1. The General Linear Model
Ramiro & Sinéad

2. The fMRI experiment: Desired End-Point
Statistical parametric map (SPM)
Statistical Inference

3. The fMRI experiment: Start Point
One Scanner
; One Experiment
???
; One Brain

4. An fMRI experiment Catch Jellyfish Scanner Bed Condition 1:
Word Generation
Jellyfish
Screen
Noun is presented
Verb is generated
Catch
Scanner
Bed
Healthy Volunteer

5. An fMRI experiment Fry Burger Scanner Bed Condition 1: Word Generation
Screen
Noun is presented
Verb is generated
Fry
Scanner
Bed
Healthy Volunteer

6. An fMRI experiment Swim Swim Scanner Bed Condition 2: Word Shadowing
Screen
Verb is presented
Verb is shadowed
Swim
Scanner
Bed
Healthy Volunteer

7. An fMRI experiment Strut Strut Scanner Bed Condition 2: Word Shadowing
Screen
Verb is presented
Verb is shadowed
Strut
Scanner
Bed
Healthy Volunteer

8. An fMRI experiment + Scanner Bed Baseline No Stimuli Screen
Cross-hair presented
Scanner
Bed
Healthy Volunteer

9. -> 6.7 million data points every 4.5 mins!!
An fMRI experiment …
etc.
4th TR
3rd TR
2nd TR
1st TR
Slides courtesy of Brainvoyager Innovation BV
Note: This is not data from the word generation experiment described above. Illustrative purposes only!
Time Series
Note: if your TR was sec; then this run would have had a duration of approximately 4 and a half minutes
-> 6.7 million data points every 4.5 mins!!
Example Experiment
12 slices * 64 voxels x 64 voxels = 49,152 voxels
1 voxel = 136 time points
1 run = 6.7 million data points
1 experiment = multiple runs; 6.7 million * ?

10. Option 1
We could, in principle, analyze data by voxel surfing: move the cursor over different areas and see if any of the time courses look interesting
Slice 9, Voxel 0, 0
Even where there’s no brain, there’s noise
Slice 9, Voxel 22, 7
The signal is much higher where there is brain, but there’s still noise
Slice 9, Voxel 9, 27
Here’s a voxel that responds well in condition 1 and condition 2
Slides courtesy of Brainvoyager Innovation BV
Here’s one that responds well to condition 1 stimuli only
Slice 9, Voxel 13, 41

11. GLM: ‘Voxel’ x ‘Voxel’ Time-Series Analysis
View 1: A series of volumes (scans, 3D “images”)
View 2: A multiple voxel time series
Standard hypothesis-driven statistical analysis (e.g. GLM) goes with view 2 since it is applied independently for each voxel time course (“voxel-wise” statistical analysis).
© Brainvoyager Innovation BV

12. fMRI Analysis: Overview of SPM
Statistical parametric map (SPM)
Image time-series
Kernel
Design matrix
Realignment
Smoothing
General linear model
Statistical
inference
Gaussian
field theory
Normalisation
p <0.05
Template
Parameter estimates

13. Summary of Regression
Linear regression models the linear relationship between a single dependent variable, Y, and a single independent variable, X, using the equation:
The regression coefficient, β, reflects how much of an effect X has on Y.
ε is the error term and is assumed to be independently, identically, and normally distributed (mean 0 and variance σ2)
Y = βX + c + ε

14. Summary of Regression
Multiple regression is used to determine the effect of a number of independent variables, X1, X2, X3 etc, on a single dependent variable, Y.
The different X variables are combined in a linear way and each has its own regression coefficient:
The β parameters reflect the independent contribution of each independent variable, X, to the value of the dependent variable, Y.
i.e. the amount of variance in Y that is accounted for by each X variable after all the other X variables have been accounted for.
Y = β1X1 + β2X2 +…..+ βLXL + ε

15. Regression and the GLM
As seen last week, regression and correlations methods form the basis of the GLM;
Moreover, the GLM can essentially be viewed as ‘an extension of linear multiple regression for a single dependent variable’.
Multiple Regression only looks at ONE dependent (Y) variable,
whereas, GLM allows you to analyse data from one voxel
as a linear combination of your predictors
ANOVA, t-test, F-test, etc. are also forms of the GLM.

