The general linear model
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The General Linear Model. Christophe Phillips. SPM Short Course London, May 2013. Image time-series. Statistical Parametric Map. Design matrix. Spatial filter. Realignment. Smoothing. General Linear Model. Statistical Inference. RFT. Normalisation. p <0.05. Anatomical reference.

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The General Linear Model

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The general linear model

The General Linear Model

Christophe Phillips

SPM Short Course

London, May 2013


The general linear model

Image time-series

Statistical Parametric Map

Design matrix

Spatial filter

Realignment

Smoothing

General Linear Model

StatisticalInference

RFT

Normalisation

p <0.05

Anatomicalreference

Parameter estimates


The general linear model

A very simple fMRI experiment

One session

Passive word listening

versus rest

7 cycles of

rest and listening

Blocks of 6 scans

with 7 sec TR

Stimulus function

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


The general linear model

Modelling the measured data

Make inferences about effects of interest

Why?

  • Decompose data into effects and error

  • Form statistic using estimates of effects and error

How?

stimulus function

effects estimate

linear

model

statistic

data

error estimate


The general linear model

Model

specification

Parameter

estimation

Hypothesis

Statistic

Voxel-wise time series analysis

Time

Time

BOLD signal

single voxel

time series

SPM


The general linear model

Single voxel regression model

error

=

+

+

1

2

Time

e

x1

x2

BOLD signal


The general linear model

Mass-univariate analysis: voxel-wise GLM

X

+

y

=

  • Model is specified by

  • Design matrix X

  • Assumptions about e

N: number of scans

p: number of regressors

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


The general linear model

GLM: mass-univariate parametric analysis

  • 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..


The general linear model

Parameter estimation

Objective:

estimate parameters to minimize

=

+

Ordinary least squares estimation (OLS) (assuming i.i.d. error):

X

y


The general linear model

y

e

x2

x1

Design space defined by X

A geometric perspective on the GLM

Smallest errors (shortest error vector)

when e is orthogonal to X

Ordinary Least Squares (OLS)


The general linear model

HRF

What are the problems of this model?

  • BOLD responses have a delayed and dispersed form.

  • The BOLD signal includes substantial amounts of low-frequency noise (eg due to scanner drift).

  • Due to breathing, heartbeat & unmodeled neuronal activity, the errors are serially correlated. This violates the assumptions of the noise model in the GLM


The general linear model

Problem 1: Shape of BOLD responseSolution: Convolution model

Expected BOLD

HRF

Impulses

=

expected BOLD response

= input function impulse response function (HRF)


The general linear model

Convolution model of the BOLD response

Convolve stimulus function with a canonical hemodynamic response function (HRF):

 HRF


The general linear model

blue= data

black = mean + low-frequency drift

green= predicted response, taking into account low-frequency drift

red= predicted response, NOT taking into account low-frequency drift

Problem 2: Low-frequency noise Solution: High pass filtering

discrete cosine transform (DCT) set


The general linear model

High pass filtering

discrete cosine transform (DCT) set


The general linear model

Problem 3: Serial correlations

with

1st order autoregressive process: AR(1)

autocovariance

function


The general linear model

Multiple covariance components

enhanced noise model at voxel i

error covariance components Q

and hyperparameters

V

Q2

Q1

1

+ 2

=

Estimation of hyperparameters  with ReML (Restricted Maximum Likelihood).


The general linear model

Parameters can then be estimated using Weighted Least Squares (WLS)

Let

Then

WLS equivalent to

OLS on whitened

data and design

where


The general linear model

Contrasts &statistical parametric maps

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

Q: activation during listening ?

Null hypothesis:


Summary

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


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