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The General Linear Model (for dummies…)
Carmen Tur and Ashwani Jha 2009
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Overview of SPM
RealignmentRealignment SmoothingSmoothing
NormalisationNormalisation
General linear modelGeneral linear model
Statistical parametric map (SPM)Statistical parametric map (SPM)Image time-seriesImage time-series
Parameter estimatesParameter estimates
Design matrixDesign matrix
TemplateTemplate
KernelKernel
Gaussian Gaussian field theoryfield theory
p <0.05p <0.05
StatisticalStatisticalinferenceinference
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What is the GLM?
• It is a model (ie equation)• It provides a framework that allows us to
make refined statistical inferences taking into account:– the (preprocessed) 3D MRI images– time information (BOLD time series)– user defined experimental conditions– neurobiologically based priors (HRF)– technical / noise corrections
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How does work?
• By creating a linear model:
Collect Data
X
Y
Data
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How does work?
• By creating a linear model:
Collect Data
Generate model
X
Y
Data
Y=bx + c
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How does work?
• By creating a linear model:
Collect Data
Generate model
Fit model
X
Y
Data
Y=0.99x + 12 + e
e
N
tte
1
2 = minimum
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How does work?
• By creating a linear model:
Collect Data
Generate model
Fit model
Test modelX
Y
Data
Y=0.99x + 12 + e
e
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GLM matrix format
Y = β1 X1 + C + e 5.9 1 2 15.0 2 0 18.4 3 5 12.3 4 4 24.7 5 8 23.2 6 8 19.3 7 0 13.6 8 9 26.1 9 1 21.6 10 5 31.7 11 2
• But GLM works with lists of numbers (matrices)
Y = 0.99x + 12 + e
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GLM matrix format
• But GLM works with lists of numbers (matrices)
Y = β1 X1 + β2 X2 + C + e 5.9 1 2 15.0 2 2 18.4 3 5 12.3 4 5 24.7 5 5 23.2 6 2 19.3 7 2 13.6 8 5 26.1 9 5 21.6 10 5 31.7 11 2
Y = 0.99x + 12 + e
‘non-linear’We need to put this in… (data Y, and design matrix Xs)
sphericity assumption
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GLM matrix format
=
e+y X
N
1
N N
1 1p
p
eXy ),0(~ 2INe
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fMRI example (from SPM course)…
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Passive word listeningversus rest7 cycles of rest and listeningBlocks of 6 scanswith 7 sec TR
Question: Is there a change in the BOLD response between listening and rest?
Stimulus function
One sessionA very simple fMRI experiment
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Time
BOLD signal
Time
A closer look at the data (Y)…
Look at each voxel over time (mass univariate)
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BOLD signal
Time = + erro
r
e
2+
x2
1
x1
The rest of the model…
exxy 2211
Instead of C
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The GLM matrix format
eXy
= +
e
2
1
y X
),0(~ 2INe
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…easy!
• How to solve the model (parameter estimation)
• Assumptions (sphericity of error)
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• Actually try to estimate
• ‘best’ has lowest overall error ie the sum of squares of the error:
• But how does this apply to GLM, where X is a matrix…
^
Solving the GLM (finding )
Y=0.99x + 12 + e
e
N
tte
1
2 = minimum
^
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…need to geometrically visualise the GLM in N dimensions
= +
e
2
1
y x1 x2
N
^
^
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…need to geometrically visualise the GLM
031
01
2
121
= +e
2
1
y x1 x2
N=3
x2
x1
Design space defined by y = X
^
^^What about the actual data y?
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…need to geometrically visualise the GLM
= +e
2
1
y x1 x2
N=3
x2
x1
Design space defined by y = X
^
^^
y
031
01
2
121
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Once again in 3D..
• The design (X) can predict the data values (y) in the design space.
• The actual data y, usually lies outside this space.
