General Linear Model and f MRI Rachel Denison

General Linear Model and f. MRI 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 -- -- 6 scans per block time

The General Linear Model y = Xβ + ε β y = X + ε Observed data = Predictors * Parameters + Error eg. Image intensities Also called the design matrix. How much each predictor contributes to the observed data Variance in the data not explained by the model

y: Activity of a single voxel over time y = Xβ + ε y 1 y 2 =y … y. N One voxel at a time: time Mass Univariate BOLD signal

X in context β y = x 1 x 2 x 3 + ε Observed data = Predictors * Parameters + Error

X in context y = x 1 x 2 x 3 β 1 β 2 β 3 + ε Observed data = Predictors * Parameters + Error

X in context *β 1 y = x 1 + x 2 *β 3 + x 3 + ε A linear combination of the predictors y 1 = x 11*β 1 + x 12*β 2 + x 13*β 3 + ε 1

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

X: The Design Matrix y = Xβ + ε x 1 x 2 x 3 Covariates Parametric and factorial predictors in the same model! Parametric variation of a single variable (eg. Task difficulty = 1 -6) or measured values of a variable (eg. Movement). β is regression slope.

X: The Design Matrix y = Xβ + ε x 1 x 2 Constant Variable eg. Always = 1 Models the baseline activity

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

Summary So far… y = Xβ + ε If you like these slides … Past Mf. D presentations (esp. Elliot Freeman, 2005); past FIL SPM Short Course presentations (esp. Klaas Enno Stephan, 2007); Human Brain Function v 2

Thanks!

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 2 34 01 01 0 12 Listening Reading Rest ≈ β 1∙ 0 + β 2∙ 0 + β 3∙ 3

Parameter Estimation 2 34 01 01 0 12 Listening Reading Rest ≈ β 1∙ 1 + β 2∙ 0 + β 3∙ 4

Parameter Estimation 2 34 01 01 0 12 Listening Reading Rest ≈ β 1∙ 0. 83 + β 2∙ 0. 16 + β 3∙ 2. 98

Parameter Estimation To estimate β we need to find the least square fit for the line • β�= XTY(XTX)-1 y e e e x 2 x 1 x 3 • If X has linearly dependant columns the model will be over parameterised • Let (XTX)- denote the psuedoinverse of (XTX) then β�= (XTX)-XTY = X-Y

Hemodynamic response function Original Convolved HRF Original Convolved

T-Tests and F-Tests c [1 0 0] c [1 -1 0] • A contrast vector is used to select conditions for comparison ~ T= cβ ~ Var(cβ) • What about c [1 1 0] • A contrast matrix is used to make a simultaneous test of multiple contrasts c= ( ) F= 100 010 ~ cβ = (1 B 1 + 0 B 2 + 0 B 3) ~ ~ βc(Var[cβ])-1 cβ K

• http: //www. fil. ion. ucl. ac. uk/spm/course/slides 06/ppt/glm. ppt#374, 1, Modelling Neuroimaging Data Using the General Linear Model (GLM©Karl) Jesper Andersson KI, Stockholm & BRU, Helsinki • http: //www. fil. ion. ucl. ac. uk/spm/doc/mfd-2005/GLM_f. MRI. ppt • http: //www. fil. ion. ucl. ac. uk/spm/doc/books/hbf 2/ • Functional MRI: an introduction to methods. Jezzard, P; Matthews, PM; Smith, SM