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

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

= + y = Xβ + ε X β ε y The General Linear Model 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 y: Activity of a single voxel over time Mass Univariate … y = Xβ + ε time BOLD signal y1 y2 yN = y … One voxel at a time: Mass Univariate

X in context β y = x1 x2 x3 + ε Observed data = Predictors * Parameters + Error

= + β1 β2 β3 y x1 x2 x3 ε X in context Observed data = Predictors * Parameters + Error

*β1 *β2 *β3 y x1 x2 x3 = + + + ε y1 = x11*β1 + x12*β2 + x13*β3 + ε1 X in context *β1 *β2 *β3 y x1 x2 x3 = + + + ε A linear combination of the predictors y1 = x11*β1 + x12*β2 + x13*β3 + ε1

label different levels of an experimental factor X: The Design Matrix y = Xβ + ε x1 -- Conditions On Off Off On Use ‘dummy codes’ to label different levels of an experimental factor (eg. On = 1, Off = 0). β is ANOVA effect size. time

(eg. Task difficulty = 1-6) a variable (eg. Movement). X: The Design Matrix y = Xβ + ε x1 x2 x3 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 x1 x2 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.

If you like these slides … Summary So far… y = Xβ + ε 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

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 How much of X is needed to approximate Y,

Parameter Estimation ≈ β1∙ + β2∙ + β3∙ 3 2 3 4 1 1 1 2 Listening 1 1 1 2 Listening Reading Rest ≈ β1∙ + β2∙ + β3∙ 3

Parameter Estimation ≈ β1∙ + β2∙ + β3∙ 1 4 2 3 4 1 1 1 2 Listening 1 1 1 2 Listening Reading Rest ≈ β1∙ + β2∙ + β3∙ 1 4

Parameter Estimation ≈ β1∙ + β2∙ + β3∙ 0.83 0.16 2.98 2 3 4 1 1 1 2 1 1 1 2 Listening Reading Rest ≈ β1∙ + β2∙ + β3∙ 0.83 0.16 2.98

Parameter Estimation β = XTY(XTX)-1 y e e e x2 x3 x1 To estimate β we need to find the least square fit for the line β = XTY(XTX)-1 y e e e x2 If X has linearly dependant columns then it is rank deficient and has no inverse and the model is over parameterised, there is an infinite amount of parameter sets describing the same model. Consequently there will be an infinate number of least squares estimates that satisfy the normal equations. or inverting (XTX) using a pseudoinverse technique – which is essentially imposing a contraint An example of a constraint is removing a column for the desing matrix, SPM doesn’t use this technique to deal with over determinded models This gives us the least squares estimates for with the mimimum sums of squares In this case the estimates can be found by - imposing constraints - or inverting (XTX) using a pseudoinverse technique x3 x1 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 The glm needs to accommodate for the sluggish response of the brain, to get the best fit of the model the sitmulus function which is usually a sharp on off stimulus needs to be convolved to create a delayed blurred version that mimics the brains activity Original Convolved

T-Tests and F-Tests ( ) cβ A contrast vector is used to select conditions for comparison ~ cβ T = ~ Var(cβ) What about c [1 1 0] A contrast matrix is used to make a simultaneous test of multiple contrasts c = Enabling you to select see if there is more activation in listening condition than rest, or if we added a reading conditing if there was more activation in reading versus rest The contast gives us an estimate of the effect but how do we know if the effect is significant, to get a t value we caclulate the ration of the effect to its standard error, whih is the standard deviation of the variance of the effect. The first row compares listening to rest and the second reading to rest ( ) 1 0 0 0 1 0 cβ = (1B1 + 0B2 + 0B3) ~ ~ ~ βc(Var[cβ])-1cβ F = K

http://www. fil. ion. ucl. ac. uk/spm/course/slides06/ppt/glm http://www.fil.ion.ucl.ac.uk/spm/course/slides06/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_fMRI.ppt http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/ Functional MRI: an introduction to methods. Jezzard, P; Matthews, PM; Smith, SM