1. Design matrix, contrasts and inference
1st Level Analysis:
Design matrix, contrasts and inference
Rebecca Knight and Lorelei Howard

2. Outline
What is first level analysis?
The General Linear Model and how this relates to the Design Matrix
Regressors within the Design Matrix

3. Overview Standard template
fMRI time-series
Smoothing
kernel
General Linear Model
Design matrix
Statistical Parametric Map
Motion
correction
Parameter Estimates
Spatial
normalisation
Standard
template
Once the image has been reconstructed, realigned, spatially normalised and
smoothed….
The next step is to statistically analyse the data
Rebecca Knight

4. Key Concepts
1st level analysis – A within subjects analysis where activation is averaged across scans for an individual subject
The Between- subject analysis is referred to as a 2nd level analysis and will be described later on in this course
Design Matrix – 2D, m = regressors, n = time. A dark-light colour map is used to show the value of each variable at specific time points
The Design Matrix forms part of the General linear model, the majority of statistics at the analysis stage use the GLM
Rebecca Knight

5. Y X β + E x = General Linear Model Generic Model
Dependent Variable (What you are measuring)
Independent Variable (What you are manipulating)
Relative Contribution (These need to be estimated)
Error (The difference between the observed data and that which is predicted by the model)
Aim: To explain as much of the variance in Y by using X, and thus reducing E
Y = X1β1 + X2β X n βn E
More than 1 IV ?

6. GLM Continued
How does this equation translate to the 1st level analysis ?
Each letter is replaced by a set of matrices (2D representations) Y X β + E x =
Matrix of BOLD signals
(What you collect)
Design matrix
(This is what is put into SPM)
Matrix parameters
(These need to be estimated)
Error matrix
(residual error for each voxel)
Time
Time
Time
Regressors
Voxels
Voxels
Regressors
Voxels

7. ‘Y’ in the GLM Y = Matrix of Bold signals fMRI brain scans
Voxel time course
Time
Time
(scan every 3 seconds)
Amplitude/Intensity
1 voxel = ~ 3mm³
Rebecca Knight

8. ‘X’ in the GLM
X = Design Matrix
Time
(n)
Regressors (m)

9. Regressors
Regressors – represent hypothesised contributors in your experiment. They are represented by columns in the design matrix (1column = 1 regressor)
Regressors of Interest or Experimental Regressors – represent those variables which you intentionally manipulated. The type of variable used affects how it will be represented in the design matrix
Regressors of no interest or nuisance regressors – represent those variables which you did not manipulate but you suspect may have an effect. By including nuisance regressors in your design matrix you decrease the amount of error.
E.g. - The 6 movement regressors (rotations x3 & translations x3 ) or physiological factors e.g. heart rate

10. Regressors
A dark-light colour map is used to show the value of each regressor within a specific time point
Black = 0 and illustrates when the regressor is at its smallest value
White = 1 and illustrates when the regressor is at its largest value
Grey represents intermediate values
The representation of each regressor column depends upon the type of variable specified
Time
(n)
Regressors (m)
Rebecca Knight

11. Conditions
As they indicate conditions they are referred to as indicator variables
Type of dummy code is used to identify the levels of each variable
E.g. Two levels of one variable is on/off, represented as
ON = 1
OFF = 0
Changes in the bold activation associated with the presentation of a stimulus
When you IV is presented
Red box plot of [0 1] doesn’t model the rise and falls
When you IV is absent (implicit baseline)
Fitted Box-Car

12. Modelling Haemodynamics
Changes in the bold activation associated with the presentation of a stimulus
Haemodynamic response function
Peak of intensity after stimulus onset, followed by a return to baseline then an undershoot
Box-car model is combined with the HRF to create a convolved regressor which matches the rise and fall in BOLD signal (greyscale)
Even with this, not always a perfect fit so can include temporal derivatives (shift the signal slightly) or dispersion derivatives (change width of the HRF response) *more later in this course
HRF Convolved

13. Covariates What if you variable can’t be described using conditions?
E.g Movement regressors – not simply just one state or another
The value can take any place along the X,Y,Z continuum for both rotations and translations
Covariates – Regressors that can take any of a continuous range of values (parametric)
Thus the type of variable affects the design matrix – the type of design is also important

14. Designs Block design v Event- related design
Intentionally design events of interest into blocks
Retrospectively look at when the events of interest occurred. Need to code the onset time for each regressor

15. Separating Regressors
The type of design and the type of variables used in your experiment will affect the construction of your design matrix
Another important consideration when designing your matrix is to make sure your regressors are separate
In other words, you should avoid correlations between regressors (collinear regressors) – because correlations in regressors means that variance explained by one regressor could be confused with another regressor
This is illustrated by an example using a 2 x 3 factorial design

16. Example Design Motion No Motion
High Medium Low
High Medium Low
IV 1 = Movement, 2 levels (Motion and No Motion)
IV 2 = Attentional Load, 3 levels (High, Medium or Low)

17. Example Cont.
V A C1 C2 C3
If you made each level of the variables a regressor you could get 5 columns and this would enable you to test main effects
BUT what about interactions? How can you test differences between Mh and Nl
This design matrix is flawed – regressors are correlated and therefore a presence of overlapping variance (Grey)
M N h m l
M N h m l
MN h ml

18. Orthogonal design matrix
M M M N N N
If you make each condition a regressor you create 6 columns and this would enable you to test main effects
AND it enable you to test interactions! You can test differences between Mh and Nl
This design matrix is orthogonal – regressors are NOT correlated and therefore each regressor explains separate variance
h m l h m l
M M M N N N
h m l
h m l h m l M N h m l M N Mh Mm Ml Nh Nm Nl