• The ‘error’ is difference.
ye
Design space defined by X
x1
x2 ̂ˆ Xy
^
eXy
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Solving the GLM (finding ) – ordinary least squares (OLS)
• To find minimum error:
• e has to be orthogonal to design space (X). “Project data onto model”
• ie: XTe = 0XT(y - X) = 0 XTy = XTX
eXy
ye
Design space defined by X
x1
x2 ̂ˆ Xy
N
tte
1
2 = minimum
yXXX TT 1)(ˆ eXy
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Assuming sphericity
• We assume that the error has:– a mean of 0, – is normally distributed– is independent (does not
correlate with itself)
),0(~ 2INe
e
526
97
51
21
e
=
e
frequency
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Assuming sphericity
• We assume that the error has:– a mean of 0, – is normally distributed– is independent (does not
correlate with itself)
),0(~ 2INe
e
526
97
51
21
e
=te
1te
x
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Solution
Half-way re-cap…
eXy
= +
e
2
1
Ordinary least squares
estimation (OLS) (assuming i.i.d.
error):yXXX TT 1)(ˆ
y X
GLM
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GLM
Methods for dummies 2009-10London, 4th November 2009
II partCarmen Tur
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I. BOLD responses have a delayed and dispersed form
HRF
Neural stimulus hemodynamic response
time
Neural stimulus
Hemodynamic Response Function: This is the expected BOLD signal if a neural stimulus takes place
expected BOLD response = input function impulse response function (HRF)
expected BOLD
response
Problems of this model
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I. BOLD responses have a delayed and dispersed form Solution: CONVOLUTION
HRF
Transform neural stimuli function into a expected BOLD signal with a canonical hemodynamic response function (HRF)
t
dtgftgf0
)()()(
Problems of this model
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II. The BOLD signal includes substantial amounts of low-frequency noise
Problems of this model
WHY? Multifactorial: biorhythms, coil heating, etc…
HOW MAY OUR DATA LOOK LIKE?
IntensityOf BOLD signal
Time
Real dataPredicted response, NOT taking into account low-frequency drift
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II. The BOLD signal includes substantial amounts of low-frequency noise
Problems of this model
Solution: HIGH PASS FILTERING
discrete cosine
transform (DCT) set
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STEPS so far…Interim summary: GLM so far…
1. Acquisition of our data (Y)2. Design our matrix (X)3. Assumptions of GLM 4. Correction for BOLD signal shape: convolution5. Cleaning of our data of low-frequency noise6. Estimation of βs
But our βs may still be wrong! Why?
7. Checkup of the error… Are all the assumptions of the error satisfied?
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III. The data are serially correlated Temporal autocorrelation:
in y = Xβ + e over time
0
It is…
t
e
e in time t is correlated with e in time t-1
Problems of this model
It should be…
0 t
e
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III. The data are serially correlated Temporal autocorrelation:
in y = Xβ + e over time
WHY? Multifactorial…
Problems of this model
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III. The data are serially correlated Temporal autocorrelation:
in y = Xβ + e over timeet = aet-1 + ε (assuming ε ~ N(0,σ2I))
Problems of this model
autocovariance function
Autoregressive model
in other words: the covariance of error at time t (et) and error at
time t-1 (et-1) is not zero
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III. The data are serially correlated Temporal autocorrelation:
in y = Xβ + e over timeet = aet-1 + ε (assuming ε ~ N(0,σ2I))
Problems of this model
Autoregressive model
et = aet-1 + ε
et = a (aet-2 + ε) + ε et = a2et-2 + aε + ε
et = a2(aet-3 + ε) + aε + ε et = a3et-3 + a2ε + aε + ε…
et -1= aet-2 + εet -2= aet-3 + ε…
But a is a number between 0 and 1
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Problems of this model
in other words: the covariance of error at time t (et) and error at time t-1
(et-1) is not zero
time (scans)
time(scans)
ERROR:Covariance
matrix
1 2 3 4 5 6 7 8 9 101
2
3
4
5
6
7
8
9
10
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III. The data are serially correlated Temporal autocorrelation:
in y = Xβ + e over timeet = aet-1 + ε (assuming ε ~ N(0,σ2I))
autocovariance function
Problems of this model
Autoregressive model
This violates the assumption of the error e ~ N (0, σ2I)
in other words: the covariance of error at time t (et) and error at
time t-1 (et-1) is not zero
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III. The data are serially correlated Solution:
1. Use an enhanced noise model with hyperparameters for multiple error covariance components
Problems of this model
It should be… It is…
et = aet-1 + εa ≠ 0 a = 0
et = aet-1 + εet = ε But…a? ?
et = εOr, if you
wish
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III. The data are serially correlated Solution:
1. Use an enhanced noise model with hyperparameters for multiple error covariance components
Problems of this model
It should be… It is…
et = aet-1 + εa ≠ 0 a = 0
et = aet-1 + εet = ε But…a? ?
We would like to know covariance (a, autocovariance) of error
But we can only estimate it: V V = Σ λiQ i
V = λ1Q1 + λ2Q 2 λ1 and λ2:
hyperparameters
Q1 and Q2: multiple error covariance
components
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III. The data are serially correlated Solution:
1. Use an enhanced noise model with hyperparameters for multiple error covariance components 2. Use estimated autocorrelation to specify filter matrix W for whitening the data
et = aet-1 + ε (assuming ε ~ N(0,σ2I))
WY = WXβ + We
Problems of this model
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Other problems – Physiological confounds
• head movements
• arterial pulsations (particularly bad in brain stem)
• breathing
• eye blinks (visual cortex)
• adaptation affects, fatigue, fluctuations in concentration, etc.
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Other problems – Correlated regressors
Example: y = x1β1 + x2β2 + e
When there is high (but not perfect) correlation between regressors, parameters can be estimated… But the estimates will
be inefficiently estimated (ie highly variable)
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• HRF varies substantially across voxels and subjects
• For example, latency can differ by ± 1 second
• Solution: MULTIPLE BASIS FUNCTIONS (another talk)
Other problems – Variability in the HRF
HRF could be understood as a linear combination of A, B and C
A
BC
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Model everythingImportant to model all known
variables, even if not experimentally interesting:
effects-of-interest (the regressors we are actually interested in)
+head movement, block and subject
effects…
subjects
globalactivity or movementconditions:
effects of interest
Ways to improve the model
Minimise residual error variance
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• The aim of modelling the measured data was to make inferences about effects of interest
• Contrasts allow us to make such inferences
• How? T-tests and F-tests
How to make inferences
REMEMBER!!
Another talk!!!
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Using an easy example...
SUMMARY
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Given data (image voxel, y)
Y = X . β + ε
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Different (rigid and known) predictors (regressors, design matrix, X)
x1
x2
x3
x4
x5
x6 Time
Y = X . β + ε
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Fitting our models into our data (estimation of parameters, β)
x1 x2 x3 x4 x5 x6
Y = X . β + εY = X . β + ε
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Y = X . β + ε HOW?
Fitting our models into our data (estimation of parameters, β)
Minimising residual error variance
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Minimising residual error variance
e (error) = yo - ye
Minimising the Sums of Squares of the Error differences between your predicted model and the observed data
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y = x16 + x23+ x31+ x42+ x51+ x636 + e
Y = X . β + ε
Fitting our models into our data (estimation of parameters, β)
y = x1β1 + x2β2 + x3β3 + x4β4 + x5β5 + x6β6 + e
We must pay attention to the problems that the GLM has…
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Y = X . β + ε
Making inferences:our final goal!!
The end
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REFERENCES1. Talks from previous years2. Human brain function
THANKS TO GUILLAUME FLANDIN
Many thanks for your attentionLondon 4th Nov 2009
GENERAL LINEAR MODEL – Methods for Dummies 2009-2010