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Efficiency in Experimental Design

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Explore the intricacies of efficiency in experimental design utilizing General Linear Models (GLM). This guide delves into how efficiently we can estimate parameters, particularly the β coefficients, given the design matrix X. We'll cover aspects such as Var(X), various overlapping and non-overlapping conditions, and specific contrasts such as simple effects. Understand the significance of relative efficiency in diverse experimental designs, including boxcar events and different stimulus onsets, while maintaining practical insight with minimal technical jargon.

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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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