1 / 13

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. =.

Download Presentation

Efficiency in Experimental Design

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related