2nd level analysis – design matrix,
contrasts and inference
Irma Kurniawan
MFD Jan 2009
Today’s menu
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Fixed, random, mixed effects
First to second level analysis
Behind button-clicking: Images produced and calculated
The buttons and what follows..
Contrast vectors, Levels of inference, Global effects, Small
Volume Correction
• Summary
Fixed vs. Random Effects
• Subjects can be Fixed or Random variables
• If subjects are a Fixed variable in a single design
matrix (SPM “sessions”), the error term conflates
within- and between-subject variance
– But in fMRI (unlike PET) the between-scan
variance is normally much smaller than the
between-subject variance
Multi-subject Fixed Effect model
Subject 1
Subject 2
Subject 3
• If one wishes to make an inference from a subject
sample to the population, one needs to treat subjects
as a Random variable, and needs a proper mixture of
within- and between-subject variance
• Mixed models: the experimental factors are fixed but
the ‘subject’ factor is random.
• In SPM, this is achieved by a two-stage procedure:
1) (Contrasts of) parameters are estimated from a
(Fixed Effect) model for each subject
2) Images of these contrasts become the data for
a second design matrix (usually simple t-test or
ANOVA)
Subject 4
Subject 5
Subject 6
error df ~ 300
Two-stage “Summary Statistic” approach
1st-level (within-subject)
2nd-level (between-subject)

b^2
^ 2)
(

b^3
^ 3)
(

b^4
^ 4)
(
b^5
^ 5)
(
b^6
^ 6)
(

One-sample
t-test
contrast images of cbi
b^1
^ 1)
(

N=6 subjects
(error df =5)
p < 0.001 (uncorrected)
SPM{t}
^b
pop


^ = within-subject error
w
WHEN special case of n
independent observations per
subject:
var(bpop) = 2b / N + 2w / Nn
Relationship between 1st & 2nd levels
• 1st-level analysis: Fit the
model for each
subject.Typically, one design
matrix per subject
• Define the effect of interest
for each subject with a
contrast vector.
• The contrast vector
produces a contrast image
containing the contrast of the
parameter estimates at each
voxel.
Contrast 1
Subject 1
Con image
for contrast
1 for
subject 1
Subject
2
• 2nd-level analysis: Feed the
contrast images into a GLM
that implements a statistical
test.
Contrast 2
Con image
for contrast
1 for
subject 2
Con image
for contrast
2 for
subject 1
Con image
for contrast
2 for
subject 2
You can use checkreg button to display con images
of different subjects for 1 contrast and eye-ball if they
show similar activations
Similarities between 1st & 2nd levels
• Both use the GLM model/tests and a similar SPM
machinery
• Both produce design matrices.
• The rows in the design matrices represent observations:
– 1st level: Time (condition onsets); within-subject variability
– 2nd level: subjects; between-subject variability
• The columns represent explanatory variables (EV):
– 1st level: All conditions within the experimental design
– 2nd level: The specific effects of interest
Similarities between 1st & 2nd levels
• The same tests can be used in both levels (but the questions are
different)
• .Con images: output at 1st level, both input and output at 2nd level
• 1st level: variance is within subject, 2nd level: variance is between
subject.
• There is typically only one 1st-level design matrix per subject, but
multiple 2nd level design matrices for the group – one for each
statistical test.
B1
B2
B3
A1
1
2
3
A2
4
5
6
For example:
2 X 3 design between variable A and B.
We’d have three design matrices (entering 3 different
sets of con images from 1st level analyses) for
1) main effect of A
2) main effect of B
3) interaction AxB.
Difference from behavioral analysis
• The ‘1st level analysis’ typical to behavioural data is
relatively simple:
– A single number: categorical or frequency
– A summary statistic, resulting from a simple model of the data,
typically the mean.
• SPM 1st level is an extra step in the analysis, which
models the response of one subject. The statistic
generated (β) then taken forward to the GLM.
– This is possible because βs are normally distributed.
• A series of 3-D matrices (β values, error terms)
Behind button-clicking…
• Which images are produced and calculated when
I press ‘run’?
1st level design matrix:
6 sessions per subject
The following images are created each
time an analysis is performed (1st or 2nd
level)
• beta images (with associated header),
images of estimated regression
coefficients (parameter estimate).
Combined to produce con. images.
• mask.img This defines the search
space for the statistical analysis.
• ResMS.img An image of the variance
of the error (NB: this image is used to
produce spmT images).
• RPV.img The estimated resels per
voxel (not currently used).
•All images can be displayed using
check-reg button
1st-level (within-subject)
b^1 Beta images contain values related to size of effect. A
^ 1 given voxel in each beta image will have a value related

to the size of effect for that explanatory variable.
b^2
^ 

b^3
^ 3

The ‘goodness of fit’ or error term is contained in the
ResMS file and is the same for a given voxel within
the design matrix regardless of which beta(s) is/are
being used to create a con.img.
b^4
^ 4

