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The General Linear Model (GLM) in SPM5

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Klaas Enno Stephan

Wellcome Trust Centre for Neuroimaging

University College London

With many thanks to my colleagues in the FIL methods group, particularly Stefan Kiebel, for useful slides

SPM Short Course, May 2007Wellcome Trust Centre for Neuroimaging

Overview of SPM

Statistical parametric map (SPM)

Design matrix

Image time-series

Kernel

Realignment

Smoothing

General linear model

Gaussian

field theory

Statistical

inference

Normalisation

p <0.05

Template

Parameter estimates

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?

Modelling the measured data

Why?

Make inferences about effects of interest

- 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

model

specification

parameter

estimation

hypothesis

statistic

Voxel-wise time series analysis

Time

Time

BOLD signal

single voxel

time series

SPM

Single voxel regression model

error

=

+

+

1

2

Time

e

x1

x2

BOLD signal

Mass-univariate analysis: voxel-wise GLM

=

+

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.

Examples for non-sphericity:

sphericity = iid:error covariance is scalar multiple of identity matrix:

Cov(e) = 2I

non-identity

non-identity

non-independence

Ordinary least squares (OLS):

Estimate parameters such that

minimal

Parameter estimation: Ordinary least squares

=

+

X

y

OLS parameter estimate

(assuming iid error)

A geometric perspective

y

e

Design space defined by X

x2

x1

y

x2

x2*

x1

Correlated regressors =

explained variance is shared between regressors

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

HRF

- BOLD responses have a delayed and dispersed form.

- The BOLD signal includes substantial amounts of low-frequency noise.
- The data are serially correlated (temporally autocorrelated) this violates the assumptions of the noise model in the GLM

hemodynamic response function (HRF)

Problem 1: Shape of BOLD responseSolution: Convolution model

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).

expected BOLD response

= input function impulse response function (HRF)

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

HRF

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

S = residual forming matrix of DCT set

discrete cosine transform (DCT) set

blue= data

black = mean + low-frequency drift

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

red= predicted response (with low-frequency drift explained away)

High pass filtering: example

Problem 3: Serial correlations

with

1st order autoregressive process: AR(1)

autocovariance

function

- Pre-colouring: impose some known autocorrelation structure on the data (filtering with matrix W) and use Satterthwaite correction for df’s.
- Pre-whitening: 1. Use an enhanced noise model with hyperparameters for multiple error covariance components. 2. Use estimated autocorrelation to specify matrix W and use weighted least squares (WLS) estimation.

enhanced noise model

Q2

Q1

V

1

+ 2

=

Estimation of hyperparameters with ReML (restricted maximum likelihood).

- Assumption: error covariance pattern is identical within voxels (of same tissue class), variance is not voxel-specific estimates of , but single estimate of for all voxels
- Estimation of in SPM proceeds by pooling over all voxels that survive an F-test at p<0.001 based on OLS estimates (“first pass” during estimation in SPM)

Three hyperparameters: 1, 2,

Contrasts &statistical parametric maps

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

Q: activation during listening ?

Null hypothesis:

t-statistic based on ML estimates

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

ReML-estimate

Physiological confounds

- head movements
- arterial pulsations
- breathing
- eye blinks
- adaptation affects, fatigue, fluctuations in concentration, etc.

- 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

Summary

- 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
- Estimation of multiple variance components (e.g. to account for serial correlations)
- Parametric statistics

- 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(Will’s talk)

- HRF varies substantially across voxels and subjects
- For example, latency can differ by ± 1 second
- Solution: use multiple basis functions
- See talk on er-fMRI by Christian