The general linear model for fMRI

Methods and Models in fMRI, 17.10.2017

Jakob Heinzleheinzle@biomed.ee.ethz.ch

Translational Neuromodeling Unit (TNU) Institute for Biomedical Engineering (IBT)University and ETH Zürich

Many thanks toK. E. Stephan andF. Petzschner for material

Translational Neuromodeling Unit

Overview of SPM

GLM for fMRI 2

Realignment Smoothing

Normalisation

General linear model

Statistical parametric map (SPM)Image time-series

Parameter estimates

Design matrix

Template

Kernel

Gaussian field theory

p <0.05

Statisticalinference

What is the problem we want to solve?

• We have an experimental paradigm andwant to test whether brain activity is(linearly) related to the paradigm.

• We will try to solve the problem bymodeling the data.

GLM for fMRI 3

Modelling the measured data

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stimulus function

1. Decompose data into effects anderror

2. Form statistic using estimates ofeffects and error

Make inferences about effects of interestWhy?

How?

datalinearmodel

effects estimate

error estimate

statistic

A very simple experiment

GLM for fMRI 5

time

• One session• 7 cycles of rest and listening• Blocks of 6 scans with 7 sec

TR

What is the brain‘s responseto such a stimulation?

How is brain data related to the input?

GLM for fMRI 6

time

single voxel time series

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

What we know.

What we measure.

A linear model of the data

GLM for fMRI 7

BOLD signal

Time =1 2+ + er

ror

e

y x11 x22 ey x11 x22 e

Explain your data… as a combination of experimental manipulation,confounds and errors

Single voxel regression model:regressors

x1 x2

Writing everything in matrix notation

GLM for fMRI 8

BOLD signal

Time = + erro

r

x1 x2 e

1

2

eXy Single voxel regression model:

The way it looks in SPM

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n

= + + erro

r

1

2

1

np

p1

n

1

desi

gn m

atrix

erro

r

eXy n: number of scansp: number of regressors

data

y

We need …

• … to specify the design matrix.• … specify a noise model, e.g. • … and then, estimate the parameters b

that minimize the error– Minimization of the error depends on

assumptions about the noise.

GLM for fMRI 10

et2

t1

N

),0(~ 2INe ),0(~ 2INe

Summary: Mass-univariate GLM

GLM for fMRI 11

=

e+y X

N

1

N

1 1p

p

Model is specified by1. Design matrix X2. Assumptions about e

N: number of scansp: number of regressors

eXy eXy

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

),0(~ 2INe ),0(~ 2INe

N

How to fit the model parameters.

GLM for fMRI 12

= +

e

2

1

y X

erro

r

y Xe y y

e y X

min(eTe) min((y X)T (y X ))

y Xe y y

e y X

min(eTe) min((y X)T (y X ))

Data predicted by our model

e = error between predicted and actual data

Goal is to determine the betas that minimize the quadratic error

OLS (Ordinary Least Squares)

OLS – Ordinary least squares

GLM for fMRI 13

eTe (y X )T (y X )

eTe (yT T XT )(y X )

eTe yT y yT X T XT y T XT X

eTe yT y 2T XT y T XT XeTe

2XT y 2XT X

0 2XT y 2XT X

(XT X)1 XT y

We want to minimize the quadratic error between data and model

OLS – Ordinary least squares

GLM for fMRIv 14

yXXX

XXyX

XXyXeeXXyXyyee

XXyXXyyyee

XyXyee

XyXyee

TT

TT

TTT

TTTTTT

TTTTTTT

TTTT

TT

1)(ˆ

ˆ220

ˆ22ˆ

ˆˆˆ2

ˆˆˆˆ)ˆ)(ˆ(

)ˆ()ˆ(

OLS estimate for

Summary: OLS solution

GLM for fMRI 15

eXy

= +

e

2

1

Ordinary least squaresestimation (OLS)

(assuming i.i.d. error):

yXXX TT 1)(ˆ

Objective:estimate parametersto minimize

N

tte

1

2

y X

Geometric perspective

GLM for fMRI 16

ye

Design space defined by X

x1

x2 ˆ Xy

yXXX TT 1)(ˆ yXXX TT 1)(ˆ OLS estimates

PIRRye

PIR

Rye

TT XXXXPPyy

1)(

ˆ

TT XXXXP

Pyy1)(

ˆ

Residual forming matrix R

Projection matrix P

Correlated and orthogonalized regressors

GLM for fMRI 17

x1

x2x2*

y

When x2 is orthogonalized with regard to x1, only the parameter estimate for x1 changes, not that for x2!

Correlated regressors = explained variance is shared between regressors

121

2211

exxy

121

2211

exxy

1;1 *21

*2

*211

exxy

1;1 *21

*2

*211

exxy

Design space defined by X

We are nearly there …

GLM for fMRI 18

linear model

effectsestimate

errorestimate

statistic

Problems of this model

GLM for fMRI 19

1. BOLD responses have a delayed and dispersed form (cf. Lecture 1).

HRF

2. The BOLD signal includes substantial amounts of low-frequency noise.

3. The data are serially correlated (temporally autocorrelated) this violates the assumptions of the noise model in the GLM

Summary: Mass-univariate GLM

GLM for fMRI 20

=

e+y X

N

1

N N

1 1p

p

Model is specified by1. Design matrix X2. Assumptions about e

N: number of scansp: number of regressors

eXy eXy

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

),0(~ 2INe ),0(~ 2INe

Problem 1: The BOLD response

GLM for fMRI 21

t

dtgftgf0

)()()(

The response of a linear time-invariant (LTI) system is the convolution of the input with the system's response to an impulse (delta function).

Basic math: What is a convolution?