19. β Y X + E x = Summary Matrix of BOLD signals Design matrix
Matrix parameters
Error matrix
Time
Time
Time
Regressors
Voxels
Voxels
Regressors
Voxels
Aim: To explain as much of the variance in Y by using X, and thus reducing E
β = relative contribution that each regressor has, the larger the β value = the greater the contribution
Next: Examine the effect of regressors

20. Outline What are contrasts? Why do we need contrasts? T contrasts
F contrasts
Rebecca Knight

21. Why use contrasts GLM: - Specify design matrix
- Determine β’s for each voxel for each regressor
Use contrasts to:
- Specify effects of interest
- Perform statistical evaluation of hypotheses
Contrasts used and their interpretation depends on the model
specification, which in turn depends on the design of the
experiment
Rebecca Knight

22. What is a contrast? cT = [1 0 0 0 0 …] Contrast vector of length p
cT β = 1xb1 + 0xb2 + 0xb3 + 0xb4 + 0xb
Contrast = statistical assessment of cT β
Rebecca Knight

23. Different contrasts T contrasts - Unidimensional (vectors)
- Directional
- Assess effect of one parameter OR compare specific
combinations of parameters
F contrasts
- Multidimensional (matrix)
- Non-directional
- Collection of T contrasts
Rebecca Knight

24. Example Two event-related conditions
Left Right
Two event-related conditions
The subjects press a button with either
their left or right hand, depending on
visual instruction

25. T contrasts Question: Which brain regions respond to
Left button presses?
Left Right
cT = [1 0 0 …]
cTβ = 1xb1 + 0xb2 + 0xb
identifies voxels whose activation
increases in response to Left button
presses
cT = [ …]
cTβ = -1xb1 + 0xb2 + 0xb
identifies voxels whose activation
decreases in response to Left button
presses
Rebecca Knight

26. Contrast of estimated parameters
T contrasts
H0 : cTβ = 0
Experimental Hypotheses:
- H1: cTβ > 0 ?
- H1: cTβ < 0 ?
T-test is a signal-to-noise measure
Test Statistic:
T df =
cT β
Contrast of estimated parameters
Variance estimate
SD (cTβ) =
Rebecca Knight

27. T contrasts Subtractive Logic:
“ The direct comparison of two regressors that are assumed to
differ only in one property, the IV ”
Left Right
Question: Which brain regions respond more to
Left than to Right button presses?
cT = [ …]
cTβ = 1xb1 + -1xb2 + 0xb
cT = [ …] ≠ cT= [ …]
must ensure sum of the weights = 0
Rebecca Knight

28. T contrasts
Rebecca Knight

29. SPM-t image Clearly see contralateral motor cortex response
The map of T-values:
spmT_*.img
The contrast itself
(cTβ; ie, numerator):
con_*.img
* = number in Contrast Manager
2nd Level

30. F contrasts Matrix of T contrasts Non-directional
Identify voxels showing modulation in
response to experimental task, ahead of
more specific contrasts
Left Right
Question: Which brain regions respond to
Left and/or Right button presses?
cT = …
0 1 0 …
Rebecca Knight

31. F contrasts Determines whether any one regressor OR combination of
regressors explains a significant amount of the variance in Y
NOT which regressor the effect can be attributed to
H0 : β1 = β2 = 0
H1: at least one β ≠ 0
Test Statistic:
F =
Explained variability
Error variance estimate
Rebecca Knight
Rebecca Knight

32. F contrasts
Rebecca Knight

33. SPM-F image Clearly see motor cortex response The map of F-values:
spmF_*.img
Also outputs:
ess_*.img
* = number in Contrast Manager
Rebecca Knight

34. Factorial e.g. IV 1 = Movement, 2 levels (Motion and No Motion)
IV 2 = Attentional Load, 3 levels (High, Medium or Low)
M M M N N N
h m l h m l
ME Movement
Stack of M > N contrasts for each
level of Load
Shows voxels which are more active in
M than N
(regardless of attentional load)
Rebecca Knight

35. Factorial e.g. IV 1 = Movement, 2 levels (Motion and No Motion)
IV 2 = Attentional Load, 3 levels (High, Medium or Low)
M M M N N N
h m l h m l
ME Attention
First row = h > m
Second row = m > l
Shows voxels which are more active in
h than m AND/OR m than l
(regardless of movement level)
Rebecca Knight

36. Factorial e.g. IV 1 = Movement, 2 levels (Motion and No Motion)
IV 2 = Attentional Load, 3 levels (High, Medium or Low)
M M M N N N
h m l h m l
Interaction
Shows voxels where the attentional
load elicits a brain response that is
different when there is motion, or not
Rebecca Knight

37. Inference
We’ve talked about 1st level so far… examining within subject
variability.
However, we can’t use a sample of one to extrapolate our findings to
the general population
2nd level analyses to look for effects at the group level… discussed
later in course
Rebecca Knight

38. Summary
Contrasts are statistical (t or F) tests of specific hypotheses
T contrasts:
- Compare effect of one regressor with 0
- Compare 2 or more regressors
F contrasts:
- Multidimensional contrasts
Rebecca Knight

39. Resources Huettel. Functional magnetic resonance imaging (Chap 10)
MfD Slides 2007
Human Brain Function (Chap 8)
Rik Henson and Guillaume Flandin’s slides from SPM courses
Rebecca Knight