16. Matrix Formulation Y1 = X11β1 +…+X1lβl +…+ X1LβL + ε1
Write out equation for each observation of variable Y from 1 to J:
Y1 = X11β1 +…+X1lβl +…+ X1LβL + ε1
Yj = Xj1β1 +…+Xjlβl +…+ XjLβL + εj
YJ = XJ1β1 +…+XJlβl +…+ XJLβL + εJ
Time
Can turn these simultaneous equations into matrix form to get a single equation: Y1 Yj YJ =
X11 … X1l … X1L
Xj1 … X1l … X1L β1 βj βJ + ε1 εj εJ
Y = X x β ε
Observed data
Design Matrix
Parameters
Error
= x

17. y = Xβ + ε Observed data; fMRI time course (voxel x voxel)
Image courtesy of Brainvoyager Innovation BV
Observed data;
fMRI time course
(voxel x voxel)
© Brainvoyager Innovation BV

18. y = Xβ + ε
Image courtesy of Brainvoyager Innovation BV
Design Matrix

19. functions, explanatory
y = Xβ + ε
Image courtesy of Brainvoyager Innovation BV
Predictors
Also referred to as:
regressors, reference
functions, explanatory
variables, covariates,
basic functions

20. y = Xβ + ε
Beta Weights/
Coefficients

21. y = Xβ + ε
Error/
Residuals

22. The GLM
y = Xβ + ε

23. fMRI Analysis: Overview of SPM
Statistical parametric map (SPM)
Image time-series
Kernel
Design matrix
Realignment
Smoothing
General linear model
Statistical
inference
Gaussian
field theory
Normalisation
p <0.05
Template
Parameter estimates

24. Single Voxel Time Series
The Observed Data
y = Xβ + ε
Time
Single Voxel Time Series
GLM = Mass Univariate ‘Voxel X Voxel’
Time Series Analysis
The observed fMRI time course in a specific voxel
(= the dependent variable)

25. = y y = Xβ + ε The Observed Data … time BOLD signal y1 y2 yN
Design Matrix Contains/embodies all available knowledge.
Note: a predictor time course is typically obtained by the convolution of a condition box-car time course with a standard HRF.
Each predictor time course gets an associated coefficient or beta weight.
This quantifies each predictors’ contribution in explaining the voxel’s time course (y) -> The beta weight of a condition predictor quantifies the contribution of its time course in explaining the voxel’s time course.

26. Q: what might our specified predictors be in this experiment?
The Design Matrix
y = Xβ + ε
X = a set of specified predictors (the design matrix); each of which has unique expected signal time course.
Q: what might our specified predictors be in this experiment?
»The GLM models the expected signal time courses for individual conditions.

27. The Design Matrix + + = + 1 × X1 X2 2 × 3 × X3
Provides the model of the expected signal time courses for individual conditions
Expected signal time course for our 1st predictor (X1) =
design matrix X1
1 ×
Expected signal time course for our 2nd predictor (X2) +
fMRI signal X2
2 × +
residuals
‘ X1, X2 and X3 are a series of know hypotheses about what we think is going on in the brain. Nothing unknown about the as we have put them in there.
i.e. the time series that we would have for a voxel that’s into generating words, that’s into words in general, or that doesn’t really care about words.
Observed Data: want to try to model this as a linear combination of the hypothetical time series, i.e. the time series that we would have for a voxel that’s into generating words, that’s into words in general, or that doesn’t really care about words.’ +
3 × X3
Time
a “constant” predictor (X3) is required to fit the base level of the signal time course
© Brainvoyager Innovation BV

28. The Design Matrix
The GLM models the expected signal time courses for individual conditions.
> A ‘model’ consists of a set of assumptions about what these time series look like for each of the specific conditions or categories of data.
We therefore have various ‘knows’ which we can put into our model: 1. The expected signal time course for each of our individual conditions 2. The expected signal time course of confounds in the data

