1. Methods for Dummies Random Field Theory
Dominic Oliver & Stephanie Alley

2. Structure What is the multiple comparisons problem?
Why we can’t use the classical Bonferroni approach
What is Random Field Theory?
How do we implement it in SPM?

3. Processing so far

4. Hypothesis testing and the type I error
Testing t-statistic against our null hypothesis (H0)
Probability of result arising by chance
p < 0.05
Chosen threshold (u) determines false positive rate ()
t-statistic distribution means you always run a risk of false positive or type I error t u

5. Significance of type I errors in fMRI u
Since α remains constant, the more tests you do, the more type I errors
Due to the number of tests performed in fMRI, the number of type I errors with this threshold is highly impractical
60,000 voxels per brain
60,000 t-tests
α = 0.05
60,000 x 0.05 = 3,000 type I errors

6. Why is this a problem?
3,000 is a large number but it’s still only 5% of our total voxels
Neural Correlates of Interspecies Perspective Taking in the Post-Mortem Atlantic Salmon
t(131) > 3.15, p(uncorrected) < 0.001
Bennett, Miller & Wolford, 2009

7. So what can we do about it?
Adjust the threshold (u), taking in account the number of tests being performed
Any values above new threshold are highly unlikely to be false positives t u
t > 0.5
t > 1.5
t > 2.5
t > 3.5
t > 4.5
t > 5.5
t > 6.5

8. The Bonferroni Correction
Why can’t we use it?

9. Bonferroni correction example
single-voxel probability threshold 
Number of voxels
 = PFWE / n
e.g. 100,000 t values, each with 40 d.f.
When PFWE = 0.05:
0.05/ =
t = 5.77
Voxel where t > 5.77 has only 5% chance of appearing in a volume of 100,000 t stats drawn from the null distribution
Family-wise error rate

10. So what’s the issue?
Bonferroni correction is too conservative for fMRI data
Bonferroni relies on independent values
But fMRI data is not independent due to
How the scanner collects data and reconstructs the images
Anatomy and physiology
Preprocessing (e.g. normalisation, smoothing)

11. The problem What Bonferroni needs: What we have:
A series of independent statistical tests
What we have:
A spatially dependent, continuous statistical image

12. Spatial dependence This slice is made of 100 x 100 voxels
Random values from normal distribution
10,000 independent values
Bonferroni accurate

13. Spatial dependence
Add spatial dependence by averaging across 10 x 10 squares
Still 10,000 numbers in image but only 100 are independent
Bonferroni correction 100 times too conservative

14. Smoothing Improves signal to noise ratio (SNR)
Convolve fMRI signal with Gaussian kernel
Improves signal to noise ratio (SNR)
Increased sensitivity
Incorporates anatomical and functional variations between subjects
Improves statistical validity by making distribution of errors more normal

15. Spatial dependence is increased by smoothing
When map is smoothed, each value in the image is replaced by a weighted average of itself and its neighbours
Image is blurred and independent values are reduced

16. If not Bonferroni, then what?
Idea behind Bonferroni correction in this context is sound
We need to adjust our threshold in a similar way but also take into account the influence on spatial dependence on our data
What does this? Random Field Theory

17. Random Field Theory

18. Random field theory provides a means of working with smooth statistical maps.
It allows us to determine the height threshold that will give the desired FWE.
Random fields have specified topological characteristics
Apply topological inference to detect activations in SPMs

19. Three stages of applying RFT:
1. Estimate smoothness
2. Calculate the number of resels
3. Get an estimate of the Euler Characteristic at
different thresholds
Resels = resolution elements; depend on smoothness and number of voxels (i.e. size of search region)!

