1. Rachel Denison & Marsha Quallo
General Linear Model
and fMRI
Rachel Denison & Marsha Quallo
Methods for Dummies 2007

2. Passive Listening vs. Rest
Did the experiment work?
Did the experimental manipulation
affect brain activity?
A simple experiment:
Passive Listening vs. Rest -- -- --
6 scans per block
time

3. = + y = Xβ + ε X β ε y The General Linear Model
Observed data = Predictors * Parameters Error
eg. Image intensities
Also called the
design matrix.
How much each
predictor contributes
to the observed data
Variance in the data
not explained by
the model

4. = y y: Activity of a single voxel over time Mass Univariate …
y = Xβ + ε
time
BOLD signal y1 y2 yN
= y …
One voxel at a time:
Mass
Univariate

5. X in context β y = x1 x2 x3 + ε
Observed data = Predictors * Parameters Error

6. = + β1 β2 β3 y x1 x2 x3 ε X in context
Observed data = Predictors * Parameters Error

7. *β1 *β2 *β3 y x1 x2 x3 = + + + ε y1 = x11*β1 + x12*β2 + x13*β3 + ε1
X in context
*β1
*β2
*β3 y x1 x2 x3 = + + + ε
A linear combination of the predictors
y1 = x11*β1 + x12*β2 + x13*β3 + ε1

8. label different levels of an experimental factor
X: The Design Matrix
y = Xβ + ε x1 --
Conditions
On Off
Off On
Use ‘dummy codes’ to
label different levels of
an experimental factor
(eg. On = 1, Off = 0).
β is ANOVA effect size.
time

9. (eg. Task difficulty = 1-6) a variable (eg. Movement).
X: The Design Matrix
y = Xβ + ε x1 x2 x3
Covariates
Parametric
and factorial
predictors in the
same model!
Parametric variation
of a single variable
(eg. Task difficulty = 1-6)
or measured values of
a variable (eg. Movement).
β is regression slope.

10. X: The Design Matrix y = Xβ + ε x1 x2 Constant Variable eg. Always = 1
Models the
baseline activity

11. X: The Design Matrix The design matrix should include everything
that might explain the data.
Conditions:
Effects of interest
Subjects
Global activity or movement
More complete models make for lower residual error, better stats,
and better estimates of the effects of interest.

12. If you like these slides …
Summary
So far…
y = Xβ + ε
If you like these slides …
Past MfD presentations (esp. Elliot Freeman, 2005); past FIL SPM Short Course
presentations (esp. Klaas Enno Stephan, 2007); Human Brain Function v2

13. Thanks!

14. General Linear Model: Part 2
Marsha Quallo

15. Content Parameters Error Parameter Estimation
Hemodynamic Response Function
T-Tests and F-Tests

16. Parameters Y= Xβ + ε
β: defines the contribution of each component of the design matrix to the value of Y
The best estimate of β will minimise ε
How much of X is needed to approximate Y,

17. Parameter Estimation ≈ β1∙ + β2∙ + β3∙ 3 2 3 4 1 1 1 2 Listening1 1 1 2
Listening
Reading
Rest ≈
β1∙
+ β2∙
+ β3∙ 3

18. Parameter Estimation ≈ β1∙ + β2∙ + β3∙ 1 4 2 3 4 1 1 1 2 Listening1 1 1 2
Listening
Reading
Rest ≈
β1∙
+ β2∙
+ β3∙ 1 4

19. Parameter Estimation ≈ β1∙ + β2∙ + β3∙ 0.83 0.16 2.98 2 3 4 1 1 1 21 1 1 2
Listening
Reading
Rest ≈
β1∙
+ β2∙
+ β3∙
0.83
0.16
2.98

20. Parameter Estimation β = XTY(XTX)-1 y e e e x2 x3 x1
To estimate β we need to find the least square fit for the line
β = XTY(XTX)-1 y e e e x2
If X has linearly dependant columns then it is rank deficient and has no inverse and the model is over parameterised, there is an infinite amount of parameter sets describing the same model.
Consequently there will be an infinate number of least squares estimates that satisfy the normal equations.
or inverting (XTX) using a pseudoinverse technique – which is essentially imposing a contraint
An example of a constraint is removing a column for the desing matrix, SPM doesn’t use this technique to deal with over determinded models
This gives us the least squares estimates for with the mimimum sums of squares
In this case the estimates can be found by
- imposing constraints
- or inverting (XTX) using a pseudoinverse technique x3 x1
If X has linearly dependant columns the model will be over parameterised
Let (XTX)- denote the psuedoinverse of (XTX) then β = (XTX)-XTY = X-Y

21. Hemodynamic response function
Original Convolved HRF
The glm needs to accommodate for the sluggish response of the brain, to get the best fit of the model the sitmulus function which is usually a sharp on off stimulus needs to be convolved to create a delayed blurred version that mimics the brains activity
Original
Convolved

22. T-Tests and F-Tests ( ) cβ
A contrast vector is used to select conditions for comparison ~ cβ
T = ~
Var(cβ)
What about c [1 1 0]
A contrast matrix is used to make a simultaneous test of multiple contrasts
c =
Enabling you to select see if there is more activation in listening condition than rest, or if we added a reading conditing if there was more activation in reading versus rest
The contast gives us an estimate of the effect but how do we know if the effect is significant, to get a t value we caclulate the ration of the effect to its standard error, whih is the standard deviation of the variance of the effect.
The first row compares listening to rest and the second reading to rest ( )
cβ = (1B1 + 0B2 + 0B3) ~ ~ ~
βc(Var[cβ])-1cβ
F = K

23. http://www. fil. ion. ucl. ac. uk/spm/course/slides06/ppt/glm
Neuroimaging Data Using the General Linear Model (GLM©Karl) Jesper Andersson KI, Stockholm & BRU, Helsinki
Functional MRI: an introduction to methods. Jezzard, P; Matthews, PM; Smith, SM