29. X: Predictors / Design Matrix
The Design Matrix +
1 ×
2 ×
3 × X1 X2 X3
X: Predictors / Design Matrix
Y: Observed Data +
Error =
Time
‘Observed Data: want to try to model this as a linear combination of the hypothetical time series, i.e. the time series that we would have for a voxel that’s into generating words, that’s into words in general, or that doesn’t really care about words’.
Images: © Brainvoyager Innovation BV
fMRI signal
residuals
y1 = x1*β1 + x2*β2 + x3*β3 + ε1
© Brainvoyager Innovation BV
A linear combination of the predictors

30. Aside

31. X: Predictors / Design Matrix
Hemodynamic response function
Y: Observed Data +
Error =
1 × +
2 × +
3 ×
Time X1 X2 X3
Note: a predictor time course is typically obtained by the convolution of a condition box-car time course with a standard HRF.
Images: © Brainvoyager Innovation BV
fMRI signal
residuals
X: Predictors / Design Matrix

32. Hemodynamic response function
Neural pathway
Hemodynamics
MR scanner
-> Reshape (convolve) regressors to resemble HRF
HRF basic function
Image 1: Brainvoyager Innovations BV

33. The Design Matrix = 1 × + 2 x + 3 x + X1 X2 X3 fMRI signal = data
Time
1 ×
+ 2 x
+ 3 x + X1 X2 X3
fMRI signal
= data
design matrix = model
residuals
= error
So far, we have only included (in our design matrix) the predicted signal time series for our ‘effects of interest’. We also have information about confounds in our data (i.e. ‘effects of NO interest’ such as head movement…). We need to add these additional predictor time courses in order to improve our model of the data (& thus reduce the error)

34. The Design Matrix
We therefore have various ‘knows’ which we can put into our model: 1. The expected signal time course for each of our individual conditions 2. The expected signal time course of confounds in the data
The Design Matrix Needs to Model the Expected Signal Time Course for:
Effects of Interest
each individual condition (X1, X2 …)
a ‘constant’ predictor (X3)
The Design Matrix thus embodies all available knowledge about experimentally controlled factors and potential confounds
Effects of No Interest (i.e. confounds):
each physiological confounds – head movement..
each psychological confounds – ∆ stress, attention…
Scanner Drift…

35. 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 more accurate estimates of the effects of interest.

36. y = Xβ + ε The GLM in Matrix Notation Observed data The observed fMRI
time course in a
specific voxel
(dependent variable)

37. y = Xβ + ε The General Linear Model Observed data = Predictors
Model is specified by:
Design matrix X
Assumptions about e
y = Xβ + ε
Embodies all available knowledge about experimentally controlled factors and potential confounds
Observed data
= Predictors
(the design matrix)
Note: A constant predictor is required to fit the base level of the signal time course.
Predictions can also be referred to as:
reference functions
regressors
explanatory variables
covariates
basic functions
Design Matrix Contains/embodies all available knowledge.
Note: a predictor time course is typically obtained by the convolution of a condition box-car time course with a standard HRF.
Each predictor time course gets an associated coefficient or beta weight.
This quantifies each predictors’ contribution in explaining the voxel’s time course (y) -> The beta weight of a condition predictor quantifies the contribution of its time course in explaining the voxel’s time course.
A set of specified
predictors
(each of which has
a unique expected
signal time course)
The observed fMRI
time course in a
specific voxel
(dependent variable)

38. y = Xβ + ε The General Linear Model Observed data = Predictors
Model is specified by:
Design matrix X
Assumptions about e
y = Xβ + ε
= Everything that we know
Observed data
= Predictors
But what about everything that we DON’T know???
The observed fMRI
time course in a
specific voxel
(dependent variable)
A set of specified
predictors
(each of which has
a unique expected
signal time course)
“The Unknowns”:
The Beta Values
The Error (Residuals)

39. The General Linear Model
y = Xβ + ε
1 ×
2 ×
3 ×
estimate =
Time
1 ×
+ 2 x
+ 3 x + X1 X2 X3
fMRI signal
= data
design matrix = model
residuals
= error
-> Each predictor time course gets an associated coefficient or beta weight.
-> The beta weight of a condition predictor quantifies the contribution of it’s time course (X1; X2; X3) in explaining the voxel’s time course (y).

