The General Linear Model

Guillaume FlandinWellcome Trust Centre for Neuroimaging

University College London

SPM fMRI CourseLondon, October 2012

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

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)• Analysis of Covariance

(ANCoVA)• correlation• linear regression• multiple regression

GLM: a flexible framework for parametric analyses

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

Problems of this model with fMRI time series

1. The BOLD response has a delayed and dispersed shape.

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.

Boynton et al, NeuroImage, 2012.

Scaling

Additivity

Shiftinvariance

Problem 1: BOLD responseHemodynamic response function (HRF):

Linear time-invariant (LTI) system:

u(t) x(t)hrf(t)

𝑥 (𝑡 )=𝑢 (𝑡 )∗h𝑟𝑓 (𝑡 )

¿∫𝑜

𝑡

𝑢 (𝜏 ) h𝑟𝑓 (𝑡−𝜏 )𝑑𝜏

Convolution operator:

Problem 1: BOLD responseSolution: Convolution model

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

)(eCovautocovariance

function

N

N

Problem 3: Serial correlations

𝑒 𝑁 (0 ,𝜎2 𝐼 )i.i.d:

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 Q and hyperparameters

jj

ii

QV

VC

2

),0(~ ii CNe

1 1 1ˆ ( )T TX V X X V y

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

A mass-univariate approach

Time

Summary

Estimation of the parameters

= +𝜀𝛽

𝜀 𝑁 (0 ,𝜎 2𝑉 )

��=(𝑋𝑇𝑉 −1 𝑋)−1 𝑋𝑇𝑉 −1 𝑦

noise assumptions:

WLS:

��1=3.9831

��2−7={0.6871 ,1.9598 ,1.3902 , 166.1007 , 76.4770 ,− 64.8189 }

��8=131.0040

=

�� 2=��𝑇 ��𝑁−𝑝�� 𝑁 (𝛽 ,𝜎2(𝑋𝑇 𝑋 )−1 )

Summary

References

Statistical parametric maps in functional imaging: a general linear approach, K.J. Friston et al, Human Brain Mapping, 1995.

Analysis of fMRI time-series revisited – again, K.J. Worsley and K.J. Friston, NeuroImage, 1995.

The general linear model and fMRI: Does love last forever?, J.-B. Poline and M. Brett, NeuroImage, 2012.

Linear systems analysis of the fMRI signal, G.M. Boynton et al, NeuroImage, 2012.

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

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

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

withttt aee 1 ),0(~ 2 Nt

1st order autoregressive process: AR(1)

)(eCovautocovariance

function

N

N

Problem 3: Serial correlations

discrete cosine transform (DCT) set

High pass filtering

∫ t

dtgftgf0

)()()(

Problem 1: BOLD responseSolution: Convolution model

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

=

Impulses HRF Expected BOLD