1. Variance components and Non-sphericity
Su Watkins
Bahador Bahrami
13 April 2005

2. Outline What is the sphericity assumption?
Why could it be a problem? What could we do to understand it better?
How do we measure sphericity?
How could we get rid of the problem?
Temporal Smoothing
Satterthwaite approximation (Greenhouse-Geisser)
ReML

3. Sphericity assumption
Remember the simplest case in GLM:
Y = X × β + ε Y1 Y2 Y3 … Yk X1 X2 X3 … Xk
ε 1
ε 2
ε 3 …
ε k =
× β +
The question was: Is β significantly different from 0 ?
t =
β / ε

4. Sphericity assumption
Next more complex step GLM is:
Y = X1 × β1 + X2 × β2 + ε Y1 Y2 … Yk
X11
X12
X21
X22 …
Xk1
Xk2
ε 1
ε 2
ε k β1 β2
t =
(β1 – β2) / ε = ×
The new question is: How valid is our estimate of error?

5. Sphericity assumption
Measurement error (a.k.a. variance) is
Identical
AND
Independent
across all levels of measurement
But what do Identical and Independent mean?

6. But why do we care? An example from Will Penny

7. Example I U. Noppeney et al.
Stimuli: Auditory Presentation (SOA = 4 secs) of
(i) words (e.g. book)
(ii) words spoken backwards (e.g. koob)
Subjects: (i) 12 control subjects
(ii) 11 blind subjects
Scanning: fMRI, 250 scans per subject, block design
Q. What are the regions that activate for real words relative to
reverse words in both blind and control groups?

8. Non-Identical (but independent) Error
BOLD
ε 1
ε 2
book
koob
book
koob
Blind
Control
Non-Identical (but independent) Error e1 e2

9. Error can be Independent but Non-Identical when…
1) One parameter but from different groups
e.g. patients and control groups
2) One parameter but design matrices differ across subjects
e.g. subsequent memory effect e1 e2

10. Error can be Non-Independent AND Non-Identical when…
Several parameters per subject
e.g. Repeated Measurement design
Conjunction over several parameters
e.g. Common brain activity for different cognitive processes
Complete characterization of the hemodynamic response
e.g. F-test combining HRF, temporal derivative and dispersion regressors

11. How do we measure sphericity?
Covariance Matrix
Sphericity
Non- Sphericity
0.3
3.0
5.9 45
2.7
9.43 88
5.6 34
0.04
5.0
0.32 10 1
2.12 σ2

12. Box’s measure (ε) measures the departure of Cov(εk) from spherical
Sphericity
Non- Sphericity c σ2
ε = 1
ε = 1/(k-1)