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)
General linear model
A very simple fMRI experiment
Passive word listening
7 cycles of
rest and listening
Blocks of 6 scans
with 7 sec TR
Question: Is there a change in the BOLD response between listening and rest?
Modelling the measured data
Make inferences about effects of interest
Voxel-wise time series analysis
Single voxel regression model
Mass-univariate analysis: voxel-wise GLM
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
Ordinary least squares (OLS):
Estimate parameters such that
Parameter estimation: Ordinary least squares
OLS parameter estimate
(assuming iid error)
A geometric perspective
Design space defined by X
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!
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):
Problem 2: Low-frequency noise Solution: High pass filtering
S = residual forming matrix of DCT set
discrete cosine transform (DCT) set
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
1st order autoregressive process: AR(1)
enhanced noise model
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 ?
t-statistic based on ML estimates
c = +1 0 0 0 0 0 0 0 0 0 0