02 General Linear Model

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The General Linear Model

(GLM)Ged Ridgway

Wellcome Trust Centre for NeuroimagingUniversity College London

SPM CourseVancouver, August 2010

[slides from the FIL Methods group]

Overview of SPM

Simple regression and the GLM

Passive wordlisteningversus rest

7 cycles of rest and listening

Blocks of 6 scanswith 7 sec TR

Question: Is there a change in the BOLD responsebetween listening and rest?

Stimulus function

One session

A very simple fMRI experiment

stimulusfunction

1. Decompose data into effects anderror

2. Form statistic using estimates of effects and error

Make inferences about effects of interesthy?

data linear model

effectsestimate

error estimate

statistic

Modelling the measured data

BOLD signal

single voxeltime series

Voxel-wise time series analysis

modelspecification

parameter estimation

hypothesis

statistic

BOLD signal

= β1 β2 + e

r r o r

x1 x2 e

Single voxel regression model

Example GLM factorial design models

• one sample t -test• two sample t -test• paired t -test• Analysis of Variance (ANOVA)

• Factorial designs• correlation• linear regression• multiple regression

• F -tests• fMRI time series models• etc…

GLM assumes Gaussian spherical (i.i.d.) errors

sphericity = i.i.d.error covariance isscalar multiple of

identity matrix:Cov(e) = σ 2I

Examples for non-sphericity:

non-identit

non-independence

Parameter estimation

Ordinary least squares estimation (OLS)

(assuming i.i.d. error):

Objective:estimate parametersto minimize

A geometric perspective on the GLM

Design space

defined by X

Smallest errors (shortest error vector)when e is orthogonal to X

Ordinary Least Squares (OLS)

What are the problems of this model?

BOLD responses have a delayedand dispersed form. HRF

2. The BOLD signal includes substantial amounts of low-frequency noise (eg due to scanner drift).

Due to breathing, heartbeat & unmodeled neuronal activity,the errors are serially correlated. This violates theassumptions of the noise model in the GLM

The response of a linear time-invariant (LTI) system is the convolution of the inputwith the system's response to an impulse (delta function).

Problem 1: Shape of BOLD responseSolution: Convolution model

expected BOLD response= input function ⊗ impulse response function (HRF)

Impulses HRF Expected BOLD

ConvolutionSuperposition principle

HRF convolution animations

Sliding the reversed HRF past a unit-

integral pulse and integrating theproduct recovers the HRF (hencethe name impulse response)

The step response shows a delay and

a slight overshoot, before reaching asteady-state. This gives some intuitionfor a square wave…

HRF convolution animations

Slow square wave. Main (fundamental)

frequency of output matches input, butwith delay (phase-shift) and change inamplitude (and shape; sinusoids keeptheir shape)

Fast square wave. Amplitude is

reduced, phase shift is more dramatic(output almost in anti-phase).Fourier transform of HRF yieldsmagnitude and phase responses.

P bl 2 L f i

blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account

low-frequency driftred = predicted response, NOT taking into

account low-frequency drift

Problem 2: Low-frequency noiseSolution: High pass filtering

discrete cosine

transform (DCT) set

discrete cosinetransform (DCT) set

High pass filtering

1 st order autoregressive process: AR(1)

autocovariancefunction

Problem 3: Serial correlations

Multiple covariance components

= λ 1 + λ 2

Q 1 Q 2

Estimation of hyperparameters λ with ReML (Restricted Maximum Likelihood).

enhanced noise model at voxel i error covariance components Qand hyperparameters λ

Parameters can then be estimated using Weighted Least Squares (WLS)

WLS equivalent to

OLS on whiteneddata and design

Contrasts &statistical parametric maps

Q: activation duringlistening ?

c = 1 0 0 0 0 0 0 0 0 0 0

Null hypothesis:

Summary

• Mass univariate approach.

• Fit GLMs with design matrix, X, to data at different points inspace to estimate local effect sizes,

• GLM is a very general approach

• Hemodynamic Response Function

• High pass filtering

• Temporal autocorrelation

Correlated and orthogonal regressors

When x 2 is orthogonalized withregard to x 1, only the parameter estimate for x 1 changes, not thatfor x 2!

Correlated regressors =explained variance is sharedbetween regressors

t t ti ti b d ML ti t

c = 1 0 0 0 0 0 0 0 0 0 0

estimates

t-statistic based on ML estimates

For brevity:

02 General Linear Model

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