Variance components and Non-sphericity

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Variance componentsandNon-sphericity

• Su Watkins
• Bahador Bahrami
• 13 April 2005

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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

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Sphericity assumption

• Remember the simplest case in GLM
• Y X ß e

X1
X2
X3




Xk
Y1
Y2
Y3




Yk
e 1
e 2
e 3




e k

ß
The question was Is ß significantly different
from 0 ?
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Sphericity assumption

• Next more complex step GLM is
• Y X1 ß1 X2 ß2 e

e 1
e 2





e k
Y1
Y2





Yk
X11 X12
X21 X22





Xk1 Xk2








ß1
ß2


The new question is How valid is our estimate of
error?
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Sphericity assumption

• Measurement error (a.k.a. variance) is
• Identical
• AND
• Independent
• across all levels of measurement

But what do Identical and Independent mean?
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But why do we care? An example from Will Penny
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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?
http//www.fil.ion.ucl.ac.uk/spm/course/slides03/p
pt/hier.ppt
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BOLD

e 1



e 2










book
book
koob
koob
Blind
Control
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• 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

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• 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

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How do we measure sphericity?

• Covariance Matrix

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• Boxs measure (e) measures the departure of
  Cov(ek) from spherical