40. Generation Shadowing Baseline
The Model
We have our set of hypothetical time-series
The estimation entails finding the parameter values such that the linear combination of these hypothetical time series ”best” fits the data.
Generation Shadowing Baseline
Measured X1 X2 X3
”Known” ≈
+ β3*
”Unknown” parameters
+ β2*
β1*
want to estimate (or want SPM to estimate) or model this observed time series as a linear combination of the hypothetical time series.

41. Generation Shadowing Baseline
Parameter Estimation
Finding the ”best” parameter values For a given voxel (time-series) we try to figure out just what type that is by ”modelling” it as a linear combination of the hypothetical time-series. 4 3 2 1 1 2 1 ≈
β1*
+ β2*
+ β3*
Generation Shadowing Baseline

42. Generation Shadowing Baseline
Parameter Estimation
Finding the ”best” parameter values For a given voxel (time-series) we try to figure out just what type of voxel this is by ”modelling” it as a linear combination of the hypothetical time-series. 4 3 2 1 1 2 1 ≈
β1*
+ β2*
+ β3*
Generation Shadowing Baseline
Not brilliant 3

43. Generation Shadowing Baseline
Parameter Estimation
Finding the ”best” parameter values For a given voxel (time-series) we try to figure out just what type of voxel this is by ”modelling” it as a linear combination of the hypothetical time-series. 4 3 2 1 1 2 1 ≈
β1*
+ β2*
+ β3*
Generation Shadowing Baseline
Neither that 1 4

44. Generation Shadowing Baseline
Parameter Estimation
SSE = (yi – i) 2 = (yi – Xβ) 2
Finding the ”best” parameter values For a given voxel (time-series) we try to figure out just what type of voxel this is by ”modelling” it as a linear combination of the hypothetical time-series. 4 3 2
+ β2*
+ β1* 1
β0*
+ β3*
β1* 2 ≈ ≈
Generation Shadowing Baseline
Cool!
0.83
0.16
2.98

45. Generation Shadowing Baseline
Parameter Estimation
Finding the ”best” parameter values And the nice thing is that the same model fits all the time-series, only with different parameters. 3 2 1
+ β2*
+ β1* 1
β0*
+ β3*
β1* 2 ≈ ≈ ≈
β3*
+ β1*
+ β2*
Generation Shadowing Baseline
0.68
0.82
2.17
In other words:

46. Generation Shadowing Baseline
Parameter Estimation
Finding the ”best” parameter values And the nice thing is that the same model fits all the time-series, only with different parameters. 3 2 1
+ β2*
+ β1* 1
β0*
+ β3*
β1* 2 ≈ ≈ ≈ ≈
β0*
+ β1*
+ β2*
Generation Shadowing Baseline
0.03
0.06
2.04
Doesn’t care:

47. Same model for all voxels. Different parameters for each voxel.
Parameter Estimation
Plots the estimated beta 1’s over the whole brain
Same model for all voxels. Different parameters for each voxel.
beta_0001.img
...
Time-series
beta_0002.img
...
beta_0003.img

48. So far…

49. y = Xβ + ε The GLM in Matrix Notation Observed data The observed fMRI
Model is specified by:
Design matrix X
Assumptions about e
y = Xβ + ε
Observed data
Design Matrix Contains/embodies all available knowledge.
Note: a predictor time course is typically obtained by the convolution of a condition box-car time course with a standard HRF.
Each predictor time course gets an associated coefficient or beta weight.
This quantifies each predictors’ contribution in explaining the voxel’s time course (y) -> The beta weight of a condition predictor quantifies the contribution of its time course in explaining the voxel’s time course.
The observed fMRI
time course in a
specific voxel
(dependent variable)

