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![Page 1: The General Linear Model Christophe Phillips Cyclotron Research Centre University of Liège, Belgium SPM Short Course London, May 2011.](https://reader035.fdocuments.net/reader035/viewer/2022070306/5516c3a7550346f6208b5950/html5/thumbnails/1.jpg)
The General Linear Model
Christophe Phillips
Cyclotron Research Centre
University of Liège, Belgium
SPM Short Course
London, May 2011
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Normalisation
Statistical Parametric MapImage time-series
Parameter estimates
General Linear ModelRealignment Smoothing
Design matrix
Anatomicalreference
Spatial filter
StatisticalInference
RFT
p <0.05
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Passive word listeningversus rest
7 cycles of rest and listening
Blocks of 6 scanswith 7 sec TR
Question: Is there a change in the BOLD response between listening and rest?
Stimulus function
One session
A very simple fMRI experiment
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stimulus function
1. Decompose data into effects and error
2. Form statistic using estimates of effects and error
Make inferences about effects of interest
Why?
How?
datastatisti
c
Modelling the measured data
linearmode
l
effects estimate
error estimate
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Time
BOLD signal
Tim
e
single voxeltime series
single voxeltime series
Voxel-wise time series analysis
Modelspecification
Modelspecification
Parameterestimation
Parameterestimation
HypothesisHypothesis
StatisticStatistic
SPMSPM
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BOLD signal
Time =1 2+ +
err
or
x1 x2 e
Single voxel regression model
exxy 2211
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Mass-univariate analysis: voxel-wise GLM
=
e+yy XX
N
1
N N
1 1p
pModel is specified by
both1. Design matrix X2. Assumptions about
e
Model is specified by both
1. Design matrix X2. Assumptions about
e
eXy
The design matrix embodies all available knowledge about experimentally controlled factors and
potential confounds.
),0(~ 2INe
N: number of scansp: number of regressors
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• one sample t-test• two sample t-test• paired t-test• Analysis of Variance
(ANOVA)• Factorial designs• correlation• linear regression• multiple regression• F-tests• fMRI time series models• Etc..
GLM: mass-univariate parametric analysis
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Parameter estimation
eXy
= +
e
2
1
Ordinary least squares estimation
(OLS) (assuming i.i.d. error):
Ordinary least squares estimation
(OLS) (assuming i.i.d. error):
yXXX TT 1)(ˆ
Objective:estimate parameters to minimize
N
tte
1
2
y X
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A geometric perspective on the GLM
y
e
Design space defined by X
x1
x2 ˆ Xy
Smallest errors (shortest error vector)when e is orthogonal to X
Ordinary Least Squares (OLS)
0eX T
XXyX TT
0)ˆ( XyX T
yXXX TT 1)(ˆ
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x1
x2x2*
y
Correlated and orthogonal regressors
When x2 is orthogonalized w.r.t. x1, only the parameter
estimate for x1 changes, not that for x2!
Correlated regressors = explained variance is
shared between regressors
121
2211
exxy
1;1 *21
*2
*211
exxy
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eXy = +
e
2
1
y X
),0(~ 2INe
What are the problems of this model?
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What are the problems of this model?
1. BOLD responses have a delayed and dispersed form.
HRF
2. The BOLD signal includes substantial amounts of low-frequency noise (eg due to scanner drift).
3. Due to breathing, heartbeat & unmodeled neuronal activity, the errors are serially correlated. This violates the i.i.d. assumptions of the noise model in the GLM
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t
dtgftgf0
)()()(
Problem 1: Shape of BOLD responseSolution: Convolution model
expected BOLD response = input function impulse response function (HRF)
=
Impulses HRF Expected BOLD
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Convolution model of the BOLD response
Convolve stimulus function with a canonical hemodynamic response function (HRF):
HRF
t
dtgftgf0
)()()(
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blue = data
black = mean + low-frequency drift
green = predicted response, taking into account low-frequency drift
red = predicted response, NOT taking into account low-frequency drift
Problem 2: Low-frequency noise Solution: High pass filtering
discrete cosine transform (DCT) set
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discrete cosine transform (DCT) set
discrete cosine transform (DCT) set
High pass filtering
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withwithttt aee 1 ),0(~ 2 Nt
1st order autoregressive process: AR(1)
)(eCovautocovariance
function
N
N
Problem 3: Serial correlations
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Multiple covariance components
= 1 + 2
Q1 Q2
Estimation of hyperparameters with ReML (Restricted Maximum Likelihood).
V
enhanced noise model at voxel i
error covariance components Qand hyperparameters
jj
ii
QV
VC
2
),0(~ ii CNe
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1 1 1ˆ ( )T TX V X X V y
Parameters can then be estimated using Weighted Least Squares (WLS)
Let 1TW W V
1
1
ˆ ( )
ˆ ( )
T T T T
T Ts s s s
X W WX X W Wy
X X X y
Then
where
,s sX WX y Wy
WLS equivalent toOLS on whiteneddata and design
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Contrasts &statistical parametric
maps
Q: activation during listening ?
Q: activation during listening ?
c = 1 0 0 0 0 0 0 0 0 0 0
Null hypothesis:Null hypothesis: 01
)ˆ(
ˆ
T
T
cStd
ct
X
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Summary
Mass univariate approach.
Fit GLMs with design matrix, X, to data at different points in space to estimate local effect sizes,
GLM is a very general approach
Hemodynamic Response Function
High pass filtering
Temporal autocorrelation
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Thank you for your attention!
…and thanks to Guillaume for his slides .