General Linear Model and fMRI Rachel Denison & Marsha Quallo Methods for Dummies 2007

Did the experiment work ? Did the experimental manipulation affect brain activity? A simple experiment: Passive Listening vs. Rest time -- 6 scans per block

The General Linear Model Observed data = Predictors (EVs) * Parameters + Error y = Xβ + ε i.e. fMRI pixel valuesAlso called the design matrix. How much each EVcontributes to the observed data Residuals not explained by the model NNN = + X β ε y = + X β ε y 111P P

y: Activity of a single voxel over time time BOLD signal y1y2yNy1y2yN = y … One voxel at a time: Mass Univariate y = Xβ + ε

X in context = + x1x1 β ε y x2x2 x3x3 Observed data = EVs * Parameters + Error Design Matrix

X in context = + x1x1 β1β2β3β1β2β3 ε y x2x2 x3x3 Observed data = EVs * Parameters + Error

X in context = + x1x1 *β1*β1 ε y x2x2 x3x3 + *β2*β2 + *β3*β3 A linear combination of the Explanatory Variables (EVs) y 1 = x 11 *β 1 + x 12 *β 2 + x 13 *β 3 + ε 1

X: The Design Matrix time x1x1 -- On Off Off On Conditions Use ‘dummy codes’ to label different levels of an experimental factor (eg. On = 1, Off = 0). β is effect size. y = Xβ + ε

X: The Design Matrix Covariates Parametric variation of a single variable (eg. Task difficulty = 1-6) or measured values of a variable (eg. Movement). β is regression slope. x3x3 x1x1 Parametric and factorial predictors in the same model! y = Xβ + ε x2x2

X: The Design Matrix Constant Variable eg. Always = 1 Models the baseline activity y = Xβ + ε x2x2 x1x1

X: The Design Matrix The design matrix should include everything that might explain the data. Subjects Global activity or movement Conditions: Effects of interest More complete models make for lower residual error, better stats, and better estimates of the effects of interest.

Summary y = Xβ + ε So far… If you like these slides … Past MfD presentations (esp. Elliot Freeman, 2005); past FIL SPM Short Course presentations (esp. Klaas Enno Stephan, 2007); Human Brain Function v2

General Linear Model: Part 2 Marsha Quallo

Content Parameters Error Parameter Estimation Hemodynamic Response Function T-Tests and F-Tests

Parameters y= Xβ + ε β: defines the contribution of each component of the design matrix to the value of y The best estimate of β will minimise ε

Parameter Estimation ≈ β1∙β1∙+ β 2 ∙+ β 3 ∙ Listening Reading Rest Guessing

Parameter Estimation ≈ β1∙β1∙+ β 2 ∙+ β 3 ∙ Listening Reading Rest Guessing

Parameter Estimation ≈ β1∙β1∙+ β 2 ∙+ β 3 ∙ Listening Reading Rest Fit to minimize Mean Square Error (MSE)

General Linear Model Minimization of Mean Square Error (MSE) y= Xβ + ε ε = y- Xβ Differentiating and setting result to zero leads β =(X T X) -1 X t y If X is a square nonsingular matrix the β =X -1 y (Called the pseudoinverse solution) (Not for fMRI)

Hemodynamic response function Original Convolved BRF Original Convolved

T-Test A contrast vector is used to select conditions for comparison C2 [1 -1 0] C1 [1 0 0] cβ cβ ~  Var(cβ) ~ T = C1 x β = (1 x β x β x β 3 ) C2 x β = (1 x β 1 -1 x β x β 3 )

Neuroimaging Data Using the General Linear Model (GLM©Karl) Jesper Andersson KI, Stockholm & BRU, Helsinki Functional MRI: an introduction to methods. Jezzard, P; Matthews, PM; Smith, SM