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Variance components and Non-sphericity

Explore the concept of sphericity assumption in Generalized Linear Models (GLM) and its implications. Learn how to measure sphericity, address issues, and interpret error estimates with relevance to various experimental designs.

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Variance components and Non-sphericity

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  1. Variance componentsandNon-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. β / ε t = Sphericity assumption Remember the simplest case in GLM: Y = X ×β + ε = ×β + The question was: Is β significantly different from 0 ?

  4. (β1 – β2) / ε t = Sphericity assumption Next more complex step GLM is: Y = X1×β1 + X2×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? http://www.fil.ion.ucl.ac.uk/spm/course/slides03/ppt/hier.ppt

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

  9. e2 e1 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

  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. 2.7 σ2 9.43 0 88 0 5.6 0 Sphericity Non- Sphericity 0.3 0 σ2 3.0 5.9 0 45 0 0 10 0 0 σ2 1 2.12 0 34 0 0.04 0 5.0 0 0.32 σ2 How do we measure sphericity? • Covariance Matrix

  12. σ2 c 0 c c 0 c 0 Sphericity Non- Sphericity c 0 c σ2 c 0 c 0 Cov(εk) = ε = 1 ε = 1/(k-1) c 0 0 c σ2 c c 0 0 c 0 c 0 c σ2 c • Box’s measure (ε) measures the departure of Cov(εk) from spherical

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