02 General Linear Model
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The General Linear Model
(GLM)Ged Ridgway
Wellcome Trust Centre for NeuroimagingUniversity College London
SPM CourseVancouver, August 2010
[slides from the FIL Methods group]
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Overview of SPM
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Simple regression and the GLM
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Passive wordlisteningversus rest
7 cycles of rest and listening
Blocks of 6 scanswith 7 sec TR
Question: Is there a change in the BOLD responsebetween listening and rest?
Stimulus function
One session
A very simple fMRI experiment
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stimulusfunction
1. Decompose data into effects anderror
2. Form statistic using estimates of effects and error
Make inferences about effects of interesthy?
How?
data linear model
effectsestimate
error estimate
statistic
Modelling the measured data
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BOLD signal
Tme
single voxeltime series
Voxel-wise time series analysis
modelspecification
parameter estimation
hypothesis
statistic
SPM
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BOLD signal
Tme
= β1 β2 + e
r r o r
x1 x2 e
Single voxel regression model
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Example GLM factorial design models
• 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…
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Example GLM factorial design models
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Example GLM factorial design models
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GLM assumes Gaussian spherical (i.i.d.) errors
sphericity = i.i.d.error covariance isscalar multiple of
identity matrix:Cov(e) = σ 2I
Examples for non-sphericity:
non-identit
non-independence
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Parameter estimation
= +
Ordinary least squares estimation (OLS)
(assuming i.i.d. error):
Objective:estimate parametersto minimize
y X
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A geometric perspective on the GLM
y
e
Design space
defined by X
x1
x2
Smallest errors (shortest error vector)when e is orthogonal to X
Ordinary Least Squares (OLS)
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What are the problems of this model?
1.
BOLD responses have a delayedand 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 theassumptions of the noise model in the GLM
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The response of a linear time-invariant (LTI) system is the convolution of the inputwith the system's response to an impulse (delta function).
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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ConvolutionSuperposition principle
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HRF
HRF convolution animations
Sliding the reversed HRF past a unit-
integral pulse and integrating theproduct recovers the HRF (hencethe name impulse response)
The step response shows a delay and
a slight overshoot, before reaching asteady-state. This gives some intuitionfor a square wave…
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HRF convolution animations
Slow square wave. Main (fundamental)
frequency of output matches input, butwith delay (phase-shift) and change inamplitude (and shape; sinusoids keeptheir shape)
Fast square wave. Amplitude is
reduced, phase shift is more dramatic(output almost in anti-phase).Fourier transform of HRF yieldsmagnitude and phase responses.
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P bl 2 L f i
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blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account
low-frequency driftred = predicted response, NOT taking into
account low-frequency drift
Problem 2: Low-frequency noiseSolution: High pass filtering
discrete cosine
transform (DCT) set
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discrete cosinetransform (DCT) set
High pass filtering
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with
1 st order autoregressive process: AR(1)
autocovariancefunction
Problem 3: Serial correlations
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Multiple covariance components
= λ 1 + λ 2
Q 1 Q 2
Estimation of hyperparameters λ with ReML (Restricted Maximum Likelihood).
V
enhanced noise model at voxel i error covariance components Qand hyperparameters λ
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Parameters can then be estimated using Weighted Least Squares (WLS)
Let
Then
where
WLS equivalent to
OLS on whiteneddata and design
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Contrasts &statistical parametric maps
Q: activation duringlistening ?
c = 1 0 0 0 0 0 0 0 0 0 0
Null hypothesis:
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Summary
• Mass univariate approach.
• Fit GLMs with design matrix, X, to data at different points inspace to estimate local effect sizes,
• GLM is a very general approach
• Hemodynamic Response Function
• High pass filtering
• Temporal autocorrelation
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x1
x2x2*
y
Correlated and orthogonal regressors
When x 2 is orthogonalized withregard to x 1, only the parameter estimate for x 1 changes, not thatfor x 2!
Correlated regressors =explained variance is sharedbetween regressors
t t ti ti b d ML ti t
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c = 1 0 0 0 0 0 0 0 0 0 0
ReML-
estimates
t-statistic based on ML estimates
For brevity: