The General Linear Model (GLM) in SPM5. Klaas Enno Stephan Wellcome Trust Centre for Neuroimaging University College London With many thanks to my colleagues in the FIL methods group, particularly Stefan Kiebel, for useful slides.
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Klaas Enno Stephan
Wellcome Trust Centre for Neuroimaging
University College London
With many thanks to my colleagues in the FIL methods group, particularly Stefan Kiebel, for useful slides
SPM Short Course, May 2007Wellcome Trust Centre for Neuroimaging
Overview of SPM
Statistical parametric map (SPM)
Design matrix
Image time-series
Kernel
Realignment
Smoothing
General linear model
Gaussian
field theory
Statistical
inference
Normalisation
p <0.05
Template
Parameter estimates
A very simple fMRI experiment
One session
Passive word listening
versus rest
7 cycles of
rest and listening
Blocks of 6 scans
with 7 sec TR
Stimulus function
Question: Is there a change in the BOLD response between listening and rest?
Modelling the measured data
Why?
Make inferences about effects of interest
How?
stimulus function
effects estimate
linear
model
statistic
data
error estimate
model
specification
parameter
estimation
hypothesis
statistic
Voxel-wise time series analysis
Time
Time
BOLD signal
single voxel
time series
SPM
Single voxel regression model
error
=
+
+
1
2
Time
e
x1
x2
BOLD signal
Mass-univariate analysis: voxel-wise GLM
=
+
y
N: number of scans
p: number of regressors
The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.
Examples for non-sphericity:
sphericity = iid:error covariance is scalar multiple of identity matrix:
Cov(e) = 2I
non-identity
non-identity
non-independence
Ordinary least squares (OLS):
Estimate parameters such that
minimal
Parameter estimation: Ordinary least squares
=
+
X
y
OLS parameter estimate
(assuming iid error)
A geometric perspective
y
e
Design space defined by X
x2
x1
y
x2
x2*
x1
Correlated regressors =
explained variance is shared between regressors
When x2 is orthogonalized with regard to x1, only the parameter estimate for x1 changes, not that for x2!
HRF
hemodynamic response function (HRF)
Problem 1: Shape of BOLD responseSolution: Convolution model
The response of a linear time-invariant (LTI) system is the convolution of the input with the system's response to an impulse (delta function).
expected BOLD response
= input function impulse response function (HRF)
Convolve stimulus function with a canonical hemodynamic response function (HRF):
HRF
Problem 2: Low-frequency noise Solution: High pass filtering
S = residual forming matrix of DCT set
discrete cosine transform (DCT) set
blue= data
black = mean + low-frequency drift
green= predicted response, taking into account low-frequency drift
red= predicted response (with low-frequency drift explained away)
High pass filtering: example
Problem 3: Serial correlations
with
1st order autoregressive process: AR(1)
autocovariance
function
enhanced noise model
Q2
Q1
V
1
+ 2
=
Estimation of hyperparameters with ReML (restricted maximum likelihood).
Three hyperparameters: 1, 2,
Contrasts &statistical parametric maps
c = 1 0 0 0 0 0 0 0 0 0 0
Q: activation during listening ?
Null hypothesis:
t-statistic based on ML estimates
c = +1 0 0 0 0 0 0 0 0 0 0
ReML-estimate
Physiological confounds
Summary