General Linear Model&

Classical InferenceGuillaume Flandin

Wellcome Trust Centre for Neuroimaging

University College London

SPM M/EEGCourse

London, May 2013

Pre-processings

GeneralLinearModel

Statistical Inference

𝑦=𝑋 𝛽+𝜀

�� 2= ��𝑇 ��𝑟𝑎𝑛𝑘(𝑋 )

Contrast cRandom

Field Theory

𝑆𝑃𝑀 {𝑇 ,𝐹 }

��= (𝑋𝑇 𝑋 )−1𝑋𝑇 𝑦

Statistical Parametric Maps

mm

mm

mm

mm

time

mm

time

freq

uenc

y

3D M/EEG sourcereconstruction,

fMRI, VBM

2D time-frequency

2D+t scalp-time

1D time

time

ERP example

Random presentation of ‘faces’ and ‘scrambled faces’

70 trials of each type 128 EEG channels

Question:is there a difference between the ERP of

‘faces’ and ‘scrambled faces’?

ERP example: channel B9

compares size of effect to its error standard deviation

sf

2

sf

11nn

t

Focus on N170

Data modelling

= b2b1 + +

Error

Faces ScrambledData

= b2b1 + +X2X1Y ••

Design matrix

= +

= b +XY •

Data

vector

Design

matri

xPara

mete

r

vector

Error

vector

b2

b1

General Linear Model

=

e+yy XX

N

1

N N

1 1p

p

Model is specified by1. Design matrix X2. Assumptions

about e

Model is specified by1. Design matrix X2. Assumptions

about e

N: number of scansp: number of regressors

N: 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

Ordinary least squares estimation

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

Ordinary least squares estimation

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

yXXX TT 1)(ˆ

Objective:estimate parameters to minimize

N

tte

1

2

= +b2

b1

eXY

Mass-univariate analysis: voxel-wise GLMEvoked response image

Tim

e

1. Transform data for all subjects and conditions2. SPM: Analyse data at each voxel

Sensor to voxeltransform

Hypothesis Testing

Null Hypothesis H0

Typically what we want to disprove (no effect).

The Alternative Hypothesis HA expresses outcome of interest.

To test an hypothesis, we construct “test statistics”.

Test Statistic T

The test statistic summarises evidence about H0.

Typically, test statistic is small in magnitude when the hypothesis H0 is true and large when false.

We need to know the distribution of T under the null hypothesis. Null Distribution of T

Hypothesis Testing

p-value:

A p-value summarises evidence against H0.

This is the chance of observing value more extreme than t under the null hypothesis.

Null Distribution of T

Significance level α:

Acceptable false positive rate α.

threshold uα

Threshold uα controls the false positive rate

t

p-value

Null Distribution of T

u

Conclusion about the hypothesis:

We reject the null hypothesis in favour of the alternative hypothesis if t > uα

)|( 0HuTp

𝑝 (𝑇>𝑡∨𝐻0 )

Contrast : specifies linear combination of parameter vector:

ERP: faces < scrambled ?=

t =

contrast ofestimated

parameters

varianceestimate

Contrast & t-test

cT = -1 +1 SPM-t over time & space

Tc

?21 ˆˆ

0?11 21 ˆˆ

0?TcTest H0:

pNTT

T

tcXXc

cT

~ˆ

ˆ

12

T-test: summary

T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate).

T-contrasts are simple combinations of the betas; the T-statistic does not depend on the scaling of the regressors or the scaling of the contrast.

H0: 0Tc vs HA: 0Tc

Alternative hypothesis:

Model comparison: Full vs. Reduced model?

Null Hypothesis H0: True model is X0 (reduced model)Null Hypothesis H0: True model is X0 (reduced model)

RSS

RSSRSSF

0

21 ,~ FRSS

ESSF

Test statistic: ratio of explained and unexplained

variability (error)

1 = rank(X) – rank(X0)2 = N – rank(X)

RSS

2ˆ fullRSS0

2ˆreduced

Full model ?

X1 X0

Or reduced model?

X0

Extra-sum-of-squares & F-test

F-test & multidimensional contrastsTests multiple linear hypotheses:

H0: True model is X0H0: True model is X0

Full or reduced model?

X1 (b3-4) X0 X0

0 0 1 0 0 0 0 1

cT =

H0: b3 = b4 = 0H0: b3 = b4 = 0 test H0 : cTb = 0 ?test H0 : cTb = 0 ?

F-test: summary

F-tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler (nested) model model comparison.

0000

0100

0010

0001

In testing uni-dimensional contrast with an F-test, for example b1 – b2, the result will be the same as testing b2 – b1. It will be exactly the square of the t-test, testing for both positive and negative effects.

F tests a weighted sum of squares of one or several combinations of the regression coefficients b.

In practice, we don’t have to explicitly separate X into [X1X2] thanks to multidimensional contrasts.

Hypotheses:

0 : Hypothesis Null 3210 H

0 oneleast at : Hypothesis eAlternativ kAH

Variability described by Variability described by

Orthogonal regressors

Variability in YTesting for Testing for

Correlated regressorsV

aria

bilit

y de

scrib

ed b

y V

ariability described by

Shared variance

Variability in Y

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for and/or

Summary Mass-univariate GLM:

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

– GLM is a very general approach(one-sample, two-sample, paired t-tests, ANCOVAs, …)

Hypothesis testing framework

– Contrasts

– t-tests

– F-tests

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

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