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Understanding General Linear Model in Data Analysis

Explore the basics of General Linear Model in statistical analysis, including regressors, parameters, residuals, and design matrices. Learn about the assumptions, errors, and solutions in GLM using Ordinary Least Squares, with practical examples and references provided.

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Understanding General Linear Model in Data Analysis

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  1. General Linear Model

  2. General Linear Model regressors β1 β2 . . . βL ε1 ε2 . . . εJ Y1 Y2 . . . YJ X11 … X1l … X1L X21… X2l… X2L . . . XJ1 … XJl… XJL = + time points time points time points regressors Y = X *β + ε Design Matrix Observed data Parameters Residuals/Error

  3. Design Matrix 0 0 0 0 0 0 0 rest task Conditions On Off Off On 1 1 1 1 1 1 1 Use ‘dummy codes’ to label different levels of an experimental factor (eg. On = 1, Off = 0). time

  4. Design Matrix 5 4 4 2 3 1 6 3 1 6 5 2 Covariates Parametric variation of a single variable (eg. Task difficulty = 1-6) or measured values of a variable (eg. Movement).

  5. Design Matrix 1 1 1 1 1 1 1 1 . . . Constant Variable Models the baseline activity (eg. Always = 1)

  6. Design Matrix Time Regressors The design matrix should include everything that might explain the data.

  7. General Linear Model regressors β1 β2 . . . βL ε1 ε2 . . . εJ Y1 Y2 . . . YJ X11 … X1l … X1L X21… X2l… X2L . . . XJ1 … XJl… XJL = + time points time points time points regressors Y = X *β + ε Design Matrix Observed data Parameters Residuals/Error

  8. Error • Independent and identically distributed iid

  9. Ordinary Least Squares Residual sum of square: The sum of the square difference between actual value and fitted value. e

  10. Ordinary Least Squares N å 2 e = minimum t = t 1 e

  11. Ordinary Least Squares Y = Xβ+e e = Y-Xβ XTe=0 => XT(Y-Xβ)=0 => XTY-XTXβ=0 => XTXβ=XTY => β=(XTX)-1XTY y e Xβ x1β1 x2β2

  12. fMRI Y = X *β + ε Observed data Design Matrix Parameters Residuals/Error

  13. Problems with the model

  14. The Convolution Model Expected BOLD HRF Impulses  =

  15. Convolve stimulus function with a canonical hemodynamic response function (HRF): OriginalConvolvedHRF  HRF

  16. Physiological Problems

  17. Noise Low-frequency noise Solution: High pass filtering

  18. blue= data black = mean + low-frequency drift green= predicted response, taking into account low-frequency drift red= predicted response, NOT taking into account low-frequency drift discrete cosine transform (DCT) set

  19. Assumptions of GLM using OLS All About Error

  20. Unbiasedness Expected value of beta = beta

  21. Normality

  22. Sphericity

  23. Homoscedasticity

  24. not

  25. Independence

  26. Autoregressive Model y = Xβ + e overtime et= aet-1 + ε autocovariance function a should = 0

  27. Thanks to… • Dr. Guillaume Flandin

  28. References • http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch7.pdf • http://www.fil.ion.ucl.ac.uk/spm/course/slides10-vancouver/02_General_Linear_Model.pdf • Previous MfD presentations

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