1. With a focus on task-based analysis and SPM12
fMRI Statistics
With a focus on task-based analysis and SPM12

2. Models
Previously, we looked at GLM models for single subjects (first level models)
We can also use a GLM to estimate group effects (second level models)
Fixed effects analysis: include all data in a single model (less used option for group analysis)
Random effects analysis*: use outputs of first level models as inputs to second level (more efficient, and generalizes)
*Strictly speaking, “mixed effects”: both random & fixed effects together

3. Fixed vs. Random Effects
In a fixed effects model:
Which brain areas are activated on average across subjects?
In a random effects model:
Which brain areas are activated in the same way across subjects?
If you only have one scan on one condition per subject, fixed is all you can do (e.g., FDG-PET)

4. Two Stage Modeling Limitations
In theory, every first level design matrix should be identical (in practice, somewhat robust to variation: Penny, 2004)
Assumes underlying error is identical across subjects (all details of first level analysis condensed to voxelwise betas/contrasts)
Limited (in SPM at least) to voxels available for all subjects

5. Covariates & Estimability
Covariates can be useful, especially in a second level model (e.g., age)
If any variable (column of X) is a linear combination of others, some betas cannot be estimated uniquely
As a result, some contrasts can be rejected as “inestimable” by SPM*
Color coding: grey (not uniquely specified) vs. white
*strictly speaking they are estimable, but without a unique solution

6. Contrast Examples +1 -1
Contrasts = linear combinations of regressors, to be compared to zero
Could compare one to zero:
a > 0
Or difference of two:
a – b > 0
aka, “a > b”
Covariates

7. The Constant Term
How the GLM constant term is included also affects what contrasts are estimable Usually better to have it “implicit”
Source: Rik Henson “GLM & RFT”

8. Regressor Scaling/Centering
Regressors in the design matrix X need to be scaled, or big ones will dominate small ones
SPM automatically scales covariates as they are added
Centering affects model error and interpretation
Overall mean (default)
No centering
Factor-based (for covariate interpretation that differs across factor levels)
Also, can specify interaction with a factor

9. Regressor Orthogonalization
“Orthogonal” regressors are independent
Inner product of vectors = 0
Not (quite) the same thing as “uncorrelated”!
Inner product of de-meaned vectors = 0
Rodgers et al. (1984) Linearly independent, orthogonal and uncorrelated variables. The American Statistician, 38:

10. Regressor Orthogonalization
If regressors are not orthogonal*
attribution of effects becomes difficult
Effect estimates (betas) are artificially reduced
Variance not assignable to one source is “lost”
*or “uncorrelated”—with default de-meaning, these are functionally identical in SPM

11. Parcellation of Variance
Orthogonal regressors account for different parts
of the variance.

12. Parcellation of Variance
Lost
Non-orthogonal regressors account for overlapping parts of the variance, and each
ends up with only its unique portion.

13. Parcellation of Variance
Lost
If the overlap is particularly severe the
effects cannot be estimated reliably: inestimable
Avoid correlated regressors!

14. Orthogonalization
How can we “orthogonalize” non-orthogonal regressors?
Avoid the issue in design (for experimental conditions)
Principal components analysis/factor analysis? (difficult to interpret)
Serial orthogonalization (used by SPM)

15. Serial Orthogonalization
When we have only one regressor, things are simple…
Y = 1X1 Y X1
Example from: Evina Chu, “Basis Functions” SPM MfD course (12/2007)

16. Serial Orthogonalization
When we have only one regressor, things are simple…
Y = 1X1
1 = 1.5 Y
This is the best estimate of Y using
only X1
X1 (vector)
1 (length)
Example from: Evina Chu, “Basis Functions” SPM MfD course (12/2007)

17. Serial Orthogonalization
Now consider adding a second regressor, one not orthogonal to the first…
Y = 1X1 + 2X2
1 = 1
2 = 1 Y X2 X1

18. Serial Orthogonalization
Now consider adding a second regressor, one not orthogonal to the first…
Y = 1X1 + 2X2
1 = 1
2 = 1
1 (length) Y
We can now estimate Y perfectly
using both Xs; however, note that
1 has dropped from 1.5 to 1.
X2 is explaining variance X1 could
also explain. X2
2 (length) X1

