The General Linear Model

The General Linear Model

Guillaume FlandinWellcome Trust Centre for Neuroimaging

University College London

SPM Short CourseLondon, 21-23 Oct 2010

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 by1. Design matrix X2. Assumptions

about eN: number of scansp: number of regressors

eXy

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

confounds.

),0(~ 2INe

• 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

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 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

Summary Mass 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