1st Level Analysis:
Design matrix, contrasts and
inference
Rebecca Knight and Lorelei Howard
Outline
 What
is first level analysis?
 The General Linear Model and how this relates to the
Design Matrix
 Regressors within the Design Matrix
Overview
fMRI time-series
Motion
correction
kernel
Design matrix
Smoothing
General Linear Model
Statistical Parametric Map
Parameter Estimates
Spatial
normalisation
Standard
template
 Once the image has been reconstructed, realigned, spatially normalised and
smoothed….
 TheRebecca
next step
is to statistically analyse the data
Knight
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
General Linear Model
Generic Model
Y
=
X
x
β
+
E
Dependent Variable Independent Variable Relative Contribution Error (The difference
(What you are
(What you are
(These need to be
between the observed
measuring)
manipulating)
estimated)
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
 More than 1 IV ?
Y = X1β1 + X2β2 + ....X n βn.... + E
GLM Continued
 How does this equation translate to the 1st level analysis ?
 Each letter is replaced by a set of matrices (2D representations)
Y
Matrix of BOLD signals
(What you collect)
Time
X
=
x
Design matrix
β
+
Matrix parameters
(This is what is put (These need to be
into SPM)
estimated)
E
Error matrix
(residual error for
each voxel)
Time
Time
Regressors
Voxels
Voxels
Regressors
Voxels
‘Y’ in the GLM
Y = Matrix of Bold signals
fMRI brain scans
Voxel time course
Time
Time
(scan every
3 seconds)
Amplitude/Intensity
Rebecca Knight
1 voxel = ~ 3mm³
‘X’ in the GLM
X = Design Matrix
Time
(n)
Regressors (m)
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
Regressors
 A dark-light colour map is used to
show the value of each regressor
within a specific time point
Time
(n)
 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
Rebecca Knight
Regressors (m)
Conditions
 As they indicate conditions they are
referred to as indicator variables
Changes in the bold activation
associated with the presentation
of a stimulus
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
When you IV is presented
OFF = 0
When you IV is absent
(implicit baseline)
Red box plot of [0 1] doesn’t
model the rise and falls
Fitted Box-Car
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
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
Designs
Block design
Intentionally design
events of interest into
blocks
v
Event- related design
Retrospectively look at when the
events of interest occurred. Need to
code the onset time for each regressor
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
Example
Design
Motion
High
Medium
No Motion
Low
High
Medium
 IV 1 = Movement, 2 levels (Motion and No Motion)
 IV 2 = Attentional Load, 3 levels (High, Medium or Low)
Low
Example Cont.
V A C1 C2 C3
M N h m l
 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
Orthogonal design matrix
M M M N N N
h m l h m l
 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
Mh
Mm
Ml
Nh
Nm
Nl
M M M N N N
h m l h m l
h
m
l
M
Mh
Mm
Ml
N
Nh
Nm
Nl
Summary
Y
Matrix of BOLD
signals
X
=
β
x
Design matrix
Matrix parameters
Time
Time
+
E
Error matrix
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
Outline
 Why do we need contrasts?
 What are contrasts?
 T contrasts
 F contrasts
Rebecca Knight
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
What is a contrast?
p
 cT = [1 0 0 0 0 …]
 Contrast vector of length p
 cT β = 1xb1 + 0xb2 + 0xb3 + 0xb4 + 0xb5 + . . .
 Contrast = statistical assessment of cT β
Rebecca Knight
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
Example
Left
Right
 Two event-related conditions
 The subjects press a button with either
their left or right hand, depending on
visual instruction
T contrasts
Left
Right
Question: Which brain regions respond to
Left button presses?
 cT = [1 0 0 …]
 cTβ = 1xb1 + 0xb2 + 0xb3 + . . .
 identifies voxels whose activation
increases in response to Left button
presses
 cT = [-1 0 0 …]
 cTβ = -1xb1 + 0xb2 + 0xb3 + . . .
 identifies voxels whose activation
decreases in response to Left button
presses
Rebecca Knight
T contrasts
 H0 : cTβ = 0
 Experimental Hypotheses:
- H1: cTβ > 0 ?
- H1: cTβ < 0 ?
 T-test is a signal-to-noise measure
 Test Statistic:
Contrast of
estimated
parameters
T df =
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Variance
estimate
cT β
=
SD (cTβ)
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 = [1 -1 0 …]
 cTβ = 1xb1 + -1xb2 + 0xb3 + . . .
 cT = [1 -1 0 …] ≠
cT= [-1 1 0 …]
 must ensure sum of the weights = 0
Rebecca Knight
T contrasts
Rebecca Knight
SPM-t image
 Clearly see contralateral motor cortex response
 The map of T-values:
spmT_*.img
 The contrast itself
(cTβ; ie, numerator):
con_*.img
2nd Level
* = number in Contrast Manager
F contrasts
 Matrix of T contrasts
Left
Right
 Non-directional
 Identify voxels showing modulation in
response to experimental task, ahead of
more specific contrasts
Question: Which brain regions respond to
Left and/or Right button presses?
 cT = 1 0 0 …
010…
Rebecca Knight
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:
Explained variability
F =
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Error variance estimate
F contrasts
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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
Factorial e.g.
 IV 1 = Movement, 2 levels (Motion and No Motion)
 IV 2 = Attentional Load, 3 levels (High, Medium or Low)
MMMNNN
h ml h ml
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
Factorial e.g.
 IV 1 = Movement, 2 levels (Motion and No Motion)
 IV 2 = Attentional Load, 3 levels (High, Medium or Low)
MMMNNN
h ml h ml
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
Factorial e.g.
 IV 1 = Movement, 2 levels (Motion and No Motion)
 IV 2 = Attentional Load, 3 levels (High, Medium or Low)
MMMNNN
h ml h ml
Interaction
• Shows voxels where the attentional
load elicits a brain response that is
different when there is motion, or not
Rebecca Knight
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
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
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