



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

Variance components

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