General Linear Model&

Classical InferenceGuillaume Flandin

Wellcome Trust Centre for NeuroimagingUniversity College London

SPM M/EEGCourseLondon, May 2013

Pre-processing

s

GeneralLinearModel

Statistical Inference

𝑦=𝑋 𝛽+𝜀

�� 2=��𝑇 ��

𝑟𝑎𝑛𝑘(𝑋 )

Contrast cRandom

Field Theory

𝑆𝑃𝑀 {𝑇 ,𝐹 }

��= (𝑋𝑇 𝑋 )−1 𝑋𝑇 𝑦

Statistical Parametric Maps

mm

mm

mm

mm

time

mm

time

frequ

ency

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

vectorDesign

matrix Parameter

vectorErro

r

vector

b2

b1

General Linear Model

=

b

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 b

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 b

Ordinary least squares estimation

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

yXXX TT 1)(ˆ b

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

bTc

?21 bb ˆˆ

0?11 21 bb ˆˆ

0?bTcTest H0:

pNTT

T

tcXXc

cT ~

ˆ

ˆ12

b

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: 0bTc vs HA: 0bTc

Alternative hypothesis:

Model comparison: Full vs. Reduced model?

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

RSSRSSRSSF

0

21 ,~ FRSSESSF

Test statistic: ratio of explained and unexplained

variability (error)

1 = rank(X) – rank(X0)2 = N – rank(X)

RSS 2ˆ full

RSS0

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 X0

Full or reduced model?

X1 (b3-4) X0 X00 0 1 0 0 0 0 1 cT =

H0: b3 = b4 = 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.

0000010000100001

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 bbbH0 oneleast at : Hypothesis eAlternativ kAH b

Variability described by Variability described by

Orthogonal regressors

Variability in YTesting for Testing for

Correlated regressorsVa

riabi

lity

desc

ribed

by

Variability described by

Shared variance

Variability in Y

Correlated regressors

Varia

bilit

y de

scrib

ed b

y Variability described by

Variability in Y

Testing for

Correlated regressors

Varia

bilit

y de

scrib

ed b

y Variability described by

Variability in Y

Testing for

Correlated regressors

Varia

bilit

y de

scrib

ed b

y Variability described by

Variability in Y

Correlated regressors

Varia

bilit

y de

scrib

ed b

y Variability described by

Variability in Y

Testing for

Correlated regressors

Varia

bilit

y de

scrib

ed b

y Variability described by

Variability in Y

Testing for

Correlated regressors

Varia

bilit

y de

scrib

ed b

y 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

b

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 yb

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

b

b

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