1. The general linear model and Statistical Parametric Mapping
Stefan Kiebel
Andrew Holmes
Wellcome Dept. of Imaging Neuroscience
Institute of Neurology, UCL, London

2. Passive word listening
fMRI example
One session
Time series of BOLD
responses in one voxel
Passive word listening
versus rest
7 cycles of
rest and listening
Each epoch 6 scans
with 7 sec TR
Stimulus function
Question: Is there a change in the BOLD response between listening and rest?

3. Why modelling? Why? Make inferences about effects of interest
Decompose data into effects and error
Form statistic using estimates of effects and error
How?
Model?
Use any available knowledge
Stimulus function
effects estimate
statistic
data
model
error estimate

4. Modelling with SPM General linear model Preprocessing SPMs
functional data
Design matrix
Contrasts
Smoothed
normalised
data
Parameter
estimates
General
linear
model
Preprocessing
SPMs
Random field theory
templates
Variance components

5. but ... not much sensitivity
Choose your model
Eyeballing
Variance
Bias
No normalisation
Lots of normalisation
default
SPM
No smoothing
Lots of smoothing
Massive model
Sparse model
Captures signal
High sensitivity
but ... not much sensitivity
but ... may miss
signal

6. Voxel by voxel Time Time model specification parameter estimation
hypothesis
statistic
Time
Intensity
single voxel
time series
SPM

7. Classical statistics General linear model etc... etc...
one sample t-test
correlation
ANOVA
multiple regression
General linear model
Fourier transform
wavelet transform
etc... etc...

8. identically distributed)
Regression model
error
e~N(0, s2I)
(error is normal and
independently and
identically distributed) = + b1 b2 +
Time x1 x2 e
Intensity
Question: Is there a change in the BOLD response between listening and rest?
Hypothesis test: b1 = 0?
(using t-statistic)

9. Regression model = + + error e~N(0, s2I) b1 b2 x1 x2 e Time Intensity
(error is normal and
independently and
identically distributed) = + b1 b2 +
Time x1 x2 e
Intensity
Stimulus function is not expected BOLD response
Data is serially correlated
What‘s wrong with this model?

10. Regression model = + +

11. Design matrix = + +

12. General Linear Model = +Y
Model is specified by
Design matrix X
Assumptions about e
N: number of scans
p: number of regressors

13. Least squares parameter estimate
Parameter estimation
Assume iid error = +
residuals
Estimate parameters
Least squares parameter estimate
such that
minimal

14. Estimation, example Assume iid error residuals and Least squares
estimate
Another
estimate

15. Convolve stimulus function with model of BOLD response
Improved model
Convolve stimulus function with model of BOLD response
Haemodynamic response function
fitted data

16. High pass filter high pass filter implemented by residuals of DCT set
discrete cosine transform set

17. data and three different models
High pass filter
data and three different models

18. Mass univariate approach= Y X +

19. autoregressive process of order 1 (AR(1))
Serial correlation
with
autoregressive process of order 1 (AR(1))
autocovariance
function
COMPUTATIONAL EFFICIENCY
To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data.
For fMRI, this procedure is ok, because there are not really many hypotheses
(HRF up or HRF down + differences/interactions)
For EEG, there are many more hypotheses to test, because there are many potential different hypotheses
Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!

20. Error covariance matrix
i.i.d.
AR(1)
sampled error
covariance matrices
(103 voxels)
COMPUTATIONAL EFFICIENCY
To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data.
For fMRI, this procedure is ok, because there are not really many hypotheses
(HRF up or HRF down + differences/interactions)
For EEG, there are many more hypotheses to test, because there are many potential different hypotheses
Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!
Serial correlation

21. Restricted Maximum Likelihood
observed Q1
ReML
estimated
correlation
matrix Q2
COMPUTATIONAL EFFICIENCY
To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data.
For fMRI, this procedure is ok, because there are not really many hypotheses
(HRF up or HRF down + differences/interactions)
For EEG, there are many more hypotheses to test, because there are many potential different hypotheses
Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!

22. Inference -- t-statistic
boxcar parameter > 0 ?
Null hypothesis:
COMPUTATIONAL EFFICIENCY
To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data.
For fMRI, this procedure is ok, because there are not really many hypotheses
(HRF up or HRF down + differences/interactions)
For EEG, there are many more hypotheses to test, because there are many potential different hypotheses
Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!

23. t-statistic -- Computations
least squares
estimates
c = X V
COMPUTATIONAL EFFICIENCY
To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data.
For fMRI, this procedure is ok, because there are not really many hypotheses
(HRF up or HRF down + differences/interactions)
For EEG, there are many more hypotheses to test, because there are many potential different hypotheses
Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!
compute df using
Satterthwaite
approximation
ReML

24. Summary General linear model: Unified and comprehensive approach
Partition data into effects (of interest and no interest) and error
Model serial correlation using
variance components
Maximum Likelihood and Restricted Maximum Likelihood estimators
t-contrast: tests for single dimension in parameter space
F-contrast: tests for multiple dimensions
inference at first or second level (fixed or random effects)
over conditions or groups: main effect, difference, interaction:
average over time window
parametric modulation with extrinsic variable
power comparison in time-frequency domain
Inference
Use classical statistics
Hypothesis test using contrast