b^5
^ 5

b^6
^ 6

^ = within-subject error
w
t masks
Mask.img
Calculated using the
intersection of 3
masks:
1) Implicit (if a zero
in any image
then masked for
all images)
default = yes
2) Thresholding
which can be i)
none, ii)
absolute, iii)
relative to global
(80%).
3) Explicit mask
(user specified)
Single subject mask
Segmentati
on of
structural
images
Group mask
Note:
You can include explicit mask at
1st- or 2nd-level.
If include at 1st-level, the
resulting group mask at 2ndlevel is the overlapping
regions of masks at 1stlevelso, will probably
much smaller than single
subject masks.
Beta value = % change
above global mean.
In this design matrix
there are 6 repetitions
of the condition so
these need to be
summed.
Con. value =
summation of all
relevant betas.
ˆ i2
ResMS.img =
residual sum of squares
or variance image and
is a measure of withinsubject error at the 1st
level or betweensubject error at the 2nd.
Con. value is combined with ResMS
value at that voxel to produce a T
statistic or spm.T.img.
Eg random noise
spmT.img
Thresholded using the
results button.
pu = 0.05
Gaussian
10mm FWHM
(2mm pixels)
spmT.img and corresponding spmF.img
So, which images?
• beta images contain information about the size of the effect
of interest.
• Information about the error variance is held in the
ResMS.img.
• beta images are linearly combined to produce relevant con.
images.
• The design matrix, contrast, constant and ResMS.img are
subjected to matrix multiplication to produce an estimate of the
st.dev. associated with each voxel in the con.img.
• The spmT.img are derived from this and are thresholded in
the results step.
The buttons and what follows..
• Specify 2nd-level
• Enter the output dir
• Enter con images from
each subject as ‘scans’
• PS: Using matlabbatch, you
can run several design
matrices for different
contrasts all at once
• Hit ‘run’
• Click ‘estimate’ (may take a
little while)
• Click ‘results’ (can ‘review’
first before this)
A few additional notes…
Effort
How to enter contrasts…
E1
E2
Reward R1
R2
R1
R2
E1 E2 E1 E2
Main effect 1
1
-1 -1
of Reward
Main effect 1
of Effort
-1
1
-1
Effort x
Reward
-1
-1
1
1
Interaction:
RE1 x RE2
= (R1E1 – R1E2) – (R2E1– R2E2)
= R1E1 – R1E2 – R2E1 + R2E2
= 1 - 1 - 1 + 1
= [ 1 -1 -1 1]
Levels of Inference
• Three levels of inference:
– extreme voxel values
 voxel-level (height) inference
– big suprathreshold clusters
voxel-level: P(t  4.37) = .048
 cluster-level (extent) inference
– many suprathreshold
clusters
n=1
2
 set-level inference
n=82
Set level: At least 3 clusters above threshold
Cluster level: At least 2 cluster with at least 82
voxels above threshold
Voxel level: at least cluster with unspecified
number of voxels above threshold
Which is more powerful?
Set > cluster > voxel level
Can use voxel level threshold for a priori
hypotheses about specific voxels.
n=32
cluster-level: P(n  82, t  u) = 0.029
set-level: P(c  3, n  k, t  u) = 0.019
Example SPM window
Global Effects
•
May be global variation from scan to scan
•
Such “global” changes in image intensity
confound local / regional changes of
experiment
global
•
Adjust for global effects (for fMRI) by:
Proportional Scaling
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•
Can improve statistics when orthogonal to
effects of interest (as here)…
…but can also worsen when effects of
interest correlated with global (as next)
Scaling
global
Global Effects
• Two types of scaling: Grand Mean scaling and Global scaling
• Grand Mean scaling is automatic, global scaling is optional
• Grand Mean scales by 100/mean over all voxels and ALL scans
(i.e, single number per session)
• Global scaling scales by 100/mean over all voxels for EACH scan
(i.e, a different scaling factor every scan)
• Problem with global scaling is that TRUE global is not (normally)
known…
• …we only estimate it by the mean over voxels
• So if there is a large signal change over many voxels, the global
estimate will be confounded by local changes
• This can produce artifactual deactivations in other regions after
global scaling
• Since most sources of global variability in fMRI are low frequency
(drift), high-pass filtering may be sufficient, and many people to not
use global scaling
Small-volume correction
• If have an a priori region of interest, no need to correct for wholebrain!
• But can correct for a Small Volume (SVC)
• Volume can be based on:
– An anatomically-defined region
– A geometric approximation to the above (eg rhomboid/sphere)
– A functionally-defined mask (based on an ORTHOGONAL contrast!)
• Extent of correction can be APPROXIMATED by a Bonferonni
correction for the number of resels…(cf. Random Field Theory
slides)
• ..but correction also depends on shape (surface area) as well as
size (volume) of region (may want to smooth volume if rough)
Example SPM window
SVC summary
•
p value associated with t and Z scores is dependent on
2
parameters:
1. Degrees of freedom.
2. How you choose to correct for multiple
comparisons.
Statistical inference: imaging vs. behavioural data
• Inference of imaging data uses some of the same
statistical tests as used for analysis of behavioral
data:
– t-tests,
– ANOVA
– The effect of covariates for the study of individualdifferences
• Some tests are more typical in imaging:
– Conjunction analysis
• Multiple comparisons poses a greater problem in
imaging (RFT; small volume correction)
With help from …
• Rik Henson’s slides.
• Debbie Talmi & Sarah White’s slides
• Alex Leff’s slides
• SPM manual (D:\spm5\man).
• Human Brain Function book
• Guillaume Flandin & Geoffrey Tan