50. The Design Matrix Needs to Model the Expected Signal Time Course for:
The General Linear Model
Model is specified by:
Design matrix X
Assumptions about e
y = Xβ + ε
The Design Matrix thus embodies all available knowledge about experimentally controlled factors and potential confounds
Observed data
= Predictors
(the design matrix)
Note: A constant predictor is required to fit the base level of the signal time course.
Predictions can also be referred to as:
reference functions
regressors
explanatory variables
covariates
basic functions
Design Matrix Contains/embodies all available knowledge.
Note: a predictor time course is typically obtained by the convolution of a condition box-car time course with a standard HRF.
Each predictor time course gets an associated coefficient or beta weight.
This quantifies each predictors’ contribution in explaining the voxel’s time course (y) -> The beta weight of a condition predictor quantifies the contribution of its time course in explaining the voxel’s time course.
A set of specified
predictors
(each of which has
a unique expected
signal time course)
The Design Matrix Needs to Model the Expected Signal Time Course for:
The observed fMRI
time course in a
specific voxel
(dependent variable)
Effects of Interest
each individual condition (X1, X2 …)
a ‘constant’ predictor (X0)
Effects of No Interest (i.e. confounds):
each physiological confounds – head movement..
each psychological confounds – ∆ stress, attention…
Scanner Drift…

51. Each predictor time course (X) gets an estimated beta weight.
The General Linear Model
Model is specified by:
Design matrix X
Assumptions about e
y = Xβ + ε
Each predictor time course (X) gets an estimated beta weight.
This weight attempts to quantify the specific predictors contribution to the specific voxels time course (Y).
Observed data
= Predictors
(the design matrix)
* Parameters
A set of specified
predictors
(each of which has
a unique expected
signal time course)
The observed fMRI
time course in a
specific voxel
(dependent variable)
Quantifies how
much each predictor
contributes to the
observed data
i.e. the voxels’ time
course (y)
Referred to as association coefficients or beta weights (β)

52. y = Xβ + ε The General Linear Model Observed data = Predictors
Model is specified by:
Design matrix X
Assumptions about e
y = Xβ + ε
Observed data
= Predictors
(the design matrix)
* Parameters
+ Error
The observed fMRI
time course in a
specific voxel
(dependent variable)
A set of specified
predictors
(each of which has
a unique expected
signal time course)
Quantifies how
much each predictor
contributes to the
observed data
i.e. the voxels’ time
course (y)
Variance in the data
not explained by the
model
(Represents the mismatch
between the observed
data and the described
Model).

53. y = Xβ + ε Error We assume that the errors are normally distributed:
Model is specified by:
Design matrix X
Assumptions about e
y = Xβ + ε
We assume that the errors are normally distributed:
Assumes we have the same error/uncertainty in each and every measurement point
And that there is no correlation between the errors in the different voxels.

54. Get an estimate of the residual error;
Model is specified by:
Design matrix X
Assumptions about e
y = Xβ + ε
Get the parameter estimates for each and every regressor/predictor (beta weights) ;
Get an estimate of the residual error;
Enables you to estimate the variance in the data (for that particular time series)

55. fMRI Analysis: Overview of SPM
Statistical parametric map (SPM)
Image time-series
Kernel
Design matrix
Realignment
Smoothing
General linear model
Statistical
inference
Gaussian
field theory
Normalisation
p <0.05
Template
Parameter estimates

56. Make inferences about effects of interest
Modelling the Measured Data
Make inferences about effects of interest
Why?
How?
Decompose the data into effects and error
Form statistic using estimates of effects and error
Data
Linear
Model
Stimulus
Function
Effect Estimate
Statistic
Error Estimate
Once we have obtained the βs & error at each voxel we can use these to do various statistical tests

57. Thanks to… Previous MfD talks: Rachel Denison & Marsha Quallo (2007)
Lύcia Garrido & Marieke Schölvinck (2006)
Elliot Freeman (2005)
Davina Bristow & Beatriz Calvo (2004)
Brainvoyager Innovation BV (for permission to use images;
Jesper Andersson. Modelling Neuroimaging Data Using the General Linear Model (GLM©Karl);
Justin Chumbley & Maria Joao