GLM for fMRI 22

t

dtgftgf0

)()()(

Solution: Convolution with the HRF

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

HRF

t

dtgftgf0

)()()(

blue = datagreen = predicted response, taking convolved with HRFred = predicted response, NOT taking into account the HRF

Problem 2: Low frequency noise

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blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account

low-frequency driftred = predicted response, NOT taking into

account low-frequency drift

Solution 2: High-pass filtering

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discrete cosine transform (DCT) set

Solution 2: High-pass filtering

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blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account

low-frequency driftred = predicted response, NOT taking into

account low-frequency drift

Linear model

Problem 3: Serial correlations

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sphericity = i.i.d.error covariance is a scalar multiple of the

identity matrix:Cov(e) = 2I

1001

)(eCov

1004

)(eCov

2112

)(eCov

Examples for non-sphericity:

non-identity

non-independence

Problem 3: Serial correlations

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Cov(e)

n: number of scans

n

n

autocovariancefunction

withttt aee 1 ),0(~ 2 Nt

1st order autoregressive process: AR(1)

Solution 3: Pre-whitening

GLM for fMRI 29

• Pre-whitening:

1. Use an enhanced noise model with multiple error covariance components, i.e. e ~ N(0,2V) instead of e ~ N(0,2I).

2. Use estimated serial correlation to specify filter matrix W for whitening the data.

WeWXWy WeWXWy

This is i.i.d

How to define W?

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• Enhanced noise model

• Remember linear transform for Gaussians

• Choose W such that error covariance becomes spherical

• Conclusion: W is a simple function of V so how do we estimate V ?

WeWXWy WeWXWy

),0(~ 2VNe ),0(~ 2VNe

),(~),,(~

22

2

aaNyaxyNx

),(~),,(~

22

2

aaNyaxyNx

2/1

2

22 ),0(~

VWIVW

VWNWe

2/1

2

22 ),0(~

VWIVW

VWNWe

Find W – multiple covariance components.

GLM for fMRI 31

),0(~ 2VNe ),0(~ 2VNe

iiQVeCovV

)(

iiQVeCovV

)(

= 1 + 2

Q1 Q2

Estimation of hyperparameters with EM (expectation maximisation) or ReML (restricted maximum likelihood). For more details see (Friston et al, Neuroimage, 16:465; 2002)

V

enhanced noise model error covariance components Qand hyperparameters

linear model

effectsestimate

errorestimate

statistic

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

Null hypothesis: 01

)ˆ(

ˆ

T

T

cStdct

)ˆ(

ˆ

T

T

cStdct

Lecture: Classical (frequentist) inference

GLM for fMRI 32

Outlook: Contrasts and statistical maps

GLM for fMRI 33

Q: activation during listening ?

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

Null hypothesis: 01

)ˆ(

ˆ

T

T

cStdct

)ˆ(

ˆ

T

T

cStdct

X

Summary of GLM

GLM for fMRI 34

WeWXWy

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

)ˆ(ˆˆ

T

T

cdtsct

cWXWXc

cdtsTT

T

)()(ˆ

)ˆ(ˆ

2

)(

ˆˆ

2

2

RtrWXWy

ReML-estimates

WyWX )(

)(2

2/1

eCovVVW

)(WXWXIRX

iiQV

TT XWXXWX 1)()( TT XWXXWX 1)()(

For brevity:

Physiological confounds

GLM for fMRI 35

• head movements

• arterial pulsations (particularly bad in brain stem)

• breathing

• eye blinks (visual cortex)

• adaptation effects, fatigue, fluctuations in concentration, etc.

Lecture: Noise models in fMRI and noise correction

Outlook – further challenges

GLM for fMRI 36

• correction for multiple comparisons

• variability in the HRF across voxels

• slice timing

• limitations of frequentist statistics Bayesian analyses

• GLM ignores interactions among voxels models of effective connectivity

These issues are discussed in future lectures.

Correction for multiple comparison

GLM for fMRI 37

• Mass-univariate approach: We apply the GLM to each of a huge number of voxels (usually > 100,000).

• Threshold of p<0.05 more than 5000 voxels significant by chance!

• Massive problem with multiple comparisons!

• Solution: Gaussian random field theory

Lecture: Multiple comparison correction

Variability in the BOLD response

GLM for fMRI 38

• HRF varies substantially across voxels and subjects

• For example, latency can differ by ± 1 second

• Solution: use multiple basis functions

• See talk on event-related fMRI

Summary

GLM for fMRI 39

• Mass-univariate approach: same GLM for each voxel

• GLM includes all known experimental effects and confounds

• Convolution with a canonical HRF

• High-pass filtering to account for low-frequency drifts, implemented by a set of cosine functions.

• Estimation of multiple variance components (e.g. to account for serial correlations)

Bibliography

GLM for fMRI 40

Friston, Ashburner, Kiebel, Nichols, Penny (2007) Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier.

• Christensen R (1996) Plane Answers to Complex Questions: The Theory of Linear Models. Springer.

• Friston KJ et al. (1995) Statistical parametric maps in functional imaging: a general linear approach. Human Brain Mapping 2: 189-210.

Supplementary slides

1. Express each function in terms of a dummy variable τ.

2. Reflect one of the functions: g(τ)→g( − τ).

3. Add a time-offset, t, which allows g(t − τ) to slide along the τ-axis.

4.Start t at -∞ and slide it all the way to +∞. Wherever the two functions intersect, find the integral of their product. In other words, compute a sliding, weighted-average of function f(τ), where the weighting function is g( − τ).

The resulting waveform (not shown here) is the convolution of functions f and g. If f(t) is a unit impulse, the result of this process is simply g(t), which is therefore called the impulse response.

Convolution step-by-step (from Wikipedia):