19. Serial Orthogonalization
Let’s orthogonalize X2 with respect to X1. This will create a new variable “X2*”.
Y = 1X1 + 2*X2*
1 = 1.5
2* = 1 Y X2
X2* X1

20. Serial Orthogonalization
Let’s orthogonalize X2 with respect to X1. This will create a new variable “X2*”.
Y = 1X1 + 2*X2*
1 = 1.5
2* = 1
1 is back to 1.5
1 (length) Y
Orthogonalization (via Gram-
Schmidt process) produces a
new variable, based on the
old one and existing variables
Serial process, so order matters! X2
2* (length)
X2* X1

21. Serial Orthogonalization in SPM
Design matrix is orthogonalized left-to-right, so order matters
Tip: put the “most important” covariates first (the ones whose meaning you don’t want to change)
If all are “nuisance regressors” that won’t be interpreted, order doesn’t matter
Can always plot the final (orthogonalized) variables

22. Significance Testing The t-test is the basic unit of SPM
Measures “signal” (here, contrast) relative to “noise” (variability)
T-value generated for each contrast at each voxel
Recall that the t-test is implemented in a GLM framework (allowing covariates)
SPM outputs spmT (“t-map”) as well as con and beta files

23. F-tests SPM can also do F-tests Produce “F-maps”
In this context, can be thought of as a generalization of t-tests
Tests multiple conditions but without the directionality of t-tests
Tells “is any combination of these variables having a significant effect?” (but not which ones/which direction)
Produce “F-maps”

24. T/F-test Comparison A B F: [1 -1] T: [1 -1] B A A B F: [1 0 0 1]
Source: Rik Henson, “SPM GLM” talk

25. The Multiple Comparisons Problem
Making “independent” assessments at each voxel leads to many, many statistical tests
If we assume p < 0.05 threshold and complete independence, 5% of voxels tested will be false positives!
~100,000 voxels –> 5,000 false positives…

26. Multiple Comparisons Correction
We can “correct” by setting a more stringent threshold
This is called setting a “family wise” threshold, and leads to a “family wise error rate”
Bonferroni correction: divide p threshold by number of tests
p < 100,000 voxels becomes
p < 0.05/100,000
p < family-wise threshold

27. Multiple Comparisons Correction
Bonferroni assumes independent tests
More generally, can use a less strict FWE correction if that doesn’t hold
SPM uses Random Field Theory (RFT)
Estimates smoothness across the brain
Smoothness implies loss of independence between neighbors, and reduces need for correction
Less smoothness —> better spatial specificity, but in the extreme RFT can become even more conservative than Bonferroni!

28. Multiple Comparisons Correction
Another option is False Discovery Rate (FDR) correction
Instead of controlling the chance of any test being a false positive (p), control the fraction of false positives (q)
Often easier to limit to, say, <5% false positives than to a <5% chance of a false positive; thus, less conservative in those cases
May or may not assume independence

29. Multiple Comparisons Correction
One other option is to change the search area
This only makes sense if done a priori!
For example, a specific hypothesis might call for investigating only regions X and Y
SPM’s results are always adjusted to the search area
Note: SPM masks in two stages, and only the first (prior to result generation) affects statistics!

30. Multiple Comparisons Correction
So far we’ve considered voxelwise correction (aka “peak level” statistics)
Can also use the extent of activation (number of contiguous activated voxels) to assess significance (aka “cluster level” statistics)
Can predict the expected number/size of clusters (using RFT)
Clusters bigger than a certain size are “significant”
SPM reports using FWE and FDR values (as with peak level)

31. Excursions, Peaks, Clusters
Excursion set: super-threshold voxels for some threshold
Peaks: local maxima in the excursion set
Clusters: sets of neighboring voxels in the excursion set
“excursion set” in black
Source: Durnez et al. 2014

32. SPM Results Reporting Clusters, peaks Sub-peaks (within each
Cluster level
(determined by
“kE”, the cluster
size)
Peak level
(determined by
t value)

33. SPM Results Visualization
Results table gives the fundamentals
Accompanied by a “glass brain” (maximum intensity projection) view
SPM results can be viewed interactively (clicking in the glass brain or table)

34. Additional SPM Plots Can also use several additional views in SPM:
Slices: several contiguous slices
Sections: several orthogonal slices
Render: surface map showing near/at surface activations
In general, want to plot what you are using to make your inferences (especially for publication)