20. 1. Estimate Smoothness
estimate
Image contains both applied and intrinsic smoothness
Estimated using the residuals (error) from GLM
In practice, smoothness is calculated using the residual values from the statistical analysis  Y = bX + e  b = beta = scaling values, X = design matrix, e = residual error
I.i.d. noise = independently and identically distributed noise  all variables have the same probability distribution and are mutual INDEPENDENT

21. 2. Calculate the number of resels Look at your estimated smoothness (FWHM)
The smoothed image contains spatial correlation, which is typical of the
output from the analysis of functional imaging data.
We now have a problem, because there is no simple way of calculating the number of independent observations in the smoothed data, so we cannot use the Bonferroni correction.
 This problem can be addressed using random field theory.
FWHM
(Full Width at Half Maximum)

22. Express your search volume in resels
Resel - A block of values that is the same size as the FWHM
Resolution element (Ri = FWHMx x FWHMy x FWHMz)
“Restores” independence of the data
A resel is simply defined as a block of values (e.g. pixels) that is the same size as the FWHM

23. 3. Get an estimate of the Euler Characteristic
Euler Characteristic – a topological property
SPM  threshold  EC
EC = number of blobs (minus number of holes)*
“Seven bridges of Kӧnisberg”
Leonhard Euler ( ), Swiss mathematician  Seven bridges of Kӧnisberg
*Not relevant: we are only interested in EC at high thresholds (when it approximates P of FEW)

24. Steps 1 and 2 yield a ‘fitted random field’ (appropriate smoothness)
Now: how likely is it to observe above threshold (‘significantly different’) local maxima (or clusters, sets) under H0?
How to find out? EC!

25. Euler Characteristic and FWE
Zt = 2.5
EC = 3
Zt = 2.75
The probability of a family wise error is approximately equivalent to the expected Euler Characteristic
Number of “above threshold blobs”
If we increase the Z score threshold to 2.75, we find that the two central blobs disappear –
because the Z scores were less than 2.75.
EC = 1

26. How to get E [EC] at different thresholds
# of resels
Expected Euler Characteristic
Z (or T) -score threshold
From this equation, it follows that the threshold depends only on the number of resels in our image.

27. Given # of resels and E[EC], we can find the appropriate z/t-threshold
E [EC] for an image of 100 resels, for Z score thresholds 0 – 5
The graph does a reasonable job of predicting the EC in our image; at a Z threshold of 2.5 it predicted an EC of 1.9, when we observed a value of 3; at Z=2.75 it predicted an EC of 1.1, for an observed EC of 1.

28. How to get E [EC] at different thresholds
# of resels
Expected Euler Characteristic
Z (or T) -score threshold
From this equation, it follows that the threshold depends only on the number of resels in our image.
From this equation, it looks like the threshold depends only on the number of resels in our image

29. Shape of search region matters, too!
#resels  E [EC]
not strictly accurate!
close approximation if ROI is large compared to the size of a resel
# + shape + size resel 
E [EC]
Matters if small or oddly shaped ROI
Example:
central 30x30 pixel box = max. 16 ‘resels’ (resel-width = 10pixel)
edge 2.5 pixel frame = same volume but max. 32 ‘resels’
Multiple comparison correction for frame must be more stringent

30. Different types of topological Inference
Topological inference can be about
Peak height (voxel level)
Regional extent (cluster level)
Number of clusters (set level)
space
intensity tα
tclus
Different height and spatial extent thresholds
Voxel-level inference based on height thresholds asks: is activation at a given voxel significantly non-zero?
Bigger framework: cluster-level and set-level inference. These require height and spatial extent thresholds to be specified by the user
Corrected p-values can then be derived that pertain to…

31. Assumptions made by RFT
Underlying fields are continuous with twice-differentiable autocorrelation function
Error fields are reasonable lattice approximation to underlying random field with multivariate Gaussian distribution
lattice representation
- Autocorrelation function need not be Gaussian
- If the data have been sufficiently smoothed and the General Linear Models correctly specified (so that the errors are indeed Gaussian) then the RFT assumptions will be met
Check: FWHM must be min. 3 voxels in each dimension
In general: if smooth enough + GLM correct (error = Gaussian)  RFT

32. Assumptions are not met if...
Data is not sufficiently smooth
Small number of subjects
High-resolution data
Non-parametric methods to get FWE (permutation)
Alternative ways of controlling for errors: FDR (false discovery rate)
E.g. in random effects analysis (chp.12) with a small number of subjects  because the resulting error fields will not be very smooth and so may violate the ‘reasonable lattice approximation’ assumption.

33. Implementation in SPM

34. Implementation in SPM

35. Activations Significant at Cluster level But not at Voxel Level

36. Sources
Human Brain Function, Chapter 14, An Introduction to Random Field Theory (Brett, Penny, Kiebel)
Former MfD Presentations
Expert Guillaume Flandin