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Efficiency in Experimental Design

Efficiency in Experimental Design. Starring …. J. Winston. P. Bentley. General Linear Model: Y = X β + e Efficiency: ability to estimate β , given X Efficiency  1  Var(X)  X T X Var( β ). It ain’t gonna get technical now is it?. X. X T. =.

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Efficiency in Experimental Design

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  1. Efficiency in Experimental Design Starring … J. Winston P. Bentley

  2. General Linear Model: Y = Xβ + e • Efficiency: ability to estimate β, given X • Efficiency  1  Var(X)  XTX Var(β) It ain’t gonna get technical now is it?

  3. . X XT = XTX A 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 D 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 A B C D A 5 0 0 0 B 0 5 0 0 C 0 0 5 4 D 0 0 4 5 A B C D 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 Non-overlapping conditions Overlapping conditions

  4. Efficiency  1  Var(X) • Var(β) •  1  1 • 1/Var(X) 1/XTX inv(XTX) XTX A B C D A 0.2 0 0 0 B 0 0.2 0 0 C 0 0 0.6 -0.4 D 0 0 -0.4 0.6 A B C D A 5 0 0 0 B 0 5 0 0 C 0 0 5 4 D 0 0 4 5

  5. Efficiency is specific to condition or contrast • Efficiency  1 • cT inv(XTX ) c inv(XTX) When c is Simple Effect, e.g. [1 0 0 0] A, B: Efficiency = 1/0.2 = 5 C, D: Efficiency = 1/0.6 = 1.7 A B C D A 0.2 0 0 0 B 0 0.2 0 0 C 0 0 0.6 -0.4 D 0 0 -0.4 0.6 When c is Contrast, e.g. [1 -1 0 0] A-B: Efficiency = 1/0.4 = 2.5 C-D: Efficiency = 1/2 = 0.5

  6. Different Designs – Boxcar Events X inv(XTX) A B C D E F 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 A B C D E F A 0.2488 0.0377 -0.0297 -0.0396 -0.0012 -0.0873 B 0.0377 0.2862 -0.0941 -0.0421 -0.0873 -0.0263 C -0.0297 -0.0941 0.2871 0.0495 -0.0297 -0.0941 D -0.0396 -0.0421 0.0495 0.2327 -0.0396 -0.0421 E -0.0012 -0.0873 -0.0297 -0.0396 0.2488 0.0377 F -0.0873 -0.0263 -0.0941 -0.0421 0.0377 0.2862 Blocked Fixed Interleaved Efficiency Simple Effects: A, B = C,D = E,F = 4 Efficiency Contrasts: A - B = C – D = E – F = 2 Random

  7. Different Designs – Haemodynamic Responses X inv(XTX) 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 Blocked 5 Fixed Interleaved 1.5 Random- Uniform Relative Efficiency 2.8 Random- Sinusoidal 3.5

  8. Different Designs – Haemodynamic Responses inv(XTX) X 10 20 30 40 50 60 70 80 Blocked 5 2.5 Relative Efficiency 2.8 3.5

  9. Different Designs – Calculated Efficiencies I wish my Blocks Were BIGGER

  10. Different SOA’s – Variable No. of Trials inv(XTX) X Random: Events = 25 Random: Events = 50 2.1 4.2 Relative Efficiency

  11. Different SOA’s – Variable Min SOA inv(XTX) X Random: Min SOA = 5 secs Random: Min SOA = 0.5 secs 7.5 10.0 Relative Efficiency

  12. But as the SOA gets smaller, the HRF- linear convolution model breaks down, and the ability to estimatesimple effects vs. baseline diminshes 1 x 1 ≠ 2

  13. Joel, can you

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