the general linear model
DESCRIPTION
The General Linear Model. Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London. SPM Short Course London, 21-23 Oct 2010. Image time-series. Statistical Parametric Map. Design matrix. Spatial filter. Realignment. Smoothing. General Linear Model. - PowerPoint PPT PresentationTRANSCRIPT
The General Linear Model
Guillaume FlandinWellcome Trust Centre for Neuroimaging
University College London
SPM Short CourseLondon, 21-23 Oct 2010
Normalisation
Statistical Parametric MapImage time-series
Parameter estimates
General Linear ModelRealignment Smoothing
Design matrix
Anatomicalreference
Spatial filter
StatisticalInference
RFT
p <0.05
Passive word listeningversus rest
7 cycles of rest and listening
Blocks of 6 scanswith 7 sec TR
Question: Is there a change in the BOLD response between listening and rest?
Stimulus function
One session
A very simple fMRI experiment
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
Time
BOLD signal
Time
single voxeltime series
Voxel-wise time series analysis
ModelspecificationParameterestimationHypothesis
Statistic
SPM
BOLD signal
Time =1 2+ +
error
x1 x2 e
Single voxel regression model
exxy 2211
Mass-univariate analysis: voxel-wise GLM
=
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
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)• Factorial designs• correlation• linear regression• multiple regression• F-tests• fMRI time series models• Etc..
GLM: mass-univariate parametric analysis
Parameter estimation
eXy
= +
e
2
1
Ordinary least squares estimation
(OLS) (assuming i.i.d. error):
yXXX TT 1)(ˆ
Objective:estimate parameters to minimize
N
tte
1
2
y X
A geometric perspective on the GLM
ye
Design space defined by X
x1
x2 ˆ Xy
Smallest errors (shortest error vector)when e is orthogonal to X
Ordinary Least Squares (OLS)
0eX T
XXyX TT
0)ˆ( XyX T
yXXX TT 1)(ˆ
What are the problems of this model?
1. BOLD responses have a delayed and dispersed form.
HRF
2. The BOLD signal includes substantial amounts of low-frequency noise (eg due to scanner drift).
3. Due to breathing, heartbeat & unmodeled neuronal activity, the errors are serially correlated. This violates the assumptions of the noise model in the GLM
t
dtgftgf0
)()()(
Problem 1: Shape of BOLD responseSolution: Convolution model
expected BOLD response = input function impulse response function (HRF)
=
Impulses HRF Expected BOLD
Convolution model of the BOLD responseConvolve stimulus function with a canonical hemodynamic response function (HRF):
HRF
t
dtgftgf0
)()()(
blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account low-frequency driftred = predicted response, NOT taking into account low-frequency drift
Problem 2: Low-frequency noise Solution: High pass filtering
discrete cosine transform (DCT) set
discrete cosine transform (DCT) set
High pass filtering
withttt aee 1 ),0(~ 2 Nt
1st order autoregressive process: AR(1)
)(eCovautocovariance
function
N
N
Problem 3: Serial correlations
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 Qand hyperparameters
jj
ii
QV
VC
2
),0(~ ii CNe
1 1 1ˆ ( )T TX V X X V y
Parameters can then be estimated using 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
Contrasts &statistical parametric
maps
Q: activation during listening ?
c = 1 0 0 0 0 0 0 0 0 0 0
Null hypothesis: 01
)ˆ(
ˆ
T
T
cStdct
X
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