Variance components and Non-sphericity

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

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

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

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?

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

But why do we care? An example from Will Penny

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/ppt/hier.ppt

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

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

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

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

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