1. 2nd level analysis – design matrix, contrasts and inference
Irma Kurniawan MFD Jan 2009

2. Today’s menu 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

3. 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
error df ~ 300
Subject 2
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 3
Subject 4
Fixed effect: A variable with fixed values
E.g. levels of an experimental variable.
Box on the right: Fixed-effect analysis; The factor ‘subject’ treated like other experimental variable in the design matrix.
Within-subject variability across condition onsets represented across rows.
Between-subject variability ignored
Case-studies approach: Fixed-effects analysis can only describe the specific sample but does not allow generalization.
Random effect: A variable with values that can vary.
E.g. the effect ‘list order’ with lists that are randomized per subject
The effect ‘Subject’ can be described as either fixed or random
Subjects in the sample are fixed
Subjects are drawn randomly from the population
Typically treated as a random effect in behavioural analysis
Random effects analysis:
Generalization to the population requires taking between-subject variability into account.
The question: Would a new subject drawn from this population show any significant activity?
Mixed models: the experimental factors are fixed but the ‘subject’ factor is random.
Mixed models take into account both within- and between- subject variability  better!
Subject 5
Subject 6

4. Two-stage “Summary Statistic” approach
1st-level (within-subject)
2nd-level (between-subject) b1 ^ b2 b3 b4 b5 b6
contrast images of cbi 
N=6 subjects
(error df =5)
One-sample
t-test ^
(1)
w = within-subject error
(2)
(3)
(4)
(5)
(6)
p < (uncorrected)
SPM{t} ^
bpop 
WHEN special case of n independent observations per subject:
var(bpop) = 2b / N + 2w / Nn

5. 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.
2nd-level analysis: Feed the contrast images into a GLM that implements a statistical test.
Con image for contrast 1 for subject 1
Con image for contrast 2 for subject 2
Con image for contrast 1 for subject 2
Con image for contrast 2 for subject 1
Contrast 1
Contrast 2
Subject 2
Subject 1
You can use checkreg button to display con images of different subjects for 1 contrast and eye-ball if they show similar activations

6. 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

7. 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.
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
main effect of A
main effect of B
interaction AxB. A1 A2 1 2 4 5 3 6 B2 B3 B1

8. 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)
A single number: categorical or frequency Whether the subject reports pain or not; the number of times the subject reported feeling pain
A summary statistic, resulting from a simple model of the data, typically the mean: The mean reaction time in a classification task.

9. Behind button-clicking…
Which images are produced and calculated when I press ‘run’?

10. 1st level design matrix:
6 sessions per subject

11. 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

12. 1st-level (within-subject)^ b2 b3 b4 b5 b6
Beta images contain values related to size of effect. A given voxel in each beta image will have a value related to the size of effect for that explanatory variable.
1 ^
2
3
4
5
6
w = within-subject error
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.

13. Explicit masks Single subject mask Group mask
Mask.img
Calculated using the intersection of 3 masks:
Implicit (if a zero in any image then masked for all images) default = yes
Thresholding which can be i) none, ii) absolute, iii) relative to global (80%).
Explicit mask (user specified)
Single subject mask
Group mask
Note:
You can include explicit mask at 1st- or 2nd-level.
If include at 1st-level, the resulting group mask at 2nd-level is the overlapping regions of masks at 1st-levelso, will probably much smaller than single subject masks.
A sensible option here is to use a segmentation of structural images to specify a within-brain mask.
Explicit masks are other images containing (implicit) masks that are to be applied to the current
analysis.
Segmentation of structural images

14. Con. value = summation of all relevant betas.
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.

15. ResMS.img =
residual sum of squares or variance image and is a measure of within-subject error at the 1st level or between-subject error at the 2nd.
Con. value is combined with ResMS value at that voxel to produce a T statistic or spm.T.img.

16. spmT.img
Thresholded using the
results button.

17. spmT.img and corresponding spmF.img

18. 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.

19. 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)

21. A few additional notes…

22. How to enter contrasts…
Effort
How to enter contrasts… E1 E2 R1
Reward R2 R1 R2 E1 E2
Main effect of Reward 1 -1
Main effect of Effort
Effort x Reward
Interaction:
RE1 x RE2
= (R1E1 – R1E2) – (R2E1– R2E2)
= R1E1 – R1E2 – R2E1 + R2E2 =
= [ ]

23. set-level: P(c  3, n  k, t  u) = 0.019
Levels of Inference
Three levels of inference:
extreme voxel values
voxel-level (height) inference
big suprathreshold clusters
cluster-level (extent) inference
many suprathreshold clusters
set-level inference
voxel-level: P(t  4.37) = .048
n=82
n=32
n=12
set-level: P(c  3, n  k, t  u) = 0.019
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.
cluster-level: P(n  82, t  u) = 0.029

24. Example SPM window

25. Global Effects May be global variation from scan to scan
Such “global” changes in image intensity confound local / regional changes of experiment
Adjust for global effects (for fMRI) by:
Proportional Scaling
Can improve statistics when orthogonal to effects of interest (as here)…
…but can also worsen when effects of interest correlated with global (as next)
global
global
Scaling

26. 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

27. Small-volume correction
If have an a priori region of interest, no need to correct for whole- brain!
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)

28. Example SPM window

29. SVC summary Degrees of freedom.
p value associated with t and Z scores is dependent on 2 parameters:
Degrees of freedom.
How you choose to correct for multiple comparisons.

30. 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 individual-differences
Some tests are more typical in imaging:
Conjunction analysis
Multiple comparisons poses a greater problem in imaging (RFT; small volume correction)

31. 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