1st Level Analysis
Design Matrix, Contrasts & Inference
Cat Sebastian and Nathalie Fontaine
University College London
1
Outline
 What is ‘1st level analysis’?
 Design matrix
 What are we testing for?
 What do all the black lines mean?
 What do we need to include?
 Contrasts
 What are they for?
 t and F contrasts
 Inferences
 How do we do that in SPM5?
A
[1
B
C
D
-1
-1
1]
2
3
What is 1st level analysis?
 1st level analysis: activation is averaged across scans
within a subject
 2nd level analysis: activation is averaged across subjects
(groups can be compared)
 What question are we asking?:
Which voxels in the brain show a pattern of activation
over conditions that is consistent with our hypothesis?
4
The Design Matrix
More on this
Time
Not so much on this
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The GLM in fMRI
Time
Y
=
X
x
β
ε
+
Observed data:
Design matrix:
Parameters:
Error:
Y is the BOLD signal at
various time points at a
single voxel
Several components which
explain the observed data,
i.e. the BOLD time series
for the voxel
The contribution of
each component of
the design matrix to
the value of Y (aim
to minimise error)
Difference between
the observed data,
Y, and that predicted
by the model, Xβ .
=

b1
+
b2
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What is Y?
Y is a matrix of BOLD
signals

Each column
represents a single
voxel sampled at
successive time
points.
Y
Time

Intensity
7
What is X (design matrix)?
 The design matrix is simply a mathematical description of
your experiment
E.g.: ‘visual stimulus on = 1’
‘visual stimulus off = 0’
 It should contain ‘regressors of interest’, i.e. variables you
have experimentally manipulated, and ‘regressors of no
interest’ – head movement, block effects.
Why?
 To minimise the error term, you want to model as much of Y
as possible using variables specified in X
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What should the model look like?
Regressors of interest
X=
Baseline Motion
E.g. of a regressor
of no interest
Usually 6 motion
regressors: 3
translations, 3
rotations
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Regressors of interest
 There are different ways to specify variables, e.g.
 Conditions:
 'dummy' codes identify different levels of
experimental factor
 e.g. integers 0 or 1: 'off' or 'on'
on
off
off
on
 Covariates:
 parametric modulation of independent
variable
 e.g. task-difficulty 1 to 6
10
Block vs. event related designs
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Modelling the baseline
This a column of ‘ones’ modelling the
constant, or mean signal (the signal is
not zero even without any stimuli or task)
SPM will model this automatically
Two eventrelated
conditions
Baseline often used as a reference (not the
same as baseline fixation)
12
From design to a design matrix: an example
 Imaging a 2x3 factorial design with factors Modality
(Auditory, Visual) and Condition (Concrete, Abstract,
Proper)
V A C1 C2 C3
C1: Concrete nouns
Visual
Auditory
You can model it
like this…but is it
the best way?
C2: Abstract nouns
C3: Proper nouns
C1: Concrete nouns
C2: Abstract nouns
C3: Proper nouns
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What can we test with this design matrix?
V A C1 C2 C3
• We can test for main effects:
- Visual > Auditory?
- Concrete > Abstract?
• But we can’t test for interactions or
simple main effects:
Visual/concrete > Visual/Abstract? etc
The design is not orthogonal…
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An orthogonal design matrix
C1
C1 C1 C2 C2 C3 C3
C2
C3
VAVAVA
V
A
Just like in SPSS, you need to cross your
variables in order to model interactions
SPM will do this for you automatically if
you have a factorial design – just input
the factors and the number of levels
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Ways to improve your model: modelling
haemodynamics
 The brain does not just
switch on and off.
HRF basic function
 Reshape (convolve)
regressors to
resemble HRF
More on
this next
week!
Original
HRF Convolved
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To return to the GLM…
Time
Y
=
=
X
x

β
ε
+
b1
+
b2
• We calculate beta values for each regressor in the design matrix
• We can then perform contrasts to see which regressors make a
significant contribution to the model
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Interim summary: design matrix
 We want X to model as much of Y as possible, making
the error term small – therefore model everything!
 This will ensure that the beta values associated with
your regressors of interest are as accurate as possible
 Make sure you specify a new regressor for each
crossed variable of interest (orthogonality)
 Additional complications (basis functions and
correlated regressors) will be covered next week
 Contrasts can then be performed...over to Nathalie
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Outline
 What is ‘1st level analysis’?
 Design matrix
 What are we testing for?
 What do all the black lines mean?
 What factors do we need to include?
 Contrasts
 What are they for?
 t and F contrasts
 Inferences
 How do we do that in SPM5?
A
[1
B
C
D
-1
-1
1]
19
What are they for?
 General Linear Model (GLM) characterises
relationships between our experimental
manipulations and the observed data
 Multiple effects all within the same design matrix
 Thus, to focus on a particular characteristic,
condition, or regressor we use contrasts
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What are they for?
 A contrast is used by SPM to test
hypotheses about the effects defined in
the design matrix, using t-tests and Ftests
 Contrast specification and the
interpretation of the results are entirely
dependent on the model specification
which in turn depends on the design of
the experiment
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Some general remarks
• Clear hypothesis / question
• Clear design to answer the research question
• The contrasts and inferences made are dependent on
choice of experimental design
• Most of the problems concerning contrast specification
come from poor design specification
• Poor design:
• Unclear about what the objective is
• Try to answer too many questions in a single model
We need to think about how the experiment is going to be
modelled and which comparisons we wish to make
BEFORE acquiring the data
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Contrasts
E.g.: Contrasts with conditions:
 The conditions that we are interested in can take on a
positive value, such as 1
 The conditions that we want to subtract from these
conditions of interest can take on a negative value,
such as -1
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Contrasts
Condition 1: Language task
Condition 2: Memory task
Condition 3: Motor task
Condition 4: Control




Contrast 1: Language minus Control: 1 0 0 -1
Contrast 2: Motor minus Memory: 0 -1 1 0
Contrast 3: Control minus Motor: 0 0 -1 1
Contrast 4: (Language + Memory) minus Control: 1 1 0 -2
 This contrast will measure areas of the brain that have significantly
increased activity in the average of the language and memory conditions,
compared with the control condition – another way of looking at this
contrast is the sum of the individual condition contrasts of 1 0 0 -1 and
0 1 0 -1.
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Contrasts - Factorial design

SIMPLE MAIN EFFECT
 A–B
 Simple main effect of motion (vs. no motion) in
the context of low load
 [ 1 -1
0
0]
LOW
A

MOTION
B
C
D
A
B
C
D
LOAD
MAIN EFFECT
 (A + B) – (C + D)
HIGH
 The main effect of low load (vs. high load)
irrelevant of motion
  Main effect of load
 [ 1 1 -1 -1]
A

NO MOTION
B
C
D
INTERACTION
 (A - B) – (C - D)
 The interaction effect of motion (vs. no motion)
greater under low (vs. high) load
 [ 1 -1 -1 1]
A
B C
D
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Contrasts
 t-test: is there a significant increase or is there a
significant decrease in a specific contrast (between
conditions) – directional
 F-test: is there a significant difference between
conditions in the contrast – non-directional
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Example
 Two event-related conditions
 The subjects press a button with either their left or right
hand depending on a visual instruction (involving some
attention)
 We are interested in finding the brain regions that
respond more to left than right motor movement
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t-contrasts
Left
Right
Mean
 t-contrasts are directional
 To find the brain regions
corresponding more to left
than right motor responses
we use the contrast:
T = [1 -1 0]
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t-contrasts
 A one dimensional contrast
contrast of
estimated
parameters
t=
c’b
t=
variance
estimate
s2c’(X’X)+c
So, for a contrast in our model of 1 -1 0:
t = (ß1x1 + ß2x-1 + ß3x0)
Estimated variance
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Brain activation: Left motor responses
This shows activation of
the contralateral motor
cortex, ipsilateral
cerebellum, etc.
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F-contrasts
Left
Right Mean
 F-contrasts are non-directional
 To test for the overall difference
(positive or negative) from the
left and right responses we use:
[1 0 0; 0 1 0]
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F-test
 To test a hypothesis about general effects, independent of the
direction of the contrast
 A collection of t-contrasts that you want to test together
Additional
variance
accounted for
by tested effects
F=
Error
variance
estimate
32
Brain activation
Areas involved in the
overall difference
(positive or negative)
from the left and right
responses
(non-directional)
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Test
Design and
contrast
SPM(t) or
SPM(F)
[1 -1 0]
t-test
[1 0 0; 0 1 0]
F-test
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Inferences about subjects and populations
 Inference about the effect in relation to:
 The within-subject variability (1st level analysis)
 The between subject variability (2nd level analysis)
 This distinction relates directly to the difference
between fixed and random-effect analyses
 Inferences based on fixed effects analyses are about the
particular subject(s) studied
 Random-effects analyses are usually more conservative but
allow the inference to be generalized to the population from
which the subjects were selected
More on this in
few weeks!
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One voxel = One test (t, F)
amplitude
General Linear Model
fitting
statistical image
Statistical image
(SPM)
Temporal series
fMRI
voxel time course
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From Poline (2005)
Choosing a statistical threshold
 Important consideration in neuroimaging = the tremendous
number of statistical tests computed for each comparison
 E.g.: if 100,000 voxels are tested at a probability threshold
of 5%, we should expect:
 5000 voxels will incorrectly appear as significant
activations
= Apparent activations by chance;
FALSE POSITIVE
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Choosing a statistical threshold
 Uncorrected threshold of p < .001
 Familywise Error (FWE)
 Bonferroni correction
 E.g.: .05/100,000 = .0000005
 False Discovery Rate (FDR)
 Adjusts the criterion used based on the amount of signal present in the data
 Reduce the number of comparisons
 E.g.: Instead of examining the entire brain, examine just a small region
IMPORTANCE of taking into account the multiple comparisons
across voxels BUT also the multiple comparisons across contrasts
(i.e., the number of contrasts tested)
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How do we do that in SPM5?
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Summary
 Contrasts are statistical (t or F) tests of specific
hypotheses
 t-contrast looks for a significant increase or
decrease in a specific contrast (directional)
 F-contrast looks for a significant difference between
conditions in the contrast (non-directional)
 Importance of having a clear design
 Inferences about subjects (1st level) and
populations (2nd level)
 Importance of considering the multiple comparisons
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References
 Human Brain Function 2, in particular Chapter 8 by Poline,
Kherif, & Penny
(http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch8.pdf)
 Introduction: Experimental design and statistical parametric
mapping, by Friston
 Linear Models and Contrasts, PowerPoint presentation by
Poline (April, 2005), SPM short course at Yale
 Previous years’ slides
 CBU Imaging Wiki (http://imaging.mrccbu.cam.ac.uk/imaging/PrinciplesStatistics) (http://imaging.mrccbu.cam.ac.uk/imaging/SpmContrasts)
 SPM5 Manual, The FIL Methods Group (2007)
 An introduction to functional MRI by de Haan & Rorden
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Thank you!
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