The General Linear Model

The General Linear Model

Christophe PhillipsCyclotron Research CentreUniversity of Liège, Belgium

SPM Short CourseLondon, May 2011

Normalisation

Statistical Parametric MapImage time-series

Parameter estimates

General Linear ModelRealignment Smoothing

Design matrix

Anatomicalreference

Spatial filter

StatisticalInference

RFT

p <0.05

Passive word listeningversus rest

7 cycles of rest and listening

Blocks of 6 scanswith 7 sec TR

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

Stimulus function

One session

A very simple fMRI experiment

stimulus function

1. Decompose data into effects and error

2. Form statistic using estimates of effects and error

Make inferences about effects of interest

Why?

How?

datastatisti

c

Modelling the measured data

linearmode

l

effects estimate

error estimate

Time

BOLD signal

Time

single voxeltime series

Voxel-wise time series analysis

ModelspecificationParameterestimationHypothesis

Statistic

SPM

BOLD signal

Time =1 2+ +

error

x1 x2 e

Single voxel regression model

exxy 2211

Mass-univariate analysis: voxel-wise GLM

=

e+y

X

N

1

N N

1 1p

pModel is specified by

both1. Design matrix X2. Assumptions about

e

eXy

The design matrix embodies all available knowledge about experimentally controlled factors and

potential confounds.

),0(~ 2INe

N: number of scansp: number of regressors

• 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: mass-univariate parametric analysis

Parameter estimation

eXy

= +

e

2

1

Ordinary least squares estimation

(OLS) (assuming i.i.d. error):

yXXX TT 1)(ˆ

Objective:estimate parameters to minimize

N

tte

1

2

y X

A geometric perspective on the GLM

ye

Design space defined by X

x1

x2 ˆ Xy

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

Ordinary Least Squares (OLS)

0eX T

XXyX TT

0)ˆ( XyX T

yXXX TT 1)(ˆ

x1

x2x2*

y

Correlated and orthogonal regressors

When x2 is orthogonalized w.r.t. x1, only the parameter

estimate for x1 changes, not that for x2!

Correlated regressors = explained variance is

shared between regressors

121

2211

exxy

1;1 *21

*2

*211

exxy

eXy = +

e

2

1

y X

),0(~ 2INe

What are the problems of this model?

What are the problems of this model?

1. BOLD responses have a delayed and dispersed form.

HRF

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

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

t

dtgftgf0

)()()(

Problem 1: Shape of BOLD responseSolution: Convolution model

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

=

Impulses HRF Expected BOLD

Convolution model of the BOLD responseConvolve stimulus function with a canonical hemodynamic response function (HRF):

HRF

t

dtgftgf0

)()()(

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 noise Solution: High pass filtering

discrete cosine transform (DCT) set

discrete cosine transform (DCT) set

High pass filtering

withttt aee 1 ),0(~ 2 Nt

1st order autoregressive process: AR(1)

)(eCovautocovariance

function

N

N

Problem 3: Serial correlations

Multiple covariance components

= 1 + 2

Q1 Q2

Estimation of hyperparameters with ReML (Restricted Maximum Likelihood).

V

enhanced noise model at voxel i

error covariance components Qand hyperparameters

jj

ii

QV

VC

2

),0(~ ii CNe

1 1 1ˆ ( )T TX V X X V y

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

Let 1TW W V

1

1

ˆ ( )ˆ ( )

T T T T

T Ts s s s

X W WX X W Wy

X X X y

Then

where

,s sX WX y Wy

WLS equivalent toOLS on whiteneddata and design

Contrasts &statistical parametric

maps

Q: activation during listening ?

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

Null hypothesis: 01

)ˆ(

ˆ

T

T

cStdct

X

SummaryMass univariate approach.

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

GLM is a very general approach

Hemodynamic Response Function

High pass filtering

Temporal autocorrelation

Thank you for your attention!

…and thanks to Guillaume for his slides .