Parametric modulation, temporal basis
functions and correlated regressors
Mkael Symmonds
Antoinette Nicolle
Methods for Dummies
21st January 2008
Outline
•
•
•
•
•
Parametric regressors: When are they useful?
Parametric regressors: What do they mean?
How to specify in SPM
Linear and non-linear effects
Correlations between regressors and orthogonalisation
Parametric Design vs Factorial Design
Factorial design:
– Experimental task has different factors that are independently
manipulated.
– Each factor may take several levels.
– Events corresponding to each level of every factor are explicitly
modelled by their own column in the design matrix.
– Assumes that each cell of the design matrix is homogeneous and that
stimuli differ only with regard to a single aspect across different levels
of an experimental factor.
– Can test for categorical or parametric effects and interactions by
appropriate selection of contrasts to isolate one cognitive process at a
time.
ON / OFF, 2 LEVELS
Parametric Design vs Factorial Design
Factorial design:
– Experimental task has different factors that are independently
manipulated.
– Each factor may take several levels.
– Events corresponding to each level of every factor are explicitly
modelled by their own column in the design matrix.
– Assumes that each cell of the design matrix is homogeneous and that
stimuli differ only with regard to a single aspect across different levels
of an experimental factor.
– Can test for categorical or parametric effects and interactions by
appropriate selection of contrasts to isolate one cognitive process at a
time.
1,2,3,4,5 – MANY LEVELS
OR CONTINUOUS
Parametric Design vs Factorial Design
Parametric design:
– Every factor or variable has its own column in the design matrix –
corresponding to one (independent) stimulus dimension of interest.
– Different levels of the factor, or the (continuous) variable are
represented numerically within one column.
– Complex stimuli with a number of stimulus dimensions can be
modelled by a set of parametric regressors tied to the presentation
of each stimulus (e.g. faces can be described in terms of
attractiveness, masculinity, symmetry, etc).
– This means that:
• 1. Statistical tests on the main effect of the condition (face presentation) will not
be biased by variance within the condition due to systematic changes in the
different stimulus dimensions  increase in power
• 2. Can look at the contribution of each stimulus dimension independently
• 3. Can test predictions about the direction and scaling of BOLD responses due
to these different dimensions.
Parametric Design vs Factorial Design
Example:
Very simple motor task - Subject presses a button then rests.
Repeats this four times, with an increasing level of force.
Hypothesis:
We will see a linear increase in activation in motor cortex as the
force increases
Model:
Factorial - regressor for each level of force of button press
Time (scans)
Regressors: 1
2
3
4
mean
Parametric Design vs Factorial Design
Example:
Very simple motor task - Subject presses a button then rests.
Repeats this four times, with an increasing level of force.
Hypothesis:
We will see a linear increase in activation in motor cortex as the
force increases
Model:
Factorial - regressor for each level of force of button press
Contrast:
1
2
3
4
0
Time (scans)
Regressors: 1
2
3
4
mean
appropriate test to
search for a linear
increase?
Parametric Design vs Factorial Design
Example:
Very simple motor task - Subject presses a button then rests.
Repeats this four times, with an increasing level of force.
Hypothesis:
We will see a linear increase in activation in motor cortex as the
force increases
Model:
Factorial - regressor for each level of force of button press
Contrast:
-3
-1
1
3
2
3
4
0
Time (scans)
Regressors: 1
mean
No – need to meancorrect to look for
differences between
regressors that can be
accounted for by a
model of linearly
increasing activation
Parametric Design vs Factorial Design
Example:
Very simple motor task - Subject presses a button then rests.
Repeats this four times, with an increasing level of force.
Hypothesis:
We will see a linear increase in activation in motor cortex as the
force increases
Model:
Parametric - regressor for each level of force of button press
Contrast:
Time (scans)
1
0
0
Main effect of button
pressing
5
3
10
2.5
15
1.5
20
0.5
25
-0.5
2
1
0
-1
0
5
10
15
20
25
30
0.5
1
1.5
2
2.5
3
3.5
Time (scans)
Regressors:
press
force
mean
30
35
Parametric Design vs Factorial Design
Example:
Very simple motor task - Subject presses a button then rests.
Repeats this four times, with an increasing level of force.
Hypothesis:
We will see a linear increase in activation in motor cortex as the
force increases
Model:
Parametric - regressor for each level of force of button press
Contrast:
Time (scans)
0
1
0
Linear effect of force
5
4
10
3
2
15
1
20
-1
0
-2
25
-3
-4
0
5
10
15
20
25
30
0.5
1
1.5
2
2.5
3
3.5
Time (scans)
Regressors:
press
force
mean
30
35
Parametric Design vs Factorial Design
Example:
Very simple motor task - Subject presses a button then rests.
Repeats this four times, with an increasing level of force.
Hypothesis:
We will see a linear increase in activation in motor cortex as the
force increases
Model:
Can look at non-linear parametric effects by adding in a column
Contrast:
Time (scans)
0
0
1
0
Quadratic effect of force
5
10
9
10
8
7
15
6
5
4
20
3
2
1
25
0
-1
0
5
10
15
20
25
30
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time (scans)
Regressors: press
force (force)2 mean
30
35
Parametric Design vs Factorial Design
Factorial design:
– OK for simple design with limited number of levels of each factor.
– But need lots of regressors if many levels – hard to handle
– If your variables of interest vary over a continuous range, it is better to
model this as a regressor with continuous values
Parametric design:
– Good for designs with continuous variables where you want to perform
a multiple-regression type of analysis
– Correlated regressors can cause problems with interpretation.
– Not as straightforward to look at interactions between factors
Specifying parametric regressors in SPM
To look at
higher-order
effects
Parametric vs Factorial Design – a real example
• Correlation of presentation rate of spoken words with neural activity
• Randomly vary the rate of presentation of words between 10 and 90
words per min
Categorisation of different forms
of rate-dependent responses
Buchel et al, Neuroimage 1998
Parametric vs Factorial Design – a real example
Zeroth order –
main effect of
word presentation
First order –
linear effect of
presentation rate
Second order –
quadratic effect of
presentation rate
Demonstration of regionally specific
word rate-dependent timecourses
Buchel et al, Neuroimage 1998
Correlated Regressors
Example of correlated regressors
• Experiment:
– Which areas of the brain are active in reward processing?
– Subjects press a button to get a reward when they spot a red dot
amongst green dots
• General Linear Model:
Y = β1X1 + β2X2 + ε
Y = BOLD response
X1 = button press (movement)
X2 = response to reward
Example of correlated regressors
Question:
Which areas of the brain are active in reward processing?
• The regressors are linearly dependent (temporally correlated colinear), so variance attributable to an individual regressor (reward)
may be confounded with another regressor (button press).
• As a result we don’t know which part of the BOLD response is
explained by movement and which by response to getting a reward
• This may lead to misinterpretations of activations in certain brain areas
– E.g. primary motor cortex involved in reward processing??
• We can’t answer our question…..
How do I deal with it?
• Avoid the problem by careful design of your
experiment.
– Can use toolboxes “Design Magic” Multicollinearity assessment for fMRI for SPM.
URL: http://www.matthijs-vink.com/tools.html
The only way you can avoid
correlated regressors in a
factorial design is to
deliberately construct your
experimental manipulations so
that they are independent
(orthogonal)
How do I deal with it?
• Avoid the problem by careful design of your
experiment.
– Can use toolboxes “Design Magic” Multicollinearity assessment for fMRI for SPM.
URL: http://www.matthijs-vink.com/tools.html
You can check for
colinearity/orthogonality in
SPM
How do I deal with it?
•
Imagine you are trying to explain data point y in terms of vector x1
y
y = 1X1
1 = 1.5
x1
How do I deal with it?
•
Adding in x2 changes the beta value of x1, as now x2 can explain some
of the variance (a linear combination of x1 and x2 perfectly explain y)
y
y = 1X1 + 2X2
1 = 1
x2
2 = 1
x1
You have changed your
inference about x1 by adding in
another correlated regressor
How do I deal with it?
•
To look at how much of y is purely explained by x1, need to
‘orthogonalise’ x2
y
y = 1X1 + 2*X2 *
1 = 1.5
x2*
x2
2 * = 1
x1
How do I deal with it?
•
SPM will automatically do serial orthogonalisation (note that this is only within
each condition, so for each condition and its associated parametric
modulators)
y
y = 1X1 + 2*X2 *
1 = 1.5
x2*
x2
2 * = 1
x1
We now have greater
power to explain y in
terms of x1 + x2 with no
residual error –
improves our
conclusions about x1
How do I deal with it?
• Parametric regressors are orthogonalised from left to right in the
design matrix automatically by SPM
• Order in which you put parametric modulators is important!!!
• Put the ‘most important’ modulators first (i.e. the ones whose meaning
you don’t want to change)
Summary
• Use parametric modulators to investigate the
effect of varying levels of a stimulus independently
from the main effect of presenting the stimulus
• Can also look at non-linear effects
• Need to be aware of problems with inference due
to correlated regressors
• Parametric modulators are serially orthogonalised
in SPM
Thanks to…
• Rik Henson’s slides:
www.mrc-cbu.cam.ac.uk/Imaging/Common/rikSPM-GLM.ppt
• Previous years’ Presentations
–
H Spiers, A Liston, H den Ouden, E Chu, and previous
• Additional references
–
–
‘Characterizing stimulus-response function using nonlinear regressors in parametric fMRI
experiments’, Buchel et al, Neuroimage, 1998
‘Using parametric regressors to disentangle properties of multi-feature processes’, Wood et al,
Behavioural and Brain Functions, 2008
Temporal Basis Functions
Methods for Dummies 21st Jan 2009
Antoinette Nicolle
What’s a basis function then…?
In linear algebra, a basis is used to describe a point in
space.
A basis function is the combining of a number of
functions to describe a more complex function.
f(t)
Fourier analysis
h1(t)
h2(t)
The complex wave at the top can be
decomposed into the sum of the three
simpler waves shown below.
f(t)=h1(t)+h2(t)+h3(t)
h3(t)
Temporal Basis Functions for fMRI
In fMRI we need to describe a function of % signal change over
time.
There are various different basis sets that we could use to
approximate the signal.
Finite Impulse
Response (FIR)
Fourier
Temporal Basis Functions for fMRI
Better though to use functions that make
use of our knowledge of the shape of the
HRF.
One gamma function alone provides a
reasonably good fit to the HRF. They are
asymmetrical and can be set at different
lags.
– However they lack an undershoot.
If we add two of them together we get the
canonical HRF.
Limits of HRF
General shape of the BOLD impulse response similar across
early sensory regions, such as V1 and S1.
Variability across higher cortical regions.
Considerable variability across people.
These types of variability can be accommodated
by expanding the HRF in terms of temporal basis
functions.
“Informed” Basis Set (Friston et al. 1998)
Canonical HRF (2 gamma
functions)
plus Multivariate Taylor
expansion in:
time (Temporal Derivative)
width (Dispersion Derivative)
The temporal derivative can
model (small) differences in the
latency of the peak response.
The dispersion derivative can
model (small) differences in the
duration of the peak response.
Comparison of the fitted response
These plots show the haemodynamic response at a single voxel. The left plot
shows the HRF as estimated using the simple model.
Lack of fit is corrected, on the right using a more flexible model with basis functions.
Which temporal basis functions…?
Putting them into your design matrix
Left
Right
Mean
1 0 0 -1 0 0 0
Thanks to…
Rik Henson’s slides:
www.mrc-cbu.cam.ac.uk/Imaging/Common/rikSPM-GLM.ppt
Previous years’ presenters’ slides
Guillaume Flandin