General Linear Model

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# General Linear Model - PowerPoint PPT Presentation

General Linear Model. L ύ cia Garrido and Marieke Sch ö lvinck ICN. Observed data. Time. Intensity. Preprocessing . Y. Y is a matrix of BOLD signals: Each column represents a single voxel sampled at successive time points. Univariate analysis. GLM in two steps :

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General Linear Model

Lύcia Garrido and Marieke Schölvinck ICN

Observed data

Time

Intensity

Preprocessing ...

Y

• Y is a matrix of BOLD signals:
• Each column represents a single voxel sampled at successive time points.
Univariate analysis

GLM in two steps:

• Does an analysis of variance separately at each voxel (univariate)
• Makes t statistic from the results of this analysis, for each voxel
Example

Y X

X can contain values quantifying experimental variable

Parameters & error

Y = βx + c + ε

• β: slope of line relating x to y
• ‘how much of x is needed to approximate y?’
• ε = residual error
• the best estimate of β minimises ε: deviations from line
• Assumed to be independently, identically and normally distributed

this line is a 'model' of the data

slope β = 0.23

Interceptc = 54.5

Multiple Regression
• Simple regression
• Multiple regression (more than one predictor/regressor/beta)
• y = β1 * x1 + β2 * x2 + c + ε
Matrix Formulation

Y = X . β + ε

• Write out equation for each observation of variable Y from 1 to J:

Y1 = X11β1 +…+X1lβl +…+ X1LβL + ε1

Yj = Xj1β1 +…+Xjlβl +…+ XjLβL + εj

YJ = XJ1β1 +…+XJlβl +…+ XJLβL + εJ

Can turn these simultaneous equations into matrix form to get a single equation:

β1

βj

βJ

ε1

εj

εJ

Y1

Yj

YJ

X11 … X1l … X1L

Xj1 … X1l… X1L

X11 … X1l… X1L

+

=

Y = X x β + ε

Observed data

Design Matrix

Parameters

Residuals/Error

GLM and fMRI

Y= X. β+ ε

Observed data:

Y is the BOLD signal at various time points at a single voxel

Design matrix:

Several components which explain the observed data, i.e. the BOLD time series for the voxel

Parameters:

Define the contribution of each component of the design matrix to the value of Y

Estimated so as to minimise the error, ε, i.e. least sums of squares

Error:

Difference between the observed data, Y, and that predicted by the model, Xβ.

Design Matrix

x1 x2 c

Matrix represents values of X

Different columns = different predictors

Parameter estimation

e = Y – Ỹ = Y - X β

S = ΣjJej2= eTe = (Y - X β )T(Y - X β )

The least square estimates are the parameter estimates which

minimize the residual sum of squares

• find derivative and solve for ∂S/∂β = 0
• β = (XTX)-1 XTY (if (XTX) is invertible)

Matlab magic: >> B = inv(X) * Y

Statistical inference
• A beta value is estimated for each column in design matrix
• Test if the slope is significantly different from zero (null hypothesis)
• t-statistic = beta / standard error of the slope
• Many betas → contrasts (contents of another talk…)
• t-tests or F-tests depending on nature of question
Continuous predictors

Y X

X can contain values quantifying experimental variable

Binary predictors

Y X

X can contain values distinguishing experimental conditions

Covariates vs. conditions
• Covariates:
• parametric modulation of independent variable
• e.g. task-difficulty 1 to 6
• Conditions:
• 'dummy' codes identify different levels of experimental factor
• e.g. integers 0 or 1: 'off' or 'on'

on off

off on

Ways to improve your model: modelling haemodynamics

HRF basic function

• Brain does not just switch on and off!
• Reshape (convolve) regressors to resemble HRF

Original

HRF Convolved

Ways to improve your model: model everything

globalactivity or movement

• Important to model all known variables, even if not experimentally interesting:
• e.g. head movement, block and subject effects
•  minimise residual error variance for better stats
• effects-of-interest are the regressors you’re actually interested in

conditions:

effects of interest

subjects

Summary
• The General Linear Model allows you to find the parameters, β, which provide the best fit with your data, Y
• The optimal parameters estimates, β, are found by minimising the Sums of Squares differences between your predicted model and the observed data
• The design matrix in SPM contains the information about the factors, X, which may explain the observed data
• Once we have obtained the βs at each voxel we can use these to do various statistical tests
Thanks to…

Previous MfD talks: Elliot Freeman (2005), Davina Bristow and Beatriz Calvo (2004)

http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch7.pdf

http://www.mrc-cbu.cam.ac.uk/Imaging/Common/spmstats.shtml

Summary

Y= X. β+ ε

Observed data:

SPM uses a mass univariate approach – that is each voxel is treated as a separate column vector of data.

Y is the BOLD signal at various time points at a single voxel

Parameters:

Define the contribution of each component of the design matrix to the value of Y

Estimated so as to minimise the error, ε, i.e. least sums of squares

Error:

Difference between the observed data, Y, and that predicted by the model, Xβ.

Not assumed to be spherical in fMRI

Design matrix:

Several components which explain the observed data, i.e. the BOLD time series for the voxel

Timing info: onset vectors, Omj, and duration vectors, Dmj

HRF, hm, describes shape of the expected BOLD response over time

Other regressors, e.